Smooth asymptotics for collapsing Calabi-Yau metrics
SSMOOTH ASYMPTOTICS FOR COLLAPSING CALABI-YAU METRICS
HANS-JOACHIM HEIN AND VALENTINO TOSATTI
Abstract.
We prove that Calabi-Yau metrics on compact Calabi-Yau manifolds whose K¨ahler classesshrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singularfibers. This follows from a stronger statement which gives an asymptotic expansion of these metrics asthe fibers shrink, in terms of explicit functions on the total space and with k -th order remainders thatsatisfy uniform C k -estimates with respect to a collapsing family of background metrics. Contents
1. Introduction 21.1. Setup 31.2. Smooth collapsing 31.3. Previous work 41.4. Smooth asymptotics 41.5. Overview of the proofs 71.6. Organization of the paper 81.7. Acknowledgments 82. H¨older norms and interpolation inequalities 92.1. D -derivatives and P -transport 92.1.1. The operator norm of P -transport 92.1.2. Commutators of D -derivatives 102.2. Definition of the H¨older norms 112.3. The main interpolation inequality 132.4. From H¨older seminorm bounds to convergence 162.5. Schauder estimates 203. Selection of obstruction functions 233.1. Approximate fiberwise Gram-Schmidt 233.2. Approximate Green operators 253.3. A simple interpolation for polynomials 293.4. Statement of the selection theorem 293.5. Proof of the Selection Theorem 3.10 313.5.1. The cases j = 0 , j (cid:62)
2: the initial list. 323.5.3. The iterative procedure 343.5.4. Iteration and conclusion 384. The asymptotic expansion theorem 404.1. Statement of the asymptotic expansion 404.2. Set-up of an inductive scheme, and initial reductions 424.2.1. The bounds that hold thanks to the induction hypothesis 424.2.2. Proving (4.6), (4.10), (4.12) and (4.13) 43
Date : February 9, 2021. a r X i v : . [ m a t h . DG ] F e b Smooth asymptotics for collapsing Calabi-Yau metrics C m + n C m × Y C m (modulo linear regularity) 494.8. Set-up of the secondary (linear) blowup argument in Case 3 504.9. Estimates on the solution components and on the background data 524.9.1. Estimates for ˜ η t,j,k γ t, , ˜ η ‡ t and ˜ η ♦ t A ∗ t,i,p,k A (cid:93)t,i,p,k η ◦ t η † t ω (cid:93)t ε t → ∞ . 594.10.1. Improved estimates after additional jet subtractions 604.10.2. The noncancellation property 614.10.3. Preliminary estimate on ˜ η ♦ t A t,i,p,k : the main claim (4.267) 684.10.5. The piece of (4.274) with d − j − αt tr ˜ ω (cid:93)t (˜ η (cid:93)t + ˜ η (cid:52) t ) 704.10.6. The piece of (4.274) with the nonlinearities 714.10.7. The piece of (4.274) with c t ˜ ω m can ∧ (Θ ∗ t Ψ ∗ t ω F ) n (˜ ω (cid:93)t ) m + n − η ♦ t η ∗ t ε t → ε t →
0. 805. Proof of the main theorems 81References 901.
Introduction
Yau’s proof of the Calabi conjecture [37] shows that compact K¨ahler manifolds with vanishing realfirst Chern class (Calabi-Yau manifolds) admit Ricci-flat K¨ahler metrics, a unique one in each K¨ahlercohomology class. The study of these metrics has since been a central topic in complex geometry, andone particularly interesting question is to understand the behavior of these metrics when the K¨ahlerclass degenerates. The case when the metrics are volume noncollapsed is by now well-understood (see[32] and references therein), but the collapsing case presents a much harder challenge. This problemwas first studied in the work of Gross-Wilson [15] on elliptically fibered K I singularfibers, where by constructing the degenerating Ricci-flat metrics via a gluing construction they showed ans-Joachim Hein and Valentino Tosatti 3 in particular that the metrics collapse locally smoothly away from the singular fibers to a canonicalK¨ahler metric with nonnegative Ricci curvature on the base. Our goal in this paper is to prove thatthis conclusion holds in complete generality in all dimensions, and to obtain a complete asymptoticexpansion for the collapsing Ricci-flat metrics locally uniformly away from the singular fibers.1.1. Setup.
Let X be a compact Calabi-Yau manifold (compact K¨ahler with c ( X ) = 0 in H ( X, R ))of dimension m + n which admits a surjective holomorphic map f : X → B with connected fibers ontoa compact K¨ahler reduced and irreducible analytic space of dimension m . Let S ⊂ X be the preimageof the singular locus of B together with the critical values of f on the regular part of B , so S and f ( S )are closed proper analytic subvarieties and f : X \ S → B \ f ( S ) is a proper holomorphic submersionwith n -dimensional Calabi-Yau fibers X b = f − ( b ). We will implicitly assume that m, n >
0, so thatthe discussion is nontrivial. The set S will be referred to as the union of the singular fibers of f .Such fiber spaces arise naturally when X is a projective Calabi-Yau manifold and L is a semiampleline bundle on X with ( L m + n ) = 0, by taking f to be the morphism given by | (cid:96)L | for (cid:96) sufficiently largeand divisible (conjecturally the semiampleness assumption can be relaxed to nefness, up to replacing L by another numerically equivalent line bundle). This gives a wealth of examples, including the ellipticfibrations of K X b , b ∈ B \ f ( S ), are diffeomorphicto a fixed Calabi-Yau n -fold Y , and f | X \ B is a locally trivial C ∞ fiber bundle. On the other hand, f | X \ B is a locally trivial holomorphic fiber bundle if and only if the smooth fibers X b are all pairwisebiholomorphic, by the Fischer-Grauert theorem [10] (in this case f is called isotrivial). Furthermore,for a Calabi-Yau fiber space f : X → B as above, it is proved in [34] and [36, Thm 3.3] that S = ∅ (i.e., B is smooth and f is a submersion) if and only if f is a holomorphic fiber bundle.Given now a fiber space f : X → B as above, fix K¨ahler metrics ω X and ω B , with ω X Ricci-flat. Forall t (cid:62) ω • t be the unique Ricci-flat K¨ahler metric on X cohomologous to f ∗ ω B + e − t ω X . Wecan write ω • t = f ∗ ω B + e − t ω X + i∂∂ψ t where ψ t are smooth functions solving the degenerating familyof complex Monge-Amp`ere equations( ω • t ) m + n = ( f ∗ ω B + e − t ω X + i∂∂ψ t ) m + n = c t e − nt ω m + nX , sup X ψ t = 0 . (1.1)Here c t > X , and as t → ∞ , c t → (cid:18) m + nn (cid:19) (cid:82) X f ∗ ω mB ∧ ω nX (cid:82) X ω m + nX > . (1.2)In particular, the total volume Vol( X, ω • t ) of X as well as the total volume Vol( X b , ω • t | X b ) of each fiberis comparable to e − nt , so we have volume collapse as t → ∞ .To describe the leading order behavior of ω • t as t → ∞ , one then produces a K¨ahler metric ω can = ω B + i∂∂ψ ∞ on B \ f ( S ) by solving the complex Monge-Amp`ere equation [28, 29] ω m can = ( ω B + i∂∂ψ ∞ ) m = c ∞ f ∗ ( ω m + nX ) , sup B ψ t = 0 , (1.3)where ψ t ∈ C ( B ) ∩ C ∞ ( B \ f ( S )) and c ∞ = (cid:82) B ω mB (cid:82) X b ω nX / (cid:82) X ω m + nX . Then Ric( ω can ) = ω WP (cid:62) ω WP measures the variation of the complex structures of the fibers.1.2. Smooth collapsing.
Our first main theorem is the following. This confirms a conjecture of thesecond-named author from roughly 10 years ago, see for instance the surveys [30, 31, 32].
Theorem A.
In the above setting, as t → ∞ , locally uniformly away from the singular fibers of f , theRicci-flat metrics ω • t converge to f ∗ ω can in the standard C ∞ topology of tensor fields. Smooth asymptotics for collapsing Calabi-Yau metrics
Equivalently, this means that ψ t → f ∗ ψ ∞ in C k loc ( X \ S, g X ) for all k (cid:62)
0, i.e., that we have a prioriestimates to all orders, independent of t , for the solution ψ t of the complex Monge-Amp`ere equation(1.1) away from the singular fibers. As observed in [19, Rmk 1.7], the analogous purely local statementfor the real or complex Monge-Amp`ere equation with ellipticity degenerating along a foliation is false,and while our proof is local on the base it crucially relies on the leaves being closed manifolds.1.3. Previous work.
There have been a number of partial results in the direction of Theorem A. Asmentioned earlier, this statement follows from [15] in the case of elliptically fibered K I fibers, where a gluing construction for ω • t is given (see also [21] for suitable higher order estimates atthe singular fibers). This was recently extended to more general singular fibers in [4], and to Lefschetzfibrations f : X → P with K ω • t near them, so it would be unrealistic to hope to prove Theorem A in this way in general.In the special situation where there are no singular fibers (so in fact f is a holomorphic fiber bundleby [34, 36]), Theorem A follows from [8, 9], where ω • t is constructed by considering a semi-Ricci-flat form ω F on X (to be discussed below) and deforming it using the implicit function theorem. For thisapproach the absence of singular fibers is crucial, as discussed in [19, § X | ψ t | (cid:54) C uniformly in t . Using this estimate,the second-named author [29] extended Yau’s second-order estimates [37] to our setting (see also [27]for the case when m = n = 1) and showed that ψ t → f ∗ ψ ∞ in C ,α loc ( X \ S, g X ) for all 0 < α <
1, andalso established that for any K (cid:98) X \ S there is a C such that C − ( f ∗ ω B + e − t ω X ) (cid:54) ω • t (cid:54) C ( f ∗ ω B + e − t ω X ) (1.4)holds on K for all t .If the smooth fibers X b are tori (or finite free quotients of tori), it is possible to “unravel” the fibersby passing to the universal cover of the preimage of a ball in the base. Applying a vertical stretchingon this cover, estimate (1.4) with some work implies that the stretched Ricci-flat metrics are uniformlyEuclidean, hence satisfy higher-order estimates by standard theory. In this way, Theorem A was provedin [13] when the fibers X b are tori, under a projectivity assumption which was removed in [18], andthen in [35] when the fibers are finite free quotients of tori. This approach crucially uses the fact thattori are covered by Euclidean space, and does not extend to more general smooth fibers.The next general result towards Theorem A was obtained in [33], where it is shown that ω • t → f ∗ ω can in C ( X \ S, g X ). This regularity appears to be a natural barrier for methods based on modificationsof Yau’s estimates [37]. However, by introducing new methods, we reproved this result in [19] andimproved the convergence to C ,α loc ( X \ S, g X ) in general (0 < α < f is isotrivial (i.e., the smooth fibers are all biholomorphic but S can be nonempty). In that work, whichis purely local on the base, we introduced a new nested blowup method and the idea of working withnon-K¨ahler Riemannian product reference metrics, deriving a contradiction to various Liouville typetheorems on cylinders. This method is also the foundation of the current paper.1.4. Smooth asymptotics.
Theorem A is in fact a simple corollary of a much stronger statement,which gives an asymptotic expansion for ω • t with strong uniform estimates on the remainders.The precise theorem that we prove is Theorem 4.1 below, whose statement we now describe in roughterms. First, thanks to (1.4), one might naively expect that ω • t satisfies higher order C k loc estimates evenwith respect to the shrinking reference metrics f ∗ ω B + e − t ω X . However, a moment’s thought revealsthat such estimates very much depend on the fiberwise restrictions of these reference metrics for t = 0,as different choices give rise to C k norms that do not remain uniformly equivalent as t → ∞ for k (cid:62) ans-Joachim Hein and Valentino Tosatti 5 Furthermore, it was shown in [33, 35] that e t ω • t | X b converges smoothly to the unique Ricci-flat K¨ahlermetric ω F,b on X b cohomologous to ω X | X b . Thus, if higher order estimates with respect to shrinkingreference metrics hold at all, this forces the reference metrics to restrict to e − t ω F,b on each X b . Wethus define a (1 , ω F on X \ S by writing ω F,b = ω X | X b + i∂∂ρ b where (cid:82) X b ρ b ω nX = 0, observingthat ρ b varies smoothly in b ∈ B \ f ( S ) and so defines a smooth function ρ on X \ S , and defining ω F = ω X + i∂∂ρ, ω (cid:92)t = f ∗ ω can + e − t ω F (1.5)on X \ S . Then the (1 , ω F is semi-Ricci-flat in the sense that it restricts to a Ricci-flat metric onevery smooth fiber, but it is not in general semipositive definite on X \ S , see [2] for a counterexample.Nevertheless, given any K (cid:98) X \ S there is a t K such that ω (cid:92)t is a K¨ahler metric on K for all t (cid:62) t K ,uniformly equivalent to f ∗ ω B + e − t ω X , and hence to ω • t by (1.4). Observe also that every closed real(1 , X \ S which is i∂∂ -cohomologous to ω X and restricts to ω F,b on each X b must be equalto ω F + i∂∂f ∗ u for some smooth function u on B \ f ( S ).From now on we work locally on the base, and simply let B denote a Euclidean coordinate ball inthe base away from f ( S ) over which f is C ∞ trivial. We are thus working on B × Y with a complexstructure J which is not in general the product one, but for which pr B is ( J, J C m )-holomorphic. Up toreplacing ψ t by ψ t − ψ ∞ − e − t ρ , we can write ω • t = ω (cid:92)t + i∂∂ψ t , where from the above-mentioned resultswe know that ψ t → C ,α ( B × Y ) (although for the purposes of the present paper it is sufficient toknow that i∂∂ψ t → f is isotrivial, we can pick the C ∞ trivialization to be holomorphic, so J is a product, the metrics ω F,b are independent of b and ω (cid:92)t is a product metric. In [19] we then proved that one does indeed haveuniform C k loc ( X \ S, g (cid:92)t ) estimates for ω • t . This was also proved in [13, 18] in the case of torus fibers if J is not a product, but then the fiberwise Ricci-flat metrics ω F,b are parallel with respect to each other ina suitable C ∞ trivialization of f , which greatly simplifies the estimates. Outside of these two specialcases, the metrics ω (cid:92)t are not products and ω F,b is not parallel with respect to ω F,b (cid:48) for any b (cid:54) = b (cid:48) withrespect to any C ∞ identification X b ∼ = X b (cid:48) . As in our previous paper [19], this forces us to introduce afamily of collapsing non-K¨ahler Riemannian product metrics g b,t = g C m + e − t g F,b .The first innovation of the present paper is to introduce a certain connection D and parallel transportoperator P adapted to the family g b,t such that, with a suitable definition of H¨older norms in termsof these objects, a sharp interpolation inequality can be proved whose constants are independent of t .Using this setup we are then also able prove the crucial new technical Theorem 2.9.At this point one might still hope to prove that ω • t has uniform C k,α ( D , P , { g b,t } b ∈ B ) bounds for all k and α , up to passing to a smaller ball in the base if necessary. This hope turns out to be justified for k = 0. The proof of this amounts to a streamlined and improved version of our previous paper [19],and a similar statement with a similar proof also holds for k = 1.However, the second innovation of the present paper is to realize that no such statement can be truefor k = 2 in general, and the obstructions roughly speaking come from the fact that the semi-Ricci-flatmetric ω (cid:92)t is itself not uniformly bounded in C ,α ( D , P , { g b,t } b ∈ B ) unless f is isotrivial or the fibers areflat. With substantial work, for any given k (cid:62)
2, we then identify explicit obstruction functions, anddecompose ω • t − ω (cid:92)t = i∂∂ψ t into a sum of finitely many terms (constructed using these obstructionfunctions together with ω F ) and a remainder, such that each of the finitely many terms is unboundedin C k,α ( D , P , { g b,t } b ∈ B ) but is almost explicit, while the remainder is not explicit at all but is indeeduniformly bounded in C k,α ( D , P , { g b,t } b ∈ B ). This is in a nutshell the content of Theorem 4.1.To deduce Theorem A from this, it suffices to check that those terms in the above decomposition of ω • t − ω (cid:92)t that come from the obstruction functions, while not uniformly bounded in C k,α ( D , P , { g b,t } b ∈ B ),are at least uniformly bounded in the C k,α norm of a t -independent metric. Smooth asymptotics for collapsing Calabi-Yau metrics
From Theorem 4.1 it is possible to extract the explicit form of the first nontrivial term in the aboveexpansion of ω • t by plugging this expansion back into the Monge-Amp`ere equation (1.1). This is exactlythe term which indirectly made us realize that starting at k = 2 a new strategy is required if f is notisotrivial and the fibers are not flat. In fact, from the explicit form of this term one can see directly whythese two special cases play a distinguished role. To state a simplified form of the result, let z , . . . , z m denote the standard coordinates on B ⊂ C m , and for any given b ∈ B and 1 (cid:54) µ (cid:54) m let A µ denotethe unique T , X b -valued (0 , X b which is harmonic with respect to the fiberwise Ricci-flatmetric ω F,b and represents the Kodaira-Spencer class of the variation of the complex structure of X b indirection ∂ z µ . Let also (cid:104)· , ·(cid:105) denote the fiberwise Ricci-flat inner product, ∆ − Y the fiberwise inverse ofthe fiberwise Ricci-flat Laplacian, and ψ the fiberwise average of a function ψ on the total space withrespect to the fiberwise Calabi-Yau volume form ω nF | X b . Then we have: Theorem B.
The expansion mentioned above begins with ω • t = f ∗ ω can + e − t ω F − e − t i∂∂ ∆ − Y ∆ − Y ( g µ ¯ ν can ( (cid:104) A µ , A ¯ ν (cid:105) − (cid:104) A µ , A ¯ ν (cid:105) )) + f ∗ error + error . (1.6) Here error is an i∂∂ -exact (1 , -form on B that goes to zero in C ( B ) whereas error is an i∂∂ -exact (1 , -form on B × Y that satisfies the following estimates. Fix any local C ∞ product coordinate systemon B × Y . For any contravariant tensor T , let T { (cid:96) } denote the component of T with (cid:96) fiber indices interms of these local coordinates. Then for some ε > and all (cid:54) j (cid:54) we have that | ( ∂ j error ) { (cid:96) }| = (cid:40) O ( e − (2+ (cid:96) − j ) t ) if (cid:54) (cid:96) < j + 2 ,O ( e − (2+ ε ) t ) if (cid:96) = j + 2 . (1.7)Thus, for 0 (cid:54) j (cid:54)
2, the components of ∂ j error with at least one base index might still swamp thecorresponding components of ∂ j of the explicit e − t term in (1.6). However, the “all fiber” componentof ∂ j error decays strictly faster than the “all fiber” component of ∂ j of that term.Recall that (cid:104) A ν , A ¯ ν (cid:105) = ω WP ,µ ¯ ν and Ric( ω can ) = ω WP . The quantity ∆ − Y ( (cid:104) A µ , A ¯ ν (cid:105) − (cid:104) A µ , A ¯ ν (cid:105) ) arisesalso in the work of Schumacher [25], and is related to the “geodesic curvature” appearing in the workof Semmes [26]. While the corresponding term in (1.6) obviously vanishes if f is isotrivial or the fibersare flat, it does not vanish in general. Indeed, consider the case when n = 2, m = 1, and the fibers are K K A can be identified withan arbitrary closed, anti-self-dual, complex-valued 2-form, and our term vanishes if and only if A hasconstant length with respect to the Ricci-flat metric. But this is false in the asymptotically cylindricalgluing limit of K §
5] because then there is a 16-dimensional space of such forms that areexponentially small on the neck, see e.g. [16, § | ∂ j error | = O j,K ( e − Kt ) for all j, K ∈ N . In the isotrivialcase, the same conclusion can be extracted from [19]. Thus, in these two cases, the expansion of ω • t actually contains no terms of finite order in e − t beyond f ∗ ω can + e − t ω F + f ∗ error . However, the bestknown result towards Theorem B in general, which also follows from [19], only states that ω • t = f ∗ ω can + e − t ω F + f ∗ error + error (cid:48) with | error (cid:48) { (cid:96) }| = O ( e − ( (cid:96) + α ) t ) (1.8)for all 0 (cid:54) (cid:96) (cid:54) α ∈ (0 , K ans-Joachim Hein and Valentino Tosatti 7 Overview of the proofs.
Having already explained how Theorem A follows from the completeform of Theorem B, i.e., Theorem 4.1, we focus here on the latter. To construct the desired expansionof the Ricci-flat metrics, we fix k and for 0 (cid:54) j (cid:54) k aim to construct the first j terms of the expansion(which, we emphasize, depend on the choice of k ) by induction on j . For this, we first need to identifythe obstruction functions G i,p mentioned above. These are smooth functions on B × Y with fiberwiseaverage zero, which are indexed over 0 (cid:54) i (cid:54) j and 1 (cid:54) p (cid:54) N i and also implicitly depend on k . Theyvanish identically for j = 0 , j (cid:62) ω (cid:92)t + i∂∂ψ t ) m + n ( ω (cid:93)t ) m + n = c t e − nt ω m + nX ( ω (cid:93)t ) m + n , (1.9)where ω (cid:93)t equals ω (cid:92)t plus small corrections that involve the obstruction functions themselves. Breakingthis apparent logical cycle requires an iterative procedure with a uniform gain at each step. Later, wewill also need the obstruction functions to be fiberwise L orthonormal, which is not immediate fromGram-Schmidt because the dimension of their fiberwise span may not be constant on the base. All ofthese problems will be solved in Section 3, specifically in Theorem 3.10.Having chosen the obstruction functions, we decompose ω • t − ω (cid:92)t inductively via γ t, := i∂∂ψ t , γ t,i,k := N i (cid:88) p =1 i∂∂ G t,k ( A t,i,p,k , G i,p ) (2 (cid:54) i (cid:54) j ) , (1.10) η t,i,k := ω • t − ω (cid:92)t − γ t, − i (cid:88) ι =2 γ t,ι,k . (1.11)Here A t,i,p,k is a function on B , approximately equal to the fiberwise L component of tr ω (cid:92)t η t,i − ,k onto G i,p , and G t,k is an approximate Green operator such that in a very rough sense,∆ ω (cid:92)t G t,k ( A t,i,p,k , G i,p ) ≈ A t,i,p,k G i,p . (1.12)Since G t,k acts as a differential rather than integral operator in the base directions (of order 2 k ), thisstatement cannot be meaningful unless A t,i,p,k behaves like a polynomial of sufficiently low degree. Forus, this assumption is justified because we are working with functions with bounded H¨older norms.Our main Theorem 4.1 will then prove uniform C j,α estimates for γ t, and uniform C j +2 ,α estimatesfor A t,i,p,k (2 (cid:54) i (cid:54) j ) with respect to the usual H¨older norms on B , uniform C j,α estimates for η t,j,k with respect to the collapsing H¨older norms on B × Y defined in Section 2, and auxiliary estimates forderivatives of A t,i,p,k of order j + 3 to 2 k + j + 2. While these auxiliary estimates are not uniform in t , they do imply a uniform C j,α ( g X ) bound for the pieces γ t,i,k (2 (cid:54) i (cid:54) j ). All of these estimates areproved by a contradiction and blowup argument in the spirit of our paper [19], which essentially dealtwith the case j = k = 0. However, the details are now vastly more complicated.If the desired estimates fail, we obtain a sequence of solutions violating these estimates more andmore strongly, and we need to study 3 cases according to how the blowup rate compares to e t . In thefast-forming and regular-forming cases, after stretching and scaling, our background geometries limitto C m + n and C m × Y respectively, and a contradiction can easily be derived from known regularity andLiouville theorems for the Monge-Amp`ere equation. The majority of the work goes into understandingthe slow-forming case, where the background geometries collapse to C m . Here we are quickly reduced toshowing that the two points where the relevant H¨older seminorm achieves its maximum cannot collide,and one can think of this as an improvement of regularity of a more linear nature.To achieve this improvement, we first split all our objects into their jets at a blowup basepoint andthe corresponding Taylor remainders, and derive precise estimates on all these pieces, which ultimately Smooth asymptotics for collapsing Calabi-Yau metrics allow us to expand and linearize the Monge-Amp`ere equation. By construction, the Taylor remainderswill have excellent convergence and growth properties, whereas the jets satisfy much worse bounds butare by definition polynomials. On the LHS of the Monge-Amp`ere equation, we then move the jets intothe reference metric ω (cid:92)t , thus creating the new reference metric ω (cid:93)t that appears in (1.9), and expandthe LHS of (1.9) as its linearization at ω (cid:93)t , plus nonlinearities, applied to the Taylor remainders of thevarious components of the original solution. The nonlinearities turn out to be negligible, as one wouldexpect at this stage of the proof. The RHS is a mixture of background objects and jets of the originalsolution components, without any helpful growth bounds but with an almost explicit structure.To actually prove that the blowup points do not collide, we again argue by contradiction and need toconsider 3 subcases, where after further stretching and scaling the background geometries converge to C m + n , C m × Y and C m , respectively. We deal with the fast-forming case by using Schauder estimatesfor the linearized PDE on balls in C m + n . While this is the easiest case conceptually, the proof is muchlonger than in the other two cases because of the large number of pieces appearing in the decompositionof the solution in this case and the complexity of the quantitative estimates that they satisfy.In the regular-forming case, we argue that the LHS of the PDE has a limit thanks to the excellentgrowth of the Taylor remainders, so the RHS must have a limit as well, necessarily of the form K ( z ) + N (cid:88) q =1 K q ( z ) H q ( y ) . (1.13)Here K , K q are polynomials on C m of degree at most j , and H q are smooth functions on Y which byour choice of the obstruction functions lie in the linear span of the restrictions G i,p | { z ∞ }× Y (2 (cid:54) i (cid:54) j ),where z ∞ ∈ B is the limit of the projections to B of the blowup basepoints. On the other hand, thedecomposition (1.10)–(1.11) and the approximate Green property (1.12) were made so that in the limit,the LHS of the PDE consists of one part coming from the Taylor remainders of γ t, ,k , . . . , γ t,j,k , whichlies in the fiberwise linear span of the functions G i,p | { z ∞ }× Y , and another part coming from the Taylorremainders of γ t, and η t,j,k , which is fiberwise L orthogonal to this span. Setting this equal to (1.13),we obtain that the Taylor remainders of order (cid:62) j + 1 of γ t, ,k , . . . , γ t,j,k are polynomials of degree (cid:54) j in the limit (an immediate contradiction), and the Taylor remainders of order (cid:62) j + 1 of γ t, and η t,j,k have trace equal to a polynomial of degree (cid:54) j in the limit (contradicting Liouville’s theorem).In the slow-forming case, Theorem 2.9 tells us directly that η t,j,k is too small to contribute to theassumed failure of Theorem 4.1, and it follows from the linearized Monge-Amp`ere equation itself that γ t, ,k , . . . , γ t,j,k are too small to contribute as well. Thus, the only remaining contribution comes from γ t, , which lives on the base, and this eventually contradicts Liouville’s theorem on C m .Lastly, while the expansion of Theorem 4.1 is not quite precise enough to identify the first nontrivialterm in Theorem B (which ultimately comes from the term γ t, , in Theorem 4.1) explicitly, this canbe done with some extra work by plugging this expansion back into the Monge-Amp`ere equation1.6. Organization of the paper.
In Section 2 we define new covariant derivatives, parallel transportoperators and H¨older norms adapted to our setting, and prove the key interpolation inequality andthe fundamental Theorem 2.9. Section 3 contains our main technical constructions of the approximateGreen operator and the selection of the obstruction functions in Theorem 3.10. Section 4 is the heartof the paper where Theorem 4.1 is proved. Theorems A and B are deduced from this in Section 5.1.7.
Acknowledgments.
We thank L. Lempert and S. Zelditch for discussions, and J.-M. Bismut andC. Margerin for encouragement. Part of this work was carried out during the second-named author’svisits to the Department of Mathematics and the Center for Mathematical Sciences and Applicationsat Harvard University, which he would like to thank for the hospitality. The authors were partiallysupported by NSF grants DMS-1745517 (H.H.) and DMS-1610278, DMS-1903147 (V.T.). ans-Joachim Hein and Valentino Tosatti 9 H¨older norms and interpolation inequalities D -derivatives and P -transport. Let f : X → B be a proper surjective holomorphic submersionwith n -dimensional Calabi-Yau fibers over the unit ball B ⊂ C m , as in the introduction, equipped witha K¨ahler form ω X on the total space, and let Y = f − (0). For each z ∈ B let ω F,z be the Calabi-Yau metric on the fiber f − ( z ) cohomologous to the restriction of ω X . Fixing an arbitrary smoothtrivialization Φ : B × Y → X , let g Y,z be the Ricci-flat Riemannian metric on Y associated to Φ ∗ ω F,z ,extended trivially to B × Y , and define a family of Riemannian product metrics on B × Y by g z,t := pr ∗ B ( g C m ) + e − t pr ∗ Y ( g Y,z ) for all z ∈ B. (2.1)In our previous work [19, § g z,t and its parallel transport on the fiber over z . As it turns out, in order to generalize this constructionto higher order derivatives and preserve all the good properties that we had in [19], we need to modifythis construction by introducing a different notion of parallel transport adapted to this setting. Definition 2.1.
For z ∈ B write ∇ z to denote the Levi-Civita connection of the product metric g z,t ,which is independent of t . Then let D denote the connection on the tangent bundle of B × Y and onall of its tensor bundles defined by setting( D τ )( x ) := ( ∇ pr B ( x ) τ )( x ) (2.2)for all tensors τ on B × Y and all x ∈ B × Y . Write D k for the associated iterated covariant derivatives.For all curves γ : [ t , t ] → B × Y and a, b ∈ [ t , t ] let P γa,b denote the associated parallel transport.It is clear that D satisfies the definition of a connection on a vector bundle. As usual, for all curves γ : [ t , t ] → B × Y and all tensor fields τ along γ we then have the useful formula ddt (cid:12)(cid:12)(cid:12)(cid:12) t = t P γt,t ( τ ( t )) = D dt (cid:12)(cid:12)(cid:12)(cid:12) t = t τ ( t ) = ∇ pr B ( γ ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12) t = t τ ( t ) . (2.3)Also note that we obtain a natural definition of P -geodesics , but in this paper we will only be using thetwo obvious families of P -geodesics: vertical paths of the form ( z , γ ( t )), where γ ( t ) is a g Y,z -geodesicin the fiber { z } × Y (we will call such a geodesic minimal if γ is minimal with respect to g Y,z ), andhorizontal paths of the form ( z ( t ) , y ), where z ( t ) is an affine segment in C m . Every two points on B × Y can be joined by concatenating two of these P -geodesics where the vertical one is minimal.We now record two technical properties of P and D which will be very useful for us later on.2.1.1. The operator norm of P -transport. Let x, x (cid:48) ∈ B × Y and let γ : [ t , t ] → B × Y be a P -geodesicfrom x to x (cid:48) , which we assume is either horizontal or minimal vertical. Let u be a tangent vector at x , which we again assume is either horizontal or vertical. Note that D ( d pr B ) = 0 because all of theconnections ∇ z are product connections on B × Y . This implies that if u is horizontal, then P γa,b u = u for all a, b ∈ [ t , t ], and if u is vertical, then P γa,b u is again vertical.Thanks to this discussion, P γa,b is block diagonal. Thus, in particular, the operator norm of P γt ,t with respect to the shrinking product metrics g z,t is fixed independent of t . Later, we will often beconsidering this picture after applying a family of stretching diffeomorphismsΣ t : B e t × Y → B × Y, ( z, y ) = Σ t (ˇ z, ˇ y ) = ( e − t ˇ z, ˇ y ) , (2.4)where it is more natural to introduce the scaled product metricsˇ g ˇ z,t := e t Σ ∗ t g z,t = pr ∗ B ( g C m ) + pr ∗ Y ( g Y,e − t ˇ z ) , (2.5)which converge smoothly on compact sets to the fixed product metric g := pr ∗ B ( g C m ) + pr ∗ Y ( g Y, ) . (2.6) Then, since P commutes with this stretching and scaling, we conclude that the operator norm of P ˇ γ from ˇ x t to ˇ x (cid:48) t (points in B e t × Y ) still has uniform bounds independent of t as long as their images x t = Σ t (ˇ x t ) , x (cid:48) t = Σ t (ˇ x (cid:48) t ) lie in a fixed relatively compact subset of B × Y (which will usually be B (cid:48) × Y where B (cid:48) ⊂ B is a concentric ball of slightly smaller radius).2.1.2. Commutators of D -derivatives. If the complex structure on B × Y is a product, then g Y,z = g Y for all z ∈ B and D is simply the product connection on B × Y given by the Euclidean derivative inthe B directions and the Levi-Civita connection of g Y in the Y directions. In particular, base and fiberderivatives with respect to D commute. In our general setting, we still have the following property. Proposition 2.2.
Let (cid:96) ∈ { , . . . , m } . Let h (cid:96) denote the (cid:96) -th standard basis vector field on B triviallyextended to B × Y . Let τ be a tensor on B × Y . For any given z ∈ B define A z(cid:96) τ := ∂∂ ˜ z (cid:96) (cid:12)(cid:12)(cid:12)(cid:12) ˜ z = z ∇ ˜ z τ. (2.7) Then it holds for all x = ( z, y ) ∈ B × Y and all v ∈ T y Y that ( D h (cid:96) , v τ )( x ) = ( D v , h (cid:96) τ )( x ) + v (cid:121) ( A z(cid:96) τ )( x ) . (2.8) Proof.
It is true by definition that ( D h (cid:96) , v τ )( x ) = v (cid:121) ( D h (cid:96) ( D τ ))( x ) . (2.9)To simplify this term, observe that the operator D h (cid:96) at x is just the ordinary (cid:96) -th partial derivative in R m . Thus, ( D h (cid:96) ( D τ ))( x ) = ∂∂ ˜ z (cid:96) (cid:12)(cid:12)(cid:12)(cid:12) ˜ z = z ( ∇ ˜ z τ )(˜ z, y ) = ( A z(cid:96) τ )( x ) + ( ∇ z h (cid:96) ( ∇ z τ ))( x ) . (2.10)This then implies that ( D h (cid:96) , v τ )( x ) = v (cid:121) ( A z(cid:96) τ )( x ) + (( ∇ z ) h (cid:96) , v τ )( x ) . (2.11)It remains to observe that ∇ z is a product connection, so that horizontal and vertical derivatives withrespect to ∇ z commute, and that(( ∇ z ) v , h (cid:96) τ )( x ) = h (cid:96) (cid:121) ( ∇ z v ( ∇ z τ ))( x ) = ( D v , h (cid:96) τ )( x ) (2.12)because ∇ z τ = D τ along the entire fiber through x . (cid:3) Going back to the definition (2.7), we can also define trivially A z v = 0 for v any vertical vector, andin this way we obtain a (3 ,
1) tensor A on B × Y (which is zero whenever the third base index is tangentto Y ). Then (2.10) also holds trivially by definition if we replace h (cid:96) by v , i.e. we can compactly write D τ = ∇ z, τ + A (cid:126) τ, (2.13)where here and in the rest of the paper (cid:126) denotes some tensorial contraction.We will need a generalization of this schematic formula to higher order derivatives, relating D j with ∇ z,j . Schematically, we write Γ z for the Christoffel symbols of ∇ z (empasizing their dependence on z ),and treating z as an extra variable ˜ z we can write symbolically D = ( ∂ usual + Γ ˜ z + ∂ ˜ z ) , (2.14)which is applied to a tensor field τ on B × Y which we think of as not depending on ˜ z (so ∂ usual areusual partial derivatives which will land on both τ and Γ ˜ z , while ∂ ˜ z will only land on Γ ˜ z ). Formallysquaring (2.14) clearly shows (2.13), where A = ∂ ˜ z Γ ˜ z . ans-Joachim Hein and Valentino Tosatti 11 We then claim that we have D j τ − ∇ z,j τ = j − (cid:88) p =0 ∇ z,p τ (cid:126) ∇ j − − p usual / ˜ z A , (2.15)where the notation ∇ j − − p usual / ˜ z A means that there are j − − p derivatives of A each of which is either ausual covariant derivative ∇ z or a ∂ ˜ z . To prove (2.15), we write the LHS as( ∂ usual + Γ ˜ z + ∂ ˜ z ) j τ − ( ∂ usual + Γ ˜ z ) j τ, (2.16)and expand this (recalling that the ∂ ˜ z operator only acts on the Christoffel symbols Γ ˜ z ). The fact thatthe derivatives go only up to j − ∂ ˜ z , and hencealso at least one Γ ˜ z (otherwise there is nothing for the ∂ ˜ z to differentiate), which generates an A , andthe number of remaining derivatives is then j − D j τ as a way ofpackaging together ∇ z,j τ with lower derivatives of τ (of order up to j −
2) contracted with some fixedbackground tensors.2.2.
Definition of the H¨older norms.
Assume that B is a ball of any radius centered at 0 ∈ C m .In our applications this will be a rescaling of the unit ball that we have been using so far. Assume thatwe have a smooth family of Riemannian metrics g Y,z on { z } × Y for z ∈ B such that there exists a C (cid:62) z ∈ B we have C − (cid:54) inj( Y, g
Y,z ) (cid:54) diam( Y, g
Y,z ) (cid:54) C , (2.17) C − g Y, (cid:54) g Y,z (cid:54) C g Y, . (2.18)Again in our later applications this will family will be simply our original family g Y,z where the variable z has been rescaled, and such a constant C will exist (up to shrinking slightly our original unit ball).Define a product Riemannian metric g := pr ∗ B ( g C m ) + pr ∗ Y ( g Y, ) . (2.19)For ease of notation, in this section we shall write B ( p, R ) := B g C m ( z, R ) × B g Y, ( y, R ) (2.20)for all p = ( z, y ) ∈ B × Y and R > Definition 2.3.
For all 0 < α <
R > p ∈ B × Y we define[ τ ] C α ( B ( p,R )) := sup | τ ( x ) − P γt ,t ( τ ( x )) | g d g ( x , x ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ : [ t , t ] → B × Y P -geodesic, eitherhorizontal or minimal vertical, and x i := γ ( t i ) ∈ B ( p, R ) for i = 0 , (2.21)for all smooth tensor fields τ on B × Y .Observe that in this definition the family of metrics g Y,z is used to define (minimal) P -geodesics and P -transport, while the fixed metric g is used to measure the distance d g ( x , x ) and to define B ( p, R ).We will often replace the metric g by a family of product metrics g t with variable fiber size, g t := pr ∗ B ( g C m ) + γ t pr ∗ Y ( g Y, ) , (2.22)where γ t > τ ] C α ( B gt ( p,R ) ,g t ) , where B g t ( p, R ) := B g C m ( z, R ) × B γ t g Y, ( y, R ). In the case when γ t → B g t ( p, R ) = B g C m ( z, R ) × Y for all t (cid:62) t R sufficiently large. Remark 2.4.
If a contravariant tensor τ satisfies (cid:107) τ (cid:107) L ∞ ( B ( p,R ) ,g ) + [ τ ] C α ( B ( p,R ) ,g ) (cid:54) A, (2.23)then (cid:107) τ (cid:107) C α ( B ( p,R ) ,g ) (cid:54) CA, (2.24)for a uniform constant C (that depends on R ), where (cid:107) τ (cid:107) C α ( B ( p,R ) ,g ) denotes the standard H¨older normof τ with respect to the metric g , defined using g -minimal geodesics and g -parallel transport. This canbe seen as follows. First, for any ( z, y ) , ( z (cid:48) , y (cid:48) ) ∈ B × Y we have thatmax( | z − z (cid:48) | , d g Y, ( y, y (cid:48) )) (cid:54) d g (( z, y ) , ( z (cid:48) , y (cid:48) )) (cid:54) | z − z (cid:48) | + d g Y, ( y, y (cid:48) ) . (2.25)This implies that if in the standard definition of the g -H¨older seminorm we only consider horizontalor vertical rather than arbitrary minimal g -geodesics, then these two seminorms are still comparableby a factor of 2. Next, we can compare this new seminorm (horizontal or vertical minimal g -geodesics, g -parallel transport) to the one defined in (2.21) exactly along the lines of [19, Lemma 3.6], using alsothat the operator norm of P is under control thanks to the discussion in Section 2.1.1. Remark 2.5.
For every contravariant tensor τ we have[ τ ] C α ( B ( p,R ) ,g ) (cid:54) CR − α (cid:107) D τ (cid:107) L ∞ ( B ( p, C R ) ,g ) . (2.26)To see this, for x , x ∈ B ( p, R ) let γ : [ t , t ] → B × Y be a P -geodesic joining them which is eitherhorizontal or minimal vertical, and use (2.3) to bound | τ ( x ) − P γt ,t ( τ ( x )) | g = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t − t ddt P γt − t,t ( τ ( t − t )) dt (cid:12)(cid:12)(cid:12)(cid:12) g = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t − t P γt − t,t (cid:20) ∇ z ( t − t ) dt τ ( t − t ) (cid:21) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g (cid:54) Cd g ( x , x ) (cid:107) D τ (cid:107) L ∞ ( B ( p, C R ) ,g ) , (2.27)applying here the estimate for the operator norm of P in Section 2.1.1 (assuming of course that B ( p, C R ) is contained in a fixed larger ball over which we have uniform control on the geometryof g Y,z ), and (2.26) follows. Also, if
R < C − then the P -geodesic γ remains inside B ( p, R ), and so in(2.26) we can replace B ( p, C R ) with B ( p, R ). Remark 2.6.
Generalizing Remark 2.4, if a contravariant tensor τ satisfies k (cid:88) j =0 (cid:107) D j τ (cid:107) L ∞ ( B ( p,R ) ,g ) + [ D k τ ] C α ( B ( p,R ) ,g ) (cid:54) A, (2.28)then (cid:107) τ (cid:107) C k,α ( B ( p,R ) ,g ) (cid:54) CA, (2.29)for a uniform constant C (that depends on R ), where (cid:107) τ (cid:107) C k,α ( B ( p,R ) ,g ) denotes standard tensor C k,α H¨older norm for the metric g . For this, we first use the conversion formula (2.15) to convert D j to ∇ z,j with lower order error terms (with up to j − τ ), which shows that k (cid:88) j =0 (cid:107)∇ z,j τ (cid:107) L ∞ ( B ( p,R ) ,g ) (cid:54) CA, (2.30)and we can further change ∇ z,j to ∇ g,j since all the metrics g Y,z are smoothly bounded. Whenconverting [ D k τ ] to [ ∇ g,k τ ] the lower order error terms that involve H¨older seminorms can also bebounded by CA (since thanks to (2.26) they can be bounded in terms of (2.30) and the radius R ). Sowe are left with the main term [ ∇ g,k τ ] C α ( B ( p,R ) ,g ) (in the sense of (2.21)) which can be converted to ans-Joachim Hein and Valentino Tosatti 13 the standard H¨older g -seminorm of ∇ g,k τ thanks to Remark 2.4, up to an acceptable error boundedby (2.30). Remark 2.7.
Clearly if in our original setup the metrics g Y,z are in fact independent of z , then the P -transport, the H¨older seminorm in (2.21) and the D -derivatives are the same as the usual paralleltransport, H¨older seminorm and covariant derivatives of the product metric g in (2.19).If on the other hand we pull back our setup via the diffeomorphisms Σ t in (2.4), then the pulled backfiber metrics g Y,e − t ˇ z converge locally smoothly to the fixed metric g Y, , and from this it follows easily(by looking at the ODE definining P , as in [19, Remark 3.7]) that in the limit our H¨older seminormbecomes the standard one for the product metric g . This clearly remains true if the metric g thatmeasures distances and norm in (2.21) is replaced by a family of product metrics g t that convergelocally smoothly to g in the limit.2.3. The main interpolation inequality.
Let again B be the unit ball in C m , suppose we havenumbers 0 < δ t (cid:54) t → ∞ ) and for t (cid:62) t Γ t : B δ − t × Y → B × Y, ( z, y ) = Γ t (ˇ z, ˇ y ) = ( δ t ˇ z, ˇ y ) . (2.31)The product metrics g t = pr ∗ B ( g C m ) + δ t pr ∗ Y ( g Y, ) , (2.32)on B × Y thus satisfy δ − t Γ ∗ t g t = g . Suppose we are given a contravariant p -tensor τ on B × Y , anddenote by ˇ τ t = δ − pt Γ ∗ t τ . Proposition 2.8.
For all k ∈ N (cid:62) and α ∈ (0 , there exists a constant C k = C k ( α, C ) such that forall smooth contravariant tensor fields τ on B × Y and for all t (cid:62) we have k (cid:88) j =1 ( R − ρ ) j (cid:107) D j ˇ τ t (cid:107) L ∞ ( B ( p,ρ )) (cid:54) C k (( R − ρ ) k + α [ D k ˇ τ t ] C α ( B ( p,R )) + (cid:107) ˇ τ t (cid:107) L ∞ ( B ( p,R )) ) (2.33) for all p ∈ B δ − t × Y , < ρ < R < d g ( p, ∂B δ − t × Y ) . This implies that k (cid:88) j =1 ( R − ρ ) j (cid:107) D j τ (cid:107) L ∞ ( B ( p,ρ ) ,g t ) (cid:54) C k (( R − ρ ) k + α [ D k τ ] C α ( B ( p,R ) ,g t ) + (cid:107) τ (cid:107) L ∞ ( B ( p,R ) ,g t ) ) , (2.34) for all p ∈ B × Y , < ρ < R < d g ( p, ∂B × Y ) . Let us remark that whenever this interpolation is applied to a tensor field which is pulled back from B , then it simply reduces to standard interpolation in Euclidean space. Proof.
It suffices to prove (2.33), since (2.34) is then obtained from it by stretching and scaling. Forease of notation we will denote ˇ τ t simply by τ .Aiming to apply [18, Lemma 3.4], for j ∈ { , . . . , k } define β j = j and f j ( ρ ) = (cid:107) D j τ (cid:107) L ∞ ( B ( p,ρ )) . Inorder to prove an inequality of the form k (cid:88) j =1 ( R − ρ ) j f j ( ρ ) (cid:54) ε k (cid:88) j =1 ( R − ρ ) j f j ( R ) + C ε ( R − ρ ) k + α [ D k τ ] C α ( B ( p,R )) + C ε (cid:107) τ (cid:107) L ∞ ( B ( p,R )) , (2.35)consider the following three cases. The constant C will always be a generic constant (cid:62) Case 1 : R − ρ < C − . Fix any j ∈ { , . . . , k } and write σ = D j − τ . Fix any x = ( z , y ) ∈ B ( p, ρ ).Let v run over a g -orthonormal basis of tangent vectors to B × Y at x which are all either horizontalor vertical. Let γ ( t ) be the unique P -geodesic with γ (0) = x and ˙ γ (0) = v , with t ∈ (0 , R − ρ ). If v is vertical, then γ ( t ) is a g Y,z -geodesic in { z } × Y , well within the injectivity radius. If v is horizontal,then γ ( t ) is an affine line in B times y . Either way, all subsegments of γ are (minimal) P -geodesics.Let x = γ ( t ) for t = ε ( R − ρ ) and ε ∈ (0 , σ ( t ) to denote the tensor field along γ ( t )induced by σ , writing z ( t ) = pr B ( γ ( t )), and using (2.3), we get that σ ( x ) − P γt , ( σ ( x )) = (cid:90) t ddt P γt − t, ( σ ( t − t )) dt = − (cid:90) t P γt − t, (cid:20) ∇ z ( t − t ) dt σ ( t − t ) (cid:21) dt. (2.36)We can rewrite the last integrand as ( ∇ z v σ )( x ) + ψ ( t − t ), where for all t ∈ [0 , t ], ψ ( t ) = P γt, (cid:20) ( ∇ z ( t )˙ γ ( t ) σ )( γ ( t )) (cid:21) − ( ∇ z v σ )( x ) . (2.37)Using the definition of the C α seminorm and the fact that γ is a (minimal) P -geodesic (so that we havethe crucial property P γt, ( ˙ γ ( t )) = v ), | ψ ( t ) | g ( x ) (cid:54) [ D σ ] C α ( B ( x ,t )) d g ( x , γ ( t )) α (cid:54) C [ D σ ] C α ( B ( x ,t )) t α . (2.38)If j < k , then we alternatively also have that ψ ( t ) = (cid:90) t dds P γs, (cid:20) ( ∇ z ( s )˙ γ ( s ) σ )( γ ( s )) (cid:21) ds = (cid:90) t P γs, (cid:20) ( D γ, ˙ γ σ )( γ ( s )) (cid:21) ds, (2.39)using that ˙ γ is P -parallel, and since the operator norm of P is under control thanks to the discussionin Section 2.1.1, we conclude that | ψ ( t ) | g ( x ) (cid:54) C (cid:107) D σ (cid:107) L ∞ ( B ( x ,t )) t. (2.40)Summarizing, for all t ∈ [0 , t ], | ψ ( t ) | g ( x ) (cid:54) (cid:40) C [ D σ ] C α ( B ( x ,t )) t α for all j,C (cid:107) D σ (cid:107) L ∞ ( B ( x ,t )) t for all j < k. (2.41)This leads to | ( ∇ z v σ )( x ) | g ( x ) t (cid:54) | σ ( x ) | g ( x ) + | P γt , ( σ ( x )) | g ( x ) + (cid:40) C [ D σ ] C α ( B ( p,ρ + t )) t α for all j,C (cid:107) D σ (cid:107) L ∞ ( B ( p,ρ + t )) t for all j < k. (2.42)Recall that v is an arbitrary element of a g -orthonormal basis at x and that σ = D j − τ . Taking thesup over all x ∈ B ( p, ρ ), we deduce that f j ( ρ ) t (cid:54) Cf j − ( ρ + t ) + (cid:40) C [ D j τ ] C α ( B ( p,ρ + t )) t α for all j,Cf j +1 ( ρ + t ) t for all j < k. (2.43)Now recall that t = ε ( R − ρ ), where ε ∈ (0 ,
1) is arbitrary. Working backwards from j = k to j = 1,decreasing and renaming ε in each step, we deduce that k (cid:88) j =1 ( R − ρ ) j f j ( ρ ) (cid:54) ε k (cid:88) j =1 ( R − ρ ) j f j ( R ) + ε ( R − ρ ) k + α [ D k τ ] C α ( B ( p,R )) + C ε (cid:107) τ (cid:107) L ∞ ( B ( p,R )) . (2.44)This is the desired inequality of type (2.35). Case 2 : R − ρ ∈ [ C − , C ]. The proof of (2.44) in this case can be reduced to Case 1. Let R (cid:48) = ρ + C and apply Case 1 to the pair of radii ( ρ, R (cid:48) ) instead of ( ρ, R ). In (2.44) with R replaced with R (cid:48) , noticethat trivially B ( x, R (cid:48) ) ⊂ B ( x, R ) and ( R (cid:48) − ρ ) k + α (cid:54) ( R − ρ ) k + α , so in order to obtain (2.44) for thepair ( ρ, R ) we only need to observe that ( R (cid:48) − ρ ) j (cid:62) (2 C ) − j ( R − ρ ) j for j = 1 , . . . , k . Case 3 : R − ρ > C . Using the same idea as in Case 1, we can prove that (2.43) still holds if f j ( ρ ) t isreplaced by (cid:107) D b ( D j − τ ) (cid:107) L ∞ ( B ( p,ρ )) t on the left-hand side. (Here and below, subscripts b and f denote ans-Joachim Hein and Valentino Tosatti 15 covariant derivatives in the horizontal and fiber directions, respectively.) This is because we are freeto take γ to be an affine line in B times a fixed point in Y , and all such paths are minimal P -geodesics(whereas a g Y,z -geodesic in { z } × Y will never be minimal on a time interval of size [0 , R − ρ ]). Onthe other hand, for all x = ( z, y ) ∈ B ( p, ρ ), | ( D f ( D j − τ ))( x ) | g ( x ) (cid:54) C [ D k − j +1 f ( D j − τ )] C α ( { z }× Y,g
Y,z ) (cid:54) C [ D k τ ] C α ( B ( p,R )) . (2.45)The first inequality holds because for all tensors σ , the derivative ( D f σ )( x ) coincides with the usual g Y,z -covariant derivative of σ restricted to { z } × Y , and because we are then in position to apply [18,Lemma 3.3]. The second inequality holds because { z } × Y ⊂ B ( p, R ) (because R − ρ > C ) andagain because P -transport along curves contained in { z } × Y coincides with the standard notion of g Y,z -parallel transport.Proceeding as in Case 1 (working backwards from j = k ), we get k (cid:88) j =1 ( R − ρ ) j f j ( ρ ) (cid:54) ε k (cid:88) j =1 ( R − ρ ) j f j ( R ) + C ε ( R − ρ ) k + α [ D k τ ] C α ( B ( p,R )) + C ε (cid:107) τ (cid:107) L ∞ ( B ( p,R )) . (2.46)The only difference to Case 1 is that we now get a large factor of C ε (rather than a small factor of ε )in front of the [ D k τ ] C α term.Proposition 2.8 now follows from [18, Lemma 3.4]. (cid:3) Let us also note that in the same setting as Proposition 2.8, for all j ∈ N and 0 < β < j + β < k + α we also have the following interpolation estimate( R − ρ ) j + β [ D j ˇ τ t ] C β ( B ( p,ρ )) (cid:54) C k (( R − ρ ) k + α [ D k ˇ τ t ] C α ( B ( p,R )) + (cid:107) ˇ τ t (cid:107) L ∞ ( B ( p,R )) ) , (2.47)for all p ∈ B δ − t × Y , 0 < ρ < R < d g ( p, ∂B δ − t × Y ), and analogously after stretching and scaling asin (2.34)( R − ρ ) j + β [ D j τ ] C β ( B ( p,ρ ) ,g t ) (cid:54) C k (( R − ρ ) k + α [ D k τ ] C α ( B ( p,R ) ,g t ) + (cid:107) τ (cid:107) L ∞ ( B ( p,R ) ,g t ) ) , (2.48)for all p ∈ B × Y , 0 < ρ < R < d g ( p, ∂B × Y ).This can be formally deduced from (2.33). Indeed, pick x, x (cid:48) ∈ B ( p, ρ ), joined by a horizontal orvertical minimal P -geodesic, and let d = d g ( x, x (cid:48) ) . If d (cid:62) ( R − ρ ) we just bound the C β difference quotient for D j ˇ τ t at x, x (cid:48) by using the triangleinequality and the boundedness of the operator norm of P from Section 2.1.1, by Cd − β (cid:107) D j ˇ τ t (cid:107) L ∞ ( B ( p,ρ )) (cid:54) ( R − ρ ) − β (cid:107) D j ˇ τ t (cid:107) L ∞ ( B ( p,ρ )) , (2.49)which can then be bounded using (2.33). If d < ( R − ρ ) and j = k we just bound trivially the C β difference quotient for D k ˇ τ t at x, x (cid:48) by d α − β [ D k ˇ τ t ] C α ( B ( p,R )) (cid:54) ( R − ρ ) α − β [ D k ˇ τ t ] C α ( B ( p,R )) . (2.50)And if d < ( R − ρ ) and j < k , then the minimal P -geodesic joining x, x (cid:48) remains inside B ( x, d ) ⊂ B ( p, ρ + ( R − ρ )), and so we can apply Remark 2.5 and bound the C β difference quotient for D j ˇ τ t at x, x (cid:48) by Cd − β (cid:107) D j +1 ˇ τ t (cid:107) L ∞ ( B ( p,ρ + ( R − ρ ))) (cid:54) C ( R − ρ ) − β (cid:107) D j +1 ˇ τ t (cid:107) L ∞ ( B ( p,ρ + ( R − ρ ))) , (2.51)which can again be bounded using (2.33). This establishes (2.47). From H¨older seminorm bounds to convergence.
Assume we are in the same setting as inSection 2.3. At each point x = ( z, y ) ∈ B × Y we have a splitting T x ( B × Y ) = T z B ⊕ T y Y , and wewill denote by f any tangent vector in the first summand and by b any tangent vector in the secondsummand. If v is any vector in C m (either real, or of type (1 , v its trivialextension to a vector field (real or of type (1 , B × Y .One more piece of notation. First, without loss we may assume that (cid:82) { z }× Y ω nF,z = 1 for all z ∈ B .For a function f on B × Y we will denote by f the function on B given by f ( z ) = (cid:90) { z }× Y f ω nF,z , (2.52)i.e. the fiber average of f with respect to the varying Calabi-Yau volume forms ω F,z on { z } × Y .The following theorem will be a cornerstone of all of our later arguments. Theorem 2.9.
Let < δ t (cid:54) be given, and define product metrics g t on B × Y by (2.32) . Given k ∈ N , < α < and radii < ρ < R such that ρδ − t (cid:62) C for all t , there is a constant C such thatfor all (1 , -forms η = i∂∂ϕ with ϕ = 0 for all z ∈ B and all (cid:54) j (cid:54) k and t (cid:62) we have (cid:107) D j η (cid:107) L ∞ ( B ρ (0) × Y,g t ) (cid:54) Cδ k + α − jt [ D k η ] C α ( B R (0) × Y,g t ) . (2.53) Also, for all < β < such that j + β < k + α we have [ D j η ] C β ( B ρ (0) × Y,g t ) (cid:54) Cδ k + α − j − βt [ D k η ] C α ( B R (0) × Y,g t ) . (2.54)Theorem 2.9 is a quantitative and much improved version of [19, Prop. 5.5]. Indeed, the proof alsorelies on our crucial [19, Lemma 3.3] and its variants in Lemmas 2.11 and 2.12 below, and on a com-mutation estimate from [19] that we include and improve here in Lemma 2.10. It also uses Proposition2.8. The idea that uniform bounds on the highest derivatives in a shrinking H¨older seminorm shouldimply quantitative decay for the lower order derivatives came to us after reading the appendix of [5]and [20], where results in this direction are proved with different techniques. Proof.
First suppose we have shown that for all k, α and 0 < ρ < R with ρ (cid:62) C δ t there is C such that (cid:107) η (cid:107) L ∞ ( B R × Y,g t ) (cid:54) Cδ k + αt [ D k η ] C α ( B R × Y,g t ) , (2.55)for all η as above and all t . Then we claim that (2.53) and (2.54) follow. Indeed, (2.55) clearly implies(2.53) when j = 0. For j >
0, note first that by assumption we have B r × Y = B ( p, r ) for any p ∈ B × Y whose first component is 0 and any r (cid:62) ρ . We then use our interpolation Proposition 2.8 in the form(2.34), letting ˜ ρ = R − min( δ t , R − ρ ) so that R − ˜ ρ (cid:54) δ t and ρ (cid:54) ˜ ρ , and interpolating and using (2.55)( R − ˜ ρ ) j (cid:107) D j η (cid:107) L ∞ ( B ˜ ρ × Y,g t ) (cid:54) C ( R − ˜ ρ ) k + α [ D k η ] C α ( B R × Y,g t ) + C (cid:107) η (cid:107) L ∞ ( B R × Y,g t ) (cid:54) Cδ k + αt [ D k η ] C α ( B R × Y,g t ) , (2.56)which implies (2.53) (by considering the two cases when δ t (cid:54) R − ρ and δ t (cid:62) R − ρ , since the constant C in (2.53) is allowed to depend on R − ρ ). As for (2.54), we apply the interpolation inequality (2.48)( R − ˜ ρ ) j + β [ D j η ] C β ( B ˜ ρ × Y,g t ) (cid:54) C ( R − ˜ ρ ) k + α [ D k η ] C α ( B R × Y,g t ) + C (cid:107) η (cid:107) L ∞ ( B R × Y,g t ) (cid:54) Cδ k + αt [ D k η ] C α ( B R × Y,g t ) , (2.57)which proves (2.54).We then establish (2.55), which in local product coordinates, with vectors of unit length in a fixedproduct metric g C m + g Y , says that for all z ∈ B R we havesup { z }× Y | η bb | (cid:54) Cδ k + αt [ D k η ] C α ( B R × Y,g t ) , (2.58) ans-Joachim Hein and Valentino Tosatti 17 sup { z }× Y | η bf | (cid:54) Cδ k +1+ αt [ D k η ] C α ( B R × Y,g t ) , (2.59)sup { z }× Y | η ff | (cid:54) Cδ k +2+ αt [ D k η ] C α ( B R × Y,g t ) , (2.60)where in (2.59) the compressed notation η bf stands for either η vh or η hv where v, h are a horizontaland a vertical vector fields of type (1 , η is real, which is why we don’t differentiate them in our notation.We first establish (2.60) by writing η = i∂∂ϕ and applying Lemma 2.11 to ( { z } × Y, g
Y,z )sup { z }× Y | i∂ f ∂ f ϕ | g Y,z (cid:54) C [ ∇ z,k f ··· f i∂∂ϕ | { z }× Y ] C α ( { z }× Y,g
Y,z ) (cid:54) Cδ k +2+ αt [ D k f ··· f i∂∂ϕ | { z }× Y ] C α ( { z }× Y,g t ) (cid:54) Cδ k +2+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) , (2.61)as claimed, where we used that by definition ∇ z,k f ··· f = D k f ··· f .To establish (2.59) we again use η = i∂∂ϕ . Fix any horizontal (1 ,
0) vector field Z = b obtainedfrom b ∈ C m by trivial extension to B × Y , and restrict the function Z ( ϕ ) to a fiber { z } × Y . Then wewish to compare its vertical ∂ with the original η bf , i.e. we wish to estimate the differenceDiff = ( Z (cid:121) i∂∂ϕ ) | { z }× Y − ∂ (cid:0) Z ( ϕ ) | { z }× Y (cid:1) . (2.62)Lemma 2.10 provides a formula for this. We can then apply Lemma 2.12 to ( { z } × Y, g
Y,z )sup { z }× Y | i∂ b ∂ f ϕ | g Y,z (cid:54) sup { z }× Y | Diff | g Y,z + sup { z }× Y | ∂ (cid:0) Z ( ϕ ) | { z }× Y (cid:1) | g Y,z (cid:54) sup { z }× Y | Diff | g Y,z + C [ ∇ z,k f ··· f ∂ (cid:0) Z ( ϕ ) | { z }× Y (cid:1) ] C α ( { z }× Y,g
Y,z ) (cid:54) (cid:107) Diff (cid:107) C k,α ( { z }× Y,g
Y,z ) + C [ ∇ z,k f ··· f ( Z (cid:121) i∂∂ϕ ) | { z }× Y ] C α ( { z }× Y,g
Y,z ) (cid:54) (cid:107) Diff (cid:107) C k,α ( { z }× Y,g
Y,z ) + Cδ k +1+ αt [ D k f ··· f ( Z (cid:121) i∂∂ϕ ) | { z }× Y ] C α ( { z }× Y,g t ) (cid:54) (cid:107) Diff (cid:107) C k,α ( { z }× Y,g
Y,z ) + Cδ k +1+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) (cid:54) C [ ∇ k f ··· f i∂∂ ( ϕ | { z }× Y )] C α ( { z }× Y,g
Y,z ) + Cδ k +1+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) (cid:54) Cδ k +2+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) + Cδ k +1+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) (cid:54) Cδ k +1+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) , (2.63)using the formula of Lemma 2.10 together with Lemma 2.11.Lastly, we discuss (2.58). Fix two vectors h , h ∈ C m , we thus obtain a global function on B × Y given by η h h . The first main point will be to use the assumption ϕ = 0 to show thatsup B | η h h | (cid:54) Cδ k +1+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) . (2.64)To prove this, let us denote by Z = h , W = h , our constant (1 ,
0) vector fields on C m extendedtrivially to C m × Y , so that we are trying to bound the fiber average of ( i∂∂ϕ )( Z, W ). To do this, wefirst claim that | ( i∂∂ϕ )( Z, W ) − Z ( W ( ϕ )) | (cid:54) Cδ k +1+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) , (2.65)To prove this, we use the schematic formula Z = ∂∂z + F ∂∂y + G ∂∂y from the proof of Lemma 2.10 below(specifically (2.77)), and similarly for W . Then the main (base-base) terms of the difference in (2.65)obviously cancel out as expected, leaving us with every possible kind of fiber, fiber-fiber, and base-fiberderivative of ϕ as the error terms (i.e., any possible combination of (1 ,
0) indices and (0 ,
1) indices,or equivalently, any possible combination of real indices). The fiber and fiber-fiber ones are boundedby Cδ k +2+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) by using Lemma 2.11 as in (2.63). The base-fiber ones are bounded by Cδ k +1+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) by Lemma 2.12 applied to ψ = Z ( ϕ ) (using additionally, again, thecommutator Lemma 2.10, as we did in (2.63) above). This establishes (2.65).Taking then the fiber average of (2.65), we are left with bounding the fiber average of Z ( W ( ϕ )). Forthis, we apply Z ( W ( · )) to the relation0 = ϕ ( z ) = (cid:90) { z }× Y ϕω nF,z , (2.66)which allows us to bound Z ( W ( ϕ )) in terms of the fiberwise L ∞ norm of ϕ and of Z ( ϕ ) , W ( ϕ ). Since ϕ = 0, we can boundsup { z }× Y | ϕ | (cid:54) sup { z }× Y |∇ f ϕ | { z }× Y | g Y,z (cid:54) C [ ∇ k f ··· f i∂∂ϕ | { z }× Y ] C α ( { z }× Y,g
Y,z ) (cid:54) Cδ k +2+ αt [ D k f ··· f i∂∂ϕ | { z }× Y ] C α ( { z }× Y,g t ) (cid:54) Cδ k +2+ αt [ D k η ] C α ( B × Y,g t ) (2.67)using Lemma 2.11. As for Z ( ϕ ), applying Z to (2.66) allows us to bounds Z ( ϕ ) by the fiberwise L ∞ norm of ϕ , and so by Cδ k +2+ αt [ D k η ] C α ( B × Y,g t ) . We can then bound the fiberwise L ∞ norm of Z ( ϕ ) in terms of Z ( ϕ ) and of the L ∞ norm of the fiberwise ∂ (cid:0) Z ( ϕ ) | { z }× Y (cid:1) , which is bounded by Cδ k +1+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) thanks to (2.63). Putting all of these together completes the proof of(2.64).Now that (2.64) is proved, we can use it to prove (2.58). Assume first that k = 0 and bound the L ∞ norm of η h h on the fiber { z } × Y in terms of its fiberwise C α seminorm and its fiberwise averagesup { z }× Y | η h h | g Y,z (cid:54) sup B | η h h | + [ η h h | { z }× Y ] C α ( { z }× Y,g
Y,z ) (cid:54) Cδ αt [ i∂∂ϕ ] C α ( B × Y,g t ) + Cδ αt [ η ] C α ( B × Y,g t ) (cid:54) Cδ αt [ η ] C α ( B × Y,g t ) , (2.68)using (2.64). If k (cid:62)
1, we instead first bound the fiberwise L ∞ norm of η h h in terms of the L ∞ normof its fiberwise gradient and its fiberwise average, and then invoke [19, Lemma 3.3]:sup { z }× Y | η h h | g Y,z (cid:54) sup B | η h h | + sup { z }× Y |∇ f η h h | { z }× Y | g Y,z (2.69) (cid:54) Cδ k +1+ αt [ D k i∂∂ϕ ] C α ( B × Y,g t ) + C [ ∇ k f ··· f η h h | { z }× Y ] C α ( { z }× Y,g
Y,z ) (2.70) (cid:54) Cδ k +1+ αt [ D k η ] C α ( B × Y,g t ) + Cδ k + αt [ D k f ··· f η h h | { z }× Y ] C α ( { z }× Y,g t ) (2.71) (cid:54) Cδ k + αt [ D k η ] C α ( B × Y,g t ) , (2.72)which concludes the proof of the Proposition. (cid:3) The following lemma was used in the proof of Theorem 2.9. Its proof is essentially contained in theproof of [19, Prop. 5.5], but we include it here for the reader’s convenience.
Lemma 2.10.
For any fixed type (1 , coordinate vector field Z on C m , ( Z (cid:121) i∂∂ϕ ) | { z }× Y − ∂ ( Z ( ϕ ) | { z }× Y ) = L (cid:126) D ( ϕ | { z }× Y ) + L (cid:126) D ( ϕ | { z }× Y ) . (2.73) Here
D, D are the gradient and Hessian with respect to any fixed local coordinate system on Y , and L , L are fixed smooth coefficient functions depending only on the coordinate system.Proof. Fix any y ∈ Y and a J Y -holomorphic chart ( y , . . . , y n ) on { z } × Y near y . Extend the functions y i to J -holomorphic functions ˆ y i on a neighborhood of ( z, y ) in the total space. The key property ofthe “fibered” chart ( z , . . . , z m , ˆ y , . . . , ˆ y n ) is that ∂ϕ∂ ˆ y p (cid:12)(cid:12)(cid:12)(cid:12) { z }× Y = ∂ ( ϕ | { z }× Y ) ∂y p (2.74) ans-Joachim Hein and Valentino Tosatti 19 for all local functions ϕ . Thus, expanding η = i∂∂ϕ in terms of this chart,( Z j (cid:121) η ) | { z }× Y = i n (cid:88) p =1 ∂∂y p (cid:18) ∂ϕ∂z j (cid:12)(cid:12)(cid:12)(cid:12) { z }× Y (cid:19) dy p (2.75)+ i n (cid:88) p,q =1 (cid:18)(cid:20) ∂ ( ϕ | { z }× Y ) ∂y q ∂y p Z j (ˆ y q ) (cid:21) dy p − (cid:20) ∂ ( ϕ | { z }× Y ) ∂y p ∂y q Z j (ˆ y q ) (cid:21) dy p (cid:19) . (2.76)This is straightforward to check.Since J is fibered, the (0 , Z j with respect to J is a section of T ∗ Y ⊗ C , and (2.74) saysthat T ∗ Y ⊗ C is generated by the vector fields ∂/∂ ˆ y p and their complex conjugates. Thus, if we expand Z j in terms of our chart, there will be no ∂/∂z k components. Since dz k ( Z j ) = δ kj , it follows that Z j = ∂∂z j − n (cid:88) p =1 (cid:18) a pj ∂∂ ˆ y p + b pj ∂∂ ˆ y p (cid:19) . (2.77)Thus, ∂∂y p (cid:18) ∂ϕ∂z j (cid:12)(cid:12)(cid:12)(cid:12) { z }× Y (cid:19) − ∂∂y p ( Z j ( ϕ ) | { z }× Y ) = n (cid:88) q =1 (cid:18) ∂a qj ∂ ˆ y p ∂ ( ϕ | { z }× Y ) ∂y q + ∂b qj ∂ ˆ y p ∂ ( ϕ | { z }× Y ) ∂y q (2.78)+ a qj ∂ ( ϕ | { z }× Y ) ∂y p ∂y q + b qj ∂ ( ϕ | { z }× Y ) ∂y p ∂y q (cid:19) . (2.79) (cid:3) Lastly, in the proof of Theorem 2.9 we used the following two variants of [19, Lemma 3.3].
Lemma 2.11.
Let ( Y, g ) be a compact K¨ahler manifold. Then for all k ∈ N (cid:62) , α ∈ (0 , there existsa constant C k = C k ( Y, g, α ) such that for all ϕ ∈ C k +2 ,α ( Y ) , k (cid:88) i = − (cid:107)∇ i +2 ϕ (cid:107) L ∞ ( Y ) (cid:54) C k [ ∇ k ∂∂ϕ ] C α ( Y ) . (2.80) Proof.
This is proved along the lines of [19, Lemma 3.3]. We may assume without loss that ϕ has averagezero. Suppose the lemma fails for some k, α , so there exists a sequence ϕ i ∈ C k +2 ,α ( Y ) with zero averagewith [ ∇ k ∂∂ϕ i ] C α ( Y ) < i (cid:80) kj = − (cid:107)∇ j +2 ϕ i (cid:107) L ∞ ( Y ) . Dividing ϕ i by (cid:80) kj = − (cid:107)∇ j +2 ϕ i (cid:107) L ∞ ( Y ) >
0, we mayfurther assume that (cid:80) kj = − (cid:107)∇ j +2 ϕ i (cid:107) L ∞ ( Y ) = 1. Claim.
There exists a constant C such that (cid:107) ϕ i (cid:107) L ∞ ( Y ) (cid:54) C for all i . Proof of the Claim.
Suppose that this is false. Then we may assume that (cid:107) ϕ i (cid:107) L ∞ ( Y ) > i . Divid-ing ϕ i by (cid:107) ϕ i (cid:107) L ∞ ( Y ) , we may further assume that (cid:107) ϕ i (cid:107) L ∞ ( Y ) = 1, (cid:80) kj = − (cid:107)∇ j +2 ϕ i (cid:107) L ∞ ( Y ) < i , and[ ∇ k ∂∂ϕ i ] C α ( Y ) < i . Using standard elliptic estimates, we obtain from these a uniform C k +2 ,α boundon ϕ i , so that by passing to a subsequence, we may then also assume that ϕ i converges to some ϕ ∈ C k +2 ,α ( Y ) in the C k +2 ,β sense for all β < α . By construction, this limit satisfies (cid:107) ϕ (cid:107) L ∞ ( Y ) = 1, ∇ ϕ = 0, and ϕ has zero average, which is a contradiction. (cid:3) Given the claim, we can again use elliptic estimates to obtain a uniform C k +2 ,α bound on ϕ i andpassing to a subsequence if needed, we may now assume that ϕ i converges to some ϕ ∈ C k +2 ,α ( Y ) inthe C k +2 ,β topology for every β < α . By construction, this limit ϕ satisfies the following properties: [ ∇ k ∂∂ϕ ] C α ( Y ) = 0, (cid:80) kj = − (cid:107)∇ j +2 ϕ (cid:107) L ∞ ( Y ) = 1, and ϕ has average zero. The first property implies that ϕ is smooth with ∇ k +1 ∂∂ϕ = 0. Thus, relying crucially on the fact that ∂Y = ∅ , (cid:90) Y |∇ k ∂∂ϕ | = (cid:90) Y (cid:104)∇ k − ∂∂ϕ, ∇ ∗ ∇ k ∂∂ϕ (cid:105) = 0 , (2.81)and working backwards we get ∇ j ∂∂ϕ = 0 for 1 (cid:54) j (cid:54) k , and then also (cid:90) Y | ∂∂ϕ | = (cid:90) Y (cid:104) ∂ϕ, ∂ ∗ ∂∂ϕ (cid:105) = 0 , (2.82)so ∂∂ϕ = 0 which implies that ϕ = 0, a contradiction. (cid:3) Lemma 2.12.
Let ( Y, g ) be a compact K¨ahler manifold. Then for all k ∈ N (cid:62) , α ∈ (0 , there existsa constant C k = C k ( Y, g, α ) such that for all ψ ∈ C k +1 ,α ( Y ) , k (cid:88) i =0 (cid:107)∇ i +1 ψ (cid:107) L ∞ ( Y ) (cid:54) C k [ ∇ k ∂ψ ] C α ( Y ) . (2.83) Proof.
Again, the proof is a simple modification of the proof of [19, Lemma 3.3]. We may assumewithout loss that ψ has average zero. Suppose the lemma fails for some k, α , so there exists a se-quence ψ i ∈ C k +1 ,α ( Y ) with zero average with [ ∇ k ∂ψ i ] C α ( Y ) < i (cid:80) kj =0 (cid:107)∇ j +1 ψ i (cid:107) L ∞ ( Y ) . Dividing ψ i by (cid:80) kj =0 (cid:107)∇ j +1 ψ i (cid:107) L ∞ ( Y ) >
0, we may further assume that (cid:80) kj =0 (cid:107)∇ j +1 ψ i (cid:107) L ∞ ( Y ) = 1. Claim.
There exists a constant C such that (cid:107) ψ i (cid:107) L ∞ ( Y ) (cid:54) C for all i . Proof of the Claim.
Suppose that this is false. Then we may assume that (cid:107) ψ i (cid:107) L ∞ ( Y ) > i . Dividing ψ i by (cid:107) ψ i (cid:107) L ∞ ( Y ) , we may further assume that (cid:107) ψ i (cid:107) L ∞ ( Y ) = 1, (cid:80) kj =0 (cid:107)∇ j +1 ψ i (cid:107) L ∞ ( Y ) < i , and [ ∇ k ∂ψ i ] C α ( Y ) < i . Using standard elliptic estimates (since we have control over the C k − ,α norm of ∆ ψ i ), we obtainfrom these a uniform C k +1 ,α bound on ψ i , so that by passing to a subsequence, we may then alsoassume that ψ i converges to some ψ ∈ C k +1 ,α ( Y ) in the C k +1 ,β sense for all β < α . By construction,this limit satisfies (cid:107) ψ (cid:107) L ∞ ( Y ) = 1, ∇ ψ = 0, and ψ has zero average, which is a contradiction. (cid:3) Given the claim, we can again use elliptic estimates to obtain a uniform C k +1 ,α bound on ψ i andpassing to a subsequence if needed, we may now assume that ψ i converges to some ψ ∈ C k +1 ,α ( Y ) inthe C k +1 ,β topology for every β < α . By construction, this limit ψ satisfies the following properties:[ ∇ k ∂ψ ] C α ( Y ) = 0, (cid:80) kj =0 (cid:107)∇ j +1 ψ (cid:107) L ∞ ( Y ) = 1, and ψ has average zero. The first property implies that ψ is smooth with ∇ k +1 ∂ψ = 0. Thus, relying crucially on the fact that ∂Y = ∅ , (cid:90) Y |∇ k ∂ψ | = (cid:90) Y (cid:104)∇ k − ∂ψ, ∇ ∗ ∇ k ∂ψ (cid:105) = 0 , (2.84)and working backwards we get ∇ j ∂ψ = 0 for 1 (cid:54) j (cid:54) k . In particular ∂∂ψ = 0 which implies that ψ = 0, a contradiction. (cid:3) Schauder estimates.
We will also need the following Schauder estimate. Let ( z t , y t ) → ( z ∞ , y ∞ )be a convergent family of points in B × Y . Consider the diffeomorphismsΛ t : B ψe t × Y → B × Y, ( z, y ) = Λ t (ˇ z, ˇ y ) = ( z t + e − t ˇ z, ˇ y ) , (2.85)defined for all t (cid:62) t ( ψ ), where ψ > d ( z ∞ , ∂B ), and letˇ J t = Λ ∗ t J denote the pullback of the given fibered complex structure J on B × Y . Then ˇ J t convergesto ˇ J ∞ = J C m + J Y,z ∞ locally smoothly. Similarly, let ˇ D t denote the pullback of the connection D , sothat ˇ D t → ˇ D ∞ = ∇ C m + ∇ g Y,z ∞ locally smoothly. Our new basepoint is (ˇ z t , ˇ y t ) = (0 , y t ) → (0 , y ∞ ). ans-Joachim Hein and Valentino Tosatti 21 Proposition 2.13.
Let U ⊂ C m × Y be a bounded open set containing (0 , y ∞ ) . Let ˇ g t , ˇ ω (cid:93)t be Riemannianresp. ˇ J t -K¨ahler metrics on U that converge locally smoothly to a Riemannian resp. ˇ J ∞ -K¨ahler metric ˇ g ∞ , ˇ ω (cid:93) ∞ on U . Then for all a ∈ N and α ∈ (0 , there exist R > , t < ∞ and C < ∞ such that forall < ρ < R (cid:54) R , t (cid:62) t , and all smooth real-valued i∂∂ -exact ˇ J t - (1 , -forms η on U we have that [ ˇ D at η ] C α ( B ˇ gt ((0 , ˇ y t ) ,ρ )) (cid:54) C [ ˇ D at tr ˇ ω (cid:93)t η ] C α ( B ˇ gt ((0 , ˇ y t ) ,R )) + C ( R − ρ ) − a − α (cid:107) η (cid:107) L ∞ ( B ˇ gt ((0 , ˇ y t ) ,R )) . (2.86) Proof.
There exists a σ > g C m + g Y,z ∞ admits a normal coordinate chartΦ : R m +2 n ⊃ B σ (0) → B g C m + g Y,z ∞ ((0 , ˇ y ∞ ) , σ ) ⊂ U (2.87)centered at the point (0 , ˇ y ∞ ). Let σ ∈ (0 , σ ] be arbitrary but fixed, to be determined later. Choose R so small and t so large that B ˇ g t ((0 , ˇ y t ) , r ) is ˇ g t -geodesically convex and moreover contained inΦ( B σ (0)) for all r (cid:54) R and t (cid:62) t . Pulling back by Φ, we may then pretend that all objects of interestare defined on (resp. contained in) the ball B σ (0) ⊂ R m +2 n . In particular, for all r (cid:54) R and t (cid:62) t ,the geodesic ball ˇ B t ( r ) := B ˇ g t ((0 , ˇ y t ) , r ) has compact closure in B σ (0). Moreover, the K¨ahler structuresˇ ω (cid:93)t , ˇ J t on B σ (0) satisfy uniform (in t ) C ∞ bounds.Fix 0 < ρ < R (cid:54) R . For all x ∈ ˇ B t ( ρ ) define ˇ B x := B ˇ g t ( x, ( R − ρ )). Let λ ˇ B x denote the concentricˇ g t -ball of λ times the radius. Then we claim that there exists a uniform C such that for all f , (cid:18) a +2 (cid:88) b =0 ( R − ρ ) b − a − − α (cid:107) ∂ b f (cid:107) L ∞ ( ˇ B x ) (cid:19) + [ ∂ a +2 f ] C α ( ˇ B x ) (cid:54) C (cid:18) ( R − ρ ) − a − − α (cid:107) f (cid:107) L ∞ (2 ˇ B x ) + (cid:18) a (cid:88) b =0 ( R − ρ ) b − a − α (cid:107) ∂ b ∆ ˇ ω (cid:93)t f (cid:107) L ∞ (2 ˇ B x ) (cid:19) + [ ∂ a ∆ ˇ ω (cid:93)t f ] C α (2 ˇ B x ) (cid:19) , (2.88)where all norms and derivatives are defined with respect to the standard Euclidean metric on R m +2 n .Indeed, after scaling and stretching by ( R − ρ ) − we can apply standard interior Schauder estimatesfor an essentially fixed pair of domains in R m +2 n and an essentially fixed operator. See [11, Thm 6.2]for the case a = 0, noting that their statement gives more than we are using here but not in a way thatwould help us directly. After scaling back to the original picture, this yields (2.88).We now eliminate the (cid:107) ∂ b ∆ ˇ ω (cid:93)t f (cid:107) L ∞ terms with b (cid:62) B x to 3 ˇ B x . Forexample, we can again stretch and scale by ( R − ρ ) − to reduce to the case of two essentially fixeddomains in R m +2 n , invoke [11, Lemma 6.32], and then again scale back to the original picture.Let η be an i∂∂ -exact (1 , η with respect to ˇ J t on B σ (0) as in the statement of the proposition.We can follow the proof of [17, Prop 3.2] to find a ˇ J t -potential f for η on 3 ˇ B x with (cid:107) f (cid:107) L ∞ (2 ˇ B x ) (cid:54) C ( R − ρ ) (cid:107) η (cid:107) L ∞ (3 ˇ B x ) . (2.89)The proof in [17] was written for a standard ball in C m + n but all the ingredients carry over to smallˇ g t -geodesic balls and the K¨ahler structure ˇ ω (cid:93)t , ˇ J t . Indeed, the standard Poincar´e lemma formula canbe made to work using radial ˇ g t -geodesics, solving the ∂ -Neumann problem only requires uniformˇ J t -plurisubharmonicity of the squared ˇ g t -distance, and Moser iteration for the ˇ ω (cid:93)t -Laplacian also onlydepends on uniform geometry bounds for ˇ g t , ˇ ω (cid:93)t , ˇ J (cid:93)t . All of these hold after increasing t and decreasing σ if necessary. Here we are only using the smoothness of the (unrelated) tensors ˇ g ∞ and ˇ ω (cid:93) ∞ , ˇ J ∞ , andare not using the normal coordinate property of Φ with respect to the metric g C m + g Y,z ∞ . Inserting (2.89) into (2.88) together with the above interpolation argument, we obtain that (cid:18) a +2 (cid:88) b =0 ( R − ρ ) b − a − − α (cid:107) ∂ b f (cid:107) L ∞ ( ˇ B x ) (cid:19) + [ ∂ a +2 f ] C α ( ˇ B x ) (cid:54) C (cid:18) ( R − ρ ) − a − α (cid:107) η (cid:107) L ∞ (3 ˇ B x ) + [ ∂ a tr ˇ ω (cid:93)t η ] C α (3 ˇ B x ) (cid:19) . (2.90)Our next claim is that this implies that (cid:18) a (cid:88) b =0 ( R − ρ ) b − a − α (cid:107) ∂ b η (cid:107) L ∞ ( ˇ B x ) (cid:19) + [ ∂ a η ] C α ( ˇ B x ) (cid:54) C (cid:18) ( R − ρ ) − a − α (cid:107) η (cid:107) L ∞ (3 ˇ B x ) + [ ∂ a tr ˇ ω (cid:93)t η ] C α (3 ˇ B x ) (cid:19) . (2.91)This is straightforward to check by writing η = − d ( df ◦ ˇ J t ) on the left-hand side of (2.91) and applyingthe Leibniz rule (discrete as well as continuous). The main terms, where no derivatives or differencequotients land on ˇ J t , can obviously be bounded by the corresponding terms (with two more derivativesof f than of η ) on the left-hand side of (2.90). All of the other terms can similarly be absorbed intothe left-hand side of (2.90) with a suitable index shift, thanks to the fact that R − ρ = O (1).Getting closer to the statement of the proposition, we now claim that (2.91) implies that (cid:18) a (cid:88) b =0 ( R − ρ ) b − a − α (cid:107) ∂ b η (cid:107) L ∞ ( ˇ B t ( ρ )) (cid:19) + [ ∂ a η ] C α ( ˇ B t ( ρ )) (cid:54) C (cid:18) ( R − ρ ) − a − α (cid:107) η (cid:107) L ∞ ( ˇ B t ( R )) + [ ∂ a tr ˇ ω (cid:93)t η ] C α ( ˇ B t ( R )) (cid:19) . (2.92)Indeed, taking the supremum of (2.91) over all x ∈ ˇ B t ( ρ ) already takes care of the (cid:107) ∂ b η (cid:107) L ∞ terms onthe left-hand side of (2.92). To bound the [ ∂ a η ] C α term, note that for x (cid:54) = x (cid:48) in ˇ B t ( ρ ) we can estimatethe C α quotient of ∂ a η at x, x (cid:48) in two ways. If x (cid:48) (cid:54)∈ ˇ B x , we can estimate it trivially using the triangleinequality, resulting in a term which is absorbed by the (cid:107) ∂ a η (cid:107) L ∞ term that we have just dealt with.On the other hand, if x (cid:48) ∈ ˇ B x , we can use (2.91) directly.In (2.92), we can replace the Euclidean metric used in the definition of the norms by ˇ g t because thesetwo metrics are uniformly equivalent. Less trivially, we may also replace ∂ by ˇ D t and the Euclideanparallel transport P implicit in the definition of the C ,α seminorm by the stretched P -transport ˇ P t .Once we know this, the proposition follows by dropping the sum over b from the left-hand side.To compare ∂ to ˇ D t , write ∂ = ˇ D t + ˇΓ t . Then ˇΓ t → ˇΓ ∞ locally smoothly, where ˇΓ ∞ (0) = 0 (recallthat 0 ∈ R m +2 n is the origin of the normal coordinate chart Φ for the metric g C m + g Y,z ∞ at (0 , ˆ y ∞ ),and that ˇ D ∞ is the Levi-Civita connection of this metric). Thus, the C ( B σ (0)) norm of ˇΓ t goes tozero as t → ∞ and σ →
0, and for all k (cid:62) C k ( B σ (0)) norm of ˇΓ t remains uniformly bounded as t → ∞ and σ →
0. By ODE estimates, the same statement holds for the C k ( B σ (0) × B σ (0)) normsof ˇ P t − P . Thus, when we replace ∂ and P by ˇ D t and ˇ P t in (2.92), the terms of highest order in η contained in the error (i.e., the C a,α seminorm terms) come with a coefficient which is o (1) as t → ∞ and σ →
0. All of the other error terms come with an explicit factor (cid:54) C ( R − ρ ) α compared to the ans-Joachim Hein and Valentino Tosatti 23 already existing terms of the same order in η in (2.92). Thus, we obtain an estimate of the form (cid:18) a (cid:88) b =0 ( R − ρ ) b − a − α (cid:107) ˇ D bt η (cid:107) L ∞ ( ˇ B t ( ρ )) (cid:19) + [ ˇ D at η ] C α ( ˇ B t ( ρ )) (cid:54) C (cid:18) ( R − ρ ) − a − α (cid:107) η (cid:107) L ∞ ( ˇ B t ( R )) + [ ˇ D at tr ˇ ω (cid:93)t η ] C α ( ˇ B t ( R )) (cid:19) + o (1) (cid:18)(cid:18) a (cid:88) b =0 ( R − ρ ) b − a − α (cid:107) ˇ D bt η (cid:107) L ∞ ( ˇ B t ( R )) (cid:19) + [ ˇ D at η ] C α ( ˇ B t ( R )) (cid:19) , (2.93)where the o (1) is uniform as t → ∞ and σ →
0. Thus, fixing σ ∈ (0 , σ ] sufficiently small and increasing t if necessary, the desired result now follows from the iteration lemma of [19, Lemma 3.4]. (cid:3) Selection of obstruction functions
The main result of this section, Theorem 3.10, roughly speaking identifies a list of smooth functions G j,p , j (cid:62) , which arise due to the unboundedness of the RHS of the Monge-Amp`ere equation in theshrinking C j,α norm. In order to even state precisely the properties that these functions satisfy, we firstneed to discuss how to turn a set of smooth functions on the total space into a fiberwise orthonormalset (with a suitable small error), which will be done in Section 3.1, and we need to construct a certainapproximate Green’s operator in Section 3.2.3.1. Approximate fiberwise Gram-Schmidt.
Let B be a ball centered at the origin in R d withreal linear coordinates z , . . . , z d . Let Y be a smooth compact manifold without boundary. Let Υ bea smooth fiberwise volume form on B × Y , with fiberwise total volume equal to 1, and for clarity wewill denote by Υ z its restriction to the fiber { z } × Y . In our later applications, we will take Υ so thatΥ z is the unit-volume fiberwise Calabi-Yau volume form. Proposition 3.1.
Suppose we are given an integer J (cid:62) and functions F , . . . , F N , H , . . . , H L ∈ C ∞ ( B × Y ) and suppose also that the functions (cid:8) H k | { z }× Y (cid:9) (cid:54) k (cid:54) L are L -orthonormal (w.r.t. Υ ) forall z ∈ B . Then we can find a concentric ball B (cid:48) ⊂ B and functions G j ∈ C ∞ ( B (cid:48) × Y ) , (cid:54) j (cid:54) N (cid:48) suchthat the functions (cid:8) G j | { z }× Y , H k | { z }× Y (cid:9) (cid:54) j (cid:54) N (cid:48) , (cid:54) k (cid:54) L are L -orthonormal (w.r.t. Υ ) for all z ∈ B (cid:48) , andthere are functions p ij , q ik , r ∈ C ∞ ( B (cid:48) ) , (cid:54) i (cid:54) N, (cid:54) j (cid:54) N (cid:48) , (cid:54) k (cid:54) L and K i,α ∈ C ∞ ( B (cid:48) × Y ) ( α ∈ N d , | α | = J + 1 ) such that on B (cid:48) × Y we have F i ( z, y ) = N (cid:48) (cid:88) j =1 p ij ( z ) G j ( z, y ) + L (cid:88) k =1 q ik ( z ) H k ( z, y ) + (cid:88) | α | = J +1 z α K i,α ( z, y ) (3.1) for (cid:54) i (cid:54) N . Remark 3.2.
The same statement holds in the real-analytic category if all the data are real-analytic,and the proof is verbatim the same.
Remark 3.3.
If we choose H to be the constant function 1, then H , . . . , H L , G , . . . , G N (cid:48) will havefiberwise average zero with respect to Υ, and if in addition some F i has fiberwise average zero, thenthe coefficient in front of H in the expansion (3.1) will be zero. Remark 3.4.
The errors z α K i,α ( z, y ) in (3.1) have the following crucial property: if Σ t are the diffeo-morphisms in (3.9) then we have that e Jt Σ ∗ t ( z α K i,α ) → , (3.2) Figure 3.1.
Source: https://xkcd.com/2407/.smoothly on B R × Y for any fixed R . Indeed, trivially,sup B R | D ι Σ ∗ t z α | (cid:54) C ( ι, R ) e − J +12 t (3.3)for all ι (cid:62)
0. Using (3.3) it is then easy to conclude that (3.2) holds.
Proof.
It is enough to prove this for one single F because if there are several then we can just processthem one by one, adding the newly gained G s to the previously given list of H s in each step. Then beginby replacing F by its fiberwise orthogonal projection onto the orthogonal complement of H , . . . , H L . Ifthis projection (which we shall call F from now on) is not identically zero on the central fiber, then wedivide F by its fiberwise L norm (after concentrically shrinking the base if necessary) and stop—theproposition is proved, with one single function G ( z, y ) := F ( z, y ) / (cid:107) F ( z, · ) (cid:107) L and with K α ≡
0. So wemay assume from now on that F vanishes identically on the central fiber.We will now construct an ordered tree such that the desired expansion can be read off by performinga depth-first traversal of the tree. An ordered tree is a tree for which the children of each node areordered, and depth-first search is a way of ordering the whole set of nodes, see Figure 3.1. In particular,it makes sense to talk about previous and subsequent nodes with respect to depth-first search. In adepth-first search, at any given node n of an ordered tree T , traversal of T proceeds by ordering thenodes of the subtree S of T whose root is n (before moving on to nodes outside of S ). Moreover, thefirst node visited after traversal of S is the first unvisited child (with respect to the given ordering ofthe children) of the most recent ancestor of n not all of whose children have been visited.The nodes of our tree come in two flavors: undecorated and decorated. An undecorated node is oneof the form ( F ), where F is a smooth function on B × Y vanishing along the central fiber { } × Y .A decorated node is one of the form ( F | G , . . . , G k ), where F is as before and G , . . . , G k are smoothfunctions on B × Y that are fiberwise orthonormal in L . By construction, G s appearing as decorationin different nodes will be fiberwise orthogonal to each other and to the H s, and the F in each node(decorated or undecorated) will be fiberwise orthogonal to the span of the H s and all the G s appearingas decoration in all previous nodes. Also by construction, k ∈ { , , . . . , d } .We put an undecorated node ( F ) as the root of the tree, where F is the function constructed in thefirst paragraph of this proof. Then we construct our tree by iterating the following procedure.1. If there is no undecorated node of depth (cid:54) J (i.e., descended from the root of the tree in atmost J generations), then stop. Otherwise, go to the first such node, ( F ), with respect to thedepth-first order of the tree that has been constructed so far, and continue as follows.2. Define functions F , . . . , F d ∈ C ∞ ( B × Y ), where for i ∈ { , . . . , d } , F i ( z, y ) := (cid:90) ∂F∂z i ( tz, y ) dt. (3.4) ans-Joachim Hein and Valentino Tosatti 25 Moreover, let O be the set of all the initial H s, union the set of all the G s from all decoratednodes that come before ( F ) in the depth-first ordering of the tree that has been constructed sofar. By induction, this is a fiberwise orthonormal set of functions in C ∞ ( B × Y ).3. On the central fiber, consider the finite-dimensional subspace V ⊂ C ∞ ( { } × Y ) spanned bythe restrictions of elements of O ∪ { F , . . . , F d } . The functions from O are an L -orthonormalsubset of V . Complete this set to a basis of V by adding elements from F , . . . , F d , which (afterreordering) we may assume are F , . . . , F d for some d ∈ { , , . . . , d } . Then, after shrinking B concentrically, F , . . . , F d will remain fiberwise linearly independent of each other and of O overevery point of B . Thus, by applying Gram-Schmidt, we can construct a fiberwise orthonormalset { G , . . . , G d } ⊂ C ∞ ( B × Y ), fiberwise orthogonal to O , such that the fiberwise span of O ∪ { G , . . . , G d } is the same as the fiberwise span of O ∪ { F , . . . , F d } (which, on the centralfiber, is V ). Change the undecorated node ( F ) to the decorated node ( F | G , . . . , G d ).4. Replace F d +1 , . . . , F d by their fiberwise orthogonal projections onto the fiberwise orthogonalcomplement of the fiberwise span of O ∪ { G , . . . , G d } . For each of the resulting functions F i ∈ C ∞ ( B × Y ) ( i ∈ { d + 1 , . . . , d } ), all of which vanish identically on the central fiber,we now create an undecorated child node ( F i ) of the decorated parent node ( F | G , . . . , G d ).Observe that the children are indeed naturally ordered at this point, even though the chosenorder is arbitrary and any choice of an order would have been fine.This procedure ends after finitely many steps because by construction the tree keeps growing in eachstep (unless the procedure ends) but can only contain at most d J +1 nodes of either kind in total. Thus,eventually, there will be no undecorated nodes of depth (cid:54) J left. In particular, at the final stage, theonly undecorated nodes will be of depth = J + 1, and will have no children.We now obtain the desired expansion of F as follows. Traverse the tree depth-first. At each decoratednode ( F | G , . . . , G k ), we may by construction write F = z F + · · · + z d F d , where each F i is either a C ∞ ( B )-linear combination of the H s, the decorating G s from previous nodes, and the decorating G sfrom the current node, or equal to the F stored in one of the (decorated or undecorated) children ofthe current node. We can ignore the contribution of F i s of the first type because this is exactly whatthe non-remainder terms of the expansion (3.1) look like. For each F i of the second type, we wait untilthe depth-first traversal has arrived at the corresponding child node, and then either do nothing (if thechild node is undecorated of depth = J + 1), or iterate the argument by further expanding F i .It remains to check that once the procedure has ended, the remainder terms are indeed of the form (cid:80) | α | = J +1 z α K α ( z, y ) with K α ∈ C ∞ ( B × Y ). This can be done by induction on J . Indeed, for J = 0 thesituation is clear. Assuming the proposition holds for J − J (cid:62)
1, consider the nodes of depth1, i.e., those (decorated) nodes ( F i | · · · ) whose parent is the root ( F | · · · ) of the tree. Each of thesenodes is the root of a subtree, and depth-first search on the whole tree restricts to depth-first searchon the subtree. Depths of nodes in the subtree are one less than their depths in the whole tree. Thus,the inductive hypothesis applies to the subtree, producing remainders of the form (cid:80) | α | = J z α K α ( z, y )in the expansion of each F i . But each F i comes with a coefficient z i in the expansion of F . (cid:3) Approximate Green operators.
Recall now the definition of ω F which was defined in theIntroduction: first, we define ω F = ω X + i∂∂ρ where ρ is the smooth function on B × Y defined by thefact that ω X | { z }× Y + i∂∂ρ ( z, · ) is the unique Ricci-flat K¨ahler metric on { z } × Y cohomologous to therestriction of ω X , and by the normalization (cid:82) { z }× Y ρ ( z, y ) ω nX ( y ) = 0.In this section we will also denote by ω can a fixed K¨ahler metric on B , which in our later applicationto Theorems A and B will be the restriction to our ball B of the solution of the Monge-Amp`ere equation(1.3). Given a smooth function H with (cid:82) { z }× Y Hω nF = 0 for all z ∈ B , and a (1 , α on B × Y wedefine a smooth function on B by P t,H ( α ) = n (pr B ) ∗ (cid:0) Hα ∧ ω n − F (cid:1) + e − t tr ω can (pr B ) ∗ ( Hα ∧ ω nF ) , (3.5)which we think of as an approximation to the fiberwise L -projection of tr ω (cid:92)t α onto span H . Note thatfor every (1 , β on B we have P t,H (pr ∗ B β ) = 0 , (3.6)since the first term in (3.5) vanishes by definition, and the second term is also seen to vanish by usingthe projection formula (pr B ) ∗ ( H pr ∗ B β ∧ ω nF ) = β ∧ (pr B ) ∗ ( Hω nF ) , and the fact that H has fiberwiseaverage zero.Let us also note that if G is a smooth function on B × Y then P t,H ( i∂∂G )( z ) = (cid:90) { z }× Y H L t ( G ) ω nF , (3.7)where we have defined L t ( G ) so that along { z } × Y it equals L t ( G ) := ∆ ω F | { z }× Y G + e − t (( (cid:80) mι =1 Z ι ∧ Z ι ) (cid:121) ( i∂∂G ∧ ω nF )) | { z }× Y ω nF | { z }× Y , (3.8)where Z ι denotes an orthonormal (1 ,
0) frame field with respect to ω can on C m , trivially extended to C m × Y .For functions A on the base and G on the total space (with fiberwise average zero), we now definea new function G t,k = G t,k ( A, G ). This should be thought of as an approximate right inverse of ∆ ω (cid:92)t applied to AG (after scaling and stretching so that the fibers have unit size). Here we want to view G as being fixed, while A is variable except for the fact that A is expected to be a polynomial or to behavelike one. The precise properties of G t,k that we need will be proved in Lemmas 3.6–3.7. Somewhatstrangely, G t,k will be a differential rather than an integral operator in the base directions.Fix any z ∈ B . The aim is to define G t,k at all points ( z , y ) in the fiber over z . To this end, weshall use the diffeomorphismsΣ t : B e t × Y → B × Y, ( z, y ) = Σ t (ˇ z, ˇ y ) = ( e − t ˇ z, ˇ y ) , (3.9)and for any function u on B × Y we will write ˇ u = Σ ∗ t u . Observe that we can write (with obviousnotation) Σ ∗ t ω F = (Σ ∗ t ω F ) bb + (Σ ∗ t ω F ) bf + (Σ ∗ t ω F ) ff = e − t ˇ ω F, bb + e − t ˇ ω F, bf + ˇ ω F, ff , (3.10)and let us also define ˇ ω can = e t Σ ∗ t ω can . In particular, pulling back (3.7) and (3.8) givesˇ P t, ˇ H ( i∂∂ ˇ G )(ˇ z ) = (cid:90) { ˇ z }× Y ˇ H ˇ L t ( ˇ G )Σ ∗ t ω nF , (3.11)and along { ˇ z } × Y ˇ L t ( ˇ G ) = ∆ Σ ∗ t ω F | { ˇ z }× Y ˇ G + (( (cid:80) mι =1 ˇ Z ι ∧ ˇ Z ι ) (cid:121) ( i∂∂ ˇ G ∧ Σ ∗ t ω nF )) | { ˇ z }× Y Σ ∗ t ω nF | { ˇ z }× Y , (3.12)where now ˇ Z ι denotes an orthonormal (1 ,
0) frame field with respect to ˇ ω can on C m (again triviallyextended to the total space). We can therefore writeˇ L t ( ˇ G ) = (1 + e − t tr ˇ ω can ˇ ω F, bb )∆ Σ ∗ t ω F | { ˇ z }× Y G + ∆ ˇ ω can G + e − t D bf G (cid:126) ˇ ω F, bf . (3.13) ans-Joachim Hein and Valentino Tosatti 27 Now we inductively define two sequences of functions ˇ u , ˇ u , . . . and ˇ v , ˇ v , . . . , where the desired G t,k ( z , y ) will be given by ˇ u k (ˇ z , ˇ y ). Given A, G as above, let ˇ A, ˇ G denote their pullbacks via Σ t , andput ˇ u := ˇ A (∆ Σ ∗ t ω F | { ˇ z }× Y ) − ˇ G, (3.14)ˇ v i := ˇ L t (ˇ u i ) − ˇ A ˇ G, (3.15)ˇ u i +1 := ˇ u i − (∆ Σ ∗ t ω F | { ˇ z }× Y ) − (ˇ v i − ˇ v i ) , (3.16)where (∆ Σ ∗ t ω F | { ˇ z }× Y ) − applied to a function with fiberwise average zero denotes the unique solutionwith fiberwise average zero of the fiberwise Poisson equation. And finally, G t,k ( z, y ) := ˇ u k (ˇ z, ˇ y ) . (3.17)We immediately observe that G t,k ( A, G ) is bilinear in its two arguments, and it has fiberwise averagezero.
Remark 3.5.
As an example, when the complex structure on B × Y is a product, ω can is just aEuclidean metric ω C m , ω F is the pullback of a fixed Ricci-flat K¨ahler metric ω Y on Y , and G is alsopulled back from Y , then we can compute that G t,k ( A, G ) = k (cid:88) (cid:96) =0 ( − (cid:96) e − (cid:96)t (∆ C m ) (cid:96) A · (∆ Y ) − (cid:96) − G, (3.18)where (∆ Y ) − G denotes the unique solution of the fiberwise Poisson equation (with zero average), seeLemma 3.7 below and Remark 3.8 for a sketch of proof. Applying the Laplacian of the product metric ω C m + e − t ω Y to this then clearly gives∆ ω C m + e − t ω Y G t,k ( A, G ) = AG + ( − k e − kt (∆ C m ) k +1 A · (∆ Y ) − k − G, (3.19)and the last term vanishes if A is a polynomial of degree < k + 2. This formula was our originalmotivation for the general construction of G t,k . Lemma 3.6.
For all functions
G, H on B × Y with fiberwise average zero, there exist functions Φ ι,k ( G, H ) independent of t on the base such that for all functions A on the base and all t , P t,H ( i∂∂ ( G t,k ( A, G ))) = A (cid:90) { z }× Y GH ω nF + e − (2 k +2) t k +2 (cid:88) ι =0 Φ ι,k ( G, H ) (cid:126) D ι A. (3.20) Proof.
Fix any point z ∈ B and aim to verify (3.20) at z . Pulling everything back by Σ t , we computeˇ P t, ˇ H ( i∂∂ ˇ G t,k )(ˇ z ) = (cid:90) { ˇ z }× Y ˇ H ˇ L t ( ˇ G t,k ) Σ ∗ t ω nF (3.21)= ˇ A (ˇ z ) (cid:90) { ˇ z }× Y ˇ G ˇ H Σ ∗ t ω nF + (cid:90) { ˇ z }× Y ˇ H ˇ v k Σ ∗ t ω nF , (3.22)where both lines hold simply by definition from (3.11) and (3.15). So we need to determine the structureof the second term in (3.22). For this, recall that by (3.14)–(3.16) and (3.13), along { ˇ z } × Y ,ˇ v = [∆ ˇ ω can + e − t (tr ˇ ω can ˇ ω F, bb )∆ Σ ∗ t ω F | { ˇ z }× Y ]( ˇ A · (∆ Σ ∗ t ω F | { ˇ z }× Y ) − ˇ G )+ e − t D bf ( ˇ A · (∆ Σ ∗ t ω F | { ˇ z }× Y ) − ˇ G ) (cid:126) ˇ ω F, bf , (3.23) and for all i (cid:62)
0, again along { ˇ z } × Y ,ˇ v i +1 = ˇ L t (ˇ u i +1 ) − ˇ A ˇ G (3.24)= ˇ v i − ˇ L t [(∆ Σ ∗ t ω F | { ˇ z }× Y ) − (ˇ v i − ˇ v i )] (3.25)= ˇ v i − [∆ ˇ ω can + e − t (tr ˇ ω can ˇ ω F, bb )∆ Σ ∗ t ω F | { ˇ z }× Y ](∆ Σ ∗ t ω F | { ˇ z }× Y ) − (ˇ v i − ˇ v i ) (3.26)+ e − t D bf [(∆ Σ ∗ t ω F | { ˇ z }× Y ) − (ˇ v i − ˇ v i )] (cid:126) ˇ ω F, bf . (3.27)It is now straightforward to prove by induction on i thatˇ v i = i +2 (cid:88) ι =0 e ( ι − (2 i +2)) t ˇΦ ι,i ( G ) (cid:126) D ι ˇ A + i (cid:88) (cid:96) =1 2 (cid:96) (cid:88) ι =0 e ( ι − (cid:96) ) t ˇΨ ι,(cid:96) ( G ) (cid:126) D ι ˇ A, (3.28)where the Φ’s live on the total space but the Ψ’s are from the base. Indeed, the base case of theinduction i = 0 follows immediately from (3.23) remembering that ˇ A is pulled back from the base, andthat the base coordinates have been stretched by a factor of e − t . As for the induction step, we assume(3.28) which gives ˇ v i − ˇ v i = i +2 (cid:88) ι =0 e ( ι − (2 i +2)) t ˇΦ ι,i ( G ) (cid:126) D ι ˇ A, (3.29)ˇ v i = i +1 (cid:88) (cid:96) =1 2 (cid:96) (cid:88) ι =0 e ( ι − (cid:96) ) t ˇΨ ι,(cid:96) ( G ) (cid:126) D ι ˇ A (3.30)where the functions ˇΦ ι,i , ˇΨ ι,(cid:96) have changed. Using (3.29) it is straightforward to check that − [∆ ˇ ω can + e − t (tr ˇ ω can ˇ ω F, bb )∆ Σ ∗ t ω F | { ˇ z }× Y ](∆ Σ ∗ t ω F | { ˇ z }× Y ) − (ˇ v i − ˇ v i )+ e − t D bf [(∆ Σ ∗ t ω F | { ˇ z }× Y ) − (ˇ v i − ˇ v i )] (cid:126) ˇ ω F, bf = i +4 (cid:88) ι =0 e ( ι − (2 i +4)) t ˇΦ ι,i ( G ) (cid:126) D ι ˇ A, (3.31)and (3.28) now follows from (3.24)–(3.27), (3.30) and (3.31).Setting i = k in (3.28) and integrating it against ˇ H over the fiber, the terms with Ψ from the secondsum disappear. To convert the result back to the undecorated picture (for i = k ), recall the usual D ι ˇ A = e − ι t D ι A . (cid:3) As a corollary of the above computations, we also obtain the following structure of G t,k itself. Lemma 3.7.
For all smooth functions G on the total space with fiberwise average zero, there existsmooth functions Φ ι,i ( G ) on the total space, independent of t , such that for all functions A on the baseand all t , G t,k ( A, G ) = k (cid:88) ι =0 k (cid:88) i = (cid:100) ι (cid:101) e − it (Φ ι,i ( G ) (cid:126) D ι A ) , (3.32) and furthermore Φ , ( G ) = (∆ ω F | {·}× Y ) − G. (3.33) Proof.
In the check picture, i.e., in the setting of the previous proof, we just need to insert the formula(3.28) for ˇ v i into the formula (3.16) for ˇ u i +1 in terms of ˇ u i and ˇ v i . Once again, the base terms with Ψin (3.28) drop out, and we are left withˇ u i +1 = ˇ u i + i +2 (cid:88) ι =0 e ( ι − (2 i +2)) t ( ˇΦ ι,i ( G ) (cid:126) D ι ˇ A ) . (3.34) ans-Joachim Hein and Valentino Tosatti 29 Hence, iterating, using (3.14), and switching the order of summation,ˇ G t,k = ˇ u k = k (cid:88) ι =0 k (cid:88) i = (cid:100) ι (cid:101) e ( ι − i ) t ( ˇΦ ι,i ( G ) (cid:126) D ι ˇ A ) . (3.35)To convert (3.35) to the undecorated picture, recall that D ι ˇ A = e − ι t D ι A .To prove (3.33), we are interested in the terms in the sum in (3.32) with i = 0, and observe thatnone of these can come from the last term in (3.34) (with index i there ranging from 0 to k −
1) sincein the undecorated picture these terms have a coefficient at least as small as e − t . Thus, the term with i = 0 in (3.32) come purely from (3.14), and (3.33) follows. (cid:3) Remark 3.8.
In the product situation described in Remark 3.5, (3.23) and (3.26) imply by inductionthat ˇ v i = ˇ v i − ˇ v i = ( − i (∆ C m ) i +1 ˇ A · (∆ Y ) − i − ˇ G. (3.36)In the proof of Lemma 3.7 we then see that in the outer sum over ι we only have the even numbers ι = 2 (cid:96) (0 (cid:54) (cid:96) (cid:54) k ) and in the inner sum over i we only see the leading term i = (cid:100) ι (cid:101) = (cid:96) , and (3.18)follows.3.3. A simple interpolation for polynomials.
The following simple lemma will be used extensivelyin this section.
Lemma 3.9.
For all d, k ∈ N (cid:62) there exists a constant C = C ( d, k ) such that for all (cid:54) ι < κ (cid:54) k ,all R > , and all polynomials p of degree at most k on R d it holds that sup B R (0) | D κ p | (cid:54) CR ι − κ sup B R (0) | D ι p | . (3.37) Proof.
It is enough to prove this for R = 1, ι = 0 and κ = 1, with D κ p = D p replaced by D e i p on theleft-hand side for any one of the standard basis vectors e , . . . , e d on R d . Indeed, the case of a general R > (cid:54) ι < κ (cid:54) k follows by considering the componentfunctions of D ι p , which are again polynomials of degree at most k , and taking their iterated derivativesin directions e , . . . , e d . To prove the simplified statement, write p ( z ) = (cid:80) | α | (cid:54) k p α z α . Then1 k sup B (0) | D e i p | = sup | z | (cid:54) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) | α | (cid:54) k α i k p α z α − e i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:88) | α | (cid:54) k | p α | =: (cid:107) p (cid:107) . (3.38)Note that (cid:107) p (cid:107) is simply the (cid:96) norm of p with respect to the standard basis of the vector space P ofpolynomials of degree at most k . Now observe that p (cid:55)→ sup B (0) | p | is also a norm on P because a realpolynomial that vanishes on an open set is the zero polynomial. So (cid:107) p (cid:107) (cid:54) C sup B (0) | p | as desired. (cid:3) Statement of the selection theorem.
In this section we have two natural numbers 0 (cid:54) j (cid:54) k ,fixed throughout, and a ball B ⊂ C m whose center is our point of interest where the later blowupargument will be centered at. We also denote by Υ = ω nF the relative Calabi-Yau volume form, and letΥ z = Υ | { z }× Y .Given t, k and two smooth functions A ∈ C ∞ ( B, R ) and G ∈ C ∞ ( B × Y, R ) with (cid:82) { z }× Y G Υ z = 0and (cid:82) { z }× Y G Υ z = 1 for all z ∈ B , we constructed in Section 3.2 a function G t,k ( A, G ) ∈ C ∞ ( B × Y, R )with (cid:82) { z }× Y G t,k ( A, G )Υ z = 0 for all z ∈ B , which is in some sense an approximate right inverse of ∆ ω (cid:92)t applied to AG (cf. Lemma 3.6).Now, for i < j we suppose that we have smooth functions G i,p ∈ C ∞ ( B × Y, R ) , (cid:54) p (cid:54) N i , whichhave fiberwise average zero and are fiberwise L orthonormal, with G i,p = 0 when i = 0 ,
1. The maingoal is to find smooth functions G j,p which satisfy a certain property that takes some work to describe. First, we let δ t be any fixed sequence of scalars which goes to zero but δ t e t → ∞ .Consider now the diffeomorphisms Σ t : B e t × Y → B × Y in (3.9), and for any (1 , α on B × Y we will denote by ˇ α = e t Σ ∗ t α .We need a few more pieces of data. Let η ‡ t be an arbitrary smooth family of (1 , B whichsatisfy that ˇ η ‡ t → A (cid:93)t,i,p,k be arbitrary polynomials of degree j on B such thatˇ A (cid:93)t,i,p,k = e t Σ ∗ t A (cid:93)t,i,p,k satisfy the bounds (cid:107) D ι ˇ A (cid:93)t,i,p,k (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Cδ ιt e − α t (3.39)for some 0 < α < (cid:54) ι (cid:54) j . With these, we define for 2 (cid:54) i (cid:54) jγ t,i,k = N i (cid:88) p =1 i∂∂ G t,k ( A (cid:93)t,i,p,k , G i,p ) , (3.40)so γ t,j,k depends on how we choose the functions G j,p . It is proved by the argument in (4.153) belowthat our assumption (3.39) on ˇ A (cid:93)t,i,p,k implies that (cid:107) D ι ˇ γ t,i,k (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Ce − α t , (3.41)for all ι (cid:62) (cid:54) i (cid:54) j (this can be just taken as an assumption for now). We also define η † t = j (cid:88) i =2 γ t,i,k , (3.42)and ω (cid:93)t = ω can + e − t ω F + η † t + η ‡ t , (3.43)which up to shrinking B slightly is a K¨ahler metric on B × Y for all t sufficiently large (using here that η ‡ t is pulled back from B and goes to zero, and the estimate (3.41) with ι = 0 for η † t ).There is one more piece of background data, namely positive constants c t which are a polynomial in e − t of degree at most m with constant coefficient (cid:0) m + nn (cid:1) .The key quantity we are interested in is then δ − j − αt (cid:32) c t ˇ ω m can ∧ (Σ ∗ t ω F ) n (ˇ ω (cid:93)t ) m + n − (cid:33) . (3.44)The following selection theorem allows us to find the desired obstruction functions: Theorem 3.10.
Suppose we are given (cid:54) j (cid:54) k and when j > we are also given smooth functions G i,p , (cid:54) i (cid:54) j − , (cid:54) p (cid:54) N i , on B × Y which are fiberwise L orthonormal, and have fiberwiseaverage zero. Then there is a concentric ball B (cid:48) ⊂ B and smooth functions G j,p , (cid:54) p (cid:54) N j , on B (cid:48) × Y (identically zero if j = 0 , ), with fiberwise average zero and so that the G i,p , (cid:54) i (cid:54) j are allfiberwise L orthonormal, with the following property: if δ t is any sequence of scalars with δ t → and δ t e t → ∞ , and if A (cid:93)t,i,p,k , η † t , η ‡ t , ω (cid:93)t are as above, and if (3.44) converges locally uniformly on C m × Y to some limiting function F , then on Σ − t ( B (cid:48) × Y ) we can write (3.44) as δ − j − αt Σ ∗ t f t, + j (cid:88) i =2 N i (cid:88) p =1 f t,i,p G i,p + o (1) , (3.45) where f t, , f t,i,p are functions pulled back from B (cid:48) which converge smoothly to zero, and o (1) is a termthat converges locally smoothly to zero. Lastly, (3.44) converges to F locally smoothly. ans-Joachim Hein and Valentino Tosatti 31 In particular, if we assume that the limit F is of the form K ( z ) + N (cid:88) q =1 K q ( z ) H q ( y ) , (3.46)where K , K q are polynomials of degree at most j and H q are smooth functions on Y with fiberwiseaverage zero, then from (3.45) it follows that the functions H q lie in the linear span of the restrictions G i,p | { }× Y , (cid:54) i (cid:54) j .We will refer to the functions G i,p as the obstruction functions. By definition they only arise for i (cid:62) Remark 3.11.
It will be clear from the proof of Theorem 3.10 that if instead of assuming that (3.44)converges locally uniformly we just assume that it is locally uniformly bounded, then the conclusion(3.45) still holds, and the quantity in (3.44) is locally uniformly bounded in all C k norms.3.5. Proof of the Selection Theorem 3.10.
The cases j = 0 , . In this case what we need to prove is simply that if (3.44) converges locallyuniformly to a limit then (3.45) holds with all G i,p = 0, and (3.44) converges locally smoothly. Bydefinition, when j < η † t = 0. Since δ t e t → ∞ and j <
2, it follows that δ − j − αt e − t → . In particular, up to an error which is O ( δ − j − αt e − t ) = o (1), we can replace c t in (3.44) by (cid:0) m + nn (cid:1) (itsleading term). Next, we can write (with obvious notation)Σ ∗ t ω F = (Σ ∗ t ω F ) bb + (Σ ∗ t ω F ) bf + (Σ ∗ t ω F ) ff = e − t ˇ ω F, bb + e − t ˇ ω F, bf + ˇ ω F, ff , (3.47)and observe that(ˇ ω (cid:93)t ) m + n = (ˇ ω can + Σ ∗ t ω F + ˇ η ‡ t ) m + n = (ˇ ω can + e − t ˇ ω F, bb + ˇ ω F, ff + ˇ η ‡ t ) m + n + O ( e − t )= (cid:18) m + nn (cid:19) (ˇ ω can + e − t ˇ ω F, bb + ˇ η ‡ t ) m (Σ ∗ t ω F ) n ff + O ( e − t ) , (3.48)and so (cid:0) m + nn (cid:1) ˇ ω m can ∧ (Σ ∗ t ω F ) n − (ˇ ω (cid:93)t ) m + n (ˇ ω (cid:93)t ) m + n = ˇ ω m can ∧ (Σ ∗ t ω F ) n − (ˇ ω can + e − t ˇ ω F, bb + ˇ η ‡ t ) m (Σ ∗ t ω F ) n ff (ˇ ω can + e − t ˇ ω F, bb + ˇ η ‡ t ) m (Σ ∗ t ω F ) n ff + O ( e − t )= ˇ ω m can − (ˇ ω can + e − t ˇ ω F, bb + ˇ η ‡ t ) m (ˇ ω can + e − t ˇ ω F, bb + ˇ η ‡ t ) m + O ( e − t )= Σ ∗ t f t + O ( e − t ) , (3.49)where the O ( e − t ) is in the locally smooth topology and f t is some family of smooth functions pulled backfrom B that converge locally smoothly to zero. Multiplying this by δ − j − αt and using that δ − j − αt e − t = o (1) we see that indeed (3.45) holds with G i,p = 0, and of course the limit of (3.44) is purely from thebase, i.e. in (3.46) we only have the term K ( z ). Lastly, since by assumption (3.44) converges locallyuniformly to F , the same is true for δ − j − αt Σ ∗ t f t , and we need to prove that this convergence is locallysmooth.For this, we work in C m and expand in Taylor series f t ( z ) = (cid:80) I a I,t z I , where the coefficients a I,t go tozero as t → ∞ , so that Σ ∗ t f t ( z ) = (cid:80) I a I,t e − | I | t z I . Split f t ( z ) = p t ( z ) + r t ( z ) where p t ( z ) = (cid:80) | I | (cid:54) j a I,t z I and r t ( z ) = (cid:80) | I | >j a I,t z I , so that δ − j − αt Σ ∗ t r t → δ − t e − t = o (1). It follows thatthe polynomials δ − j − αt Σ ∗ t p t also converges locally uniformly to F , and by Lemma 3.9 this convergenceis locally smooth as desired. The case j (cid:62) : the initial list. As in the statement of Theorem 3.10, we suppose that the smoothfunctions G i,p with 2 (cid:54) i < j have already been selected on B (cid:48) × Y (they are fiberwise orthonormal,and fiberwise orthogonal to the constants), and we now need to select the G j,p ’s. The process goesas follows: first we will select an initial list of G j,p ’s. Then we will construct a procedure that giventhese functions, it gives us some new functions to add to the list. This procedure will then need to beiterated a number of times (shrinking the ball B (cid:48) at each step without changing its notation) until thefinal list of functions is complete, and we then verify that these satisfy the conclusion of Theorem 3.10.We introduce a new index κ to remember the number of the iteration that we are in, so initially κ = 0. For each κ we suppose we are given a “list of obstruction functions at generations up to κ ” G [ q ] j,p , (cid:54) q (cid:54) κ, which are smooth functions on B (cid:48) × Y with fiberwise average zero, with this list beingthe empty list when κ = 0. Given these functions, and given also stretched polynomials ˇ A (cid:93), [ κ ] t,j,p,k ofdegree at most j satisfying the assumption (3.39), we construct as in (3.40)ˇ γ [ κ ] t,j,k = N [ κ ] j (cid:88) p =1 i∂∂ ˇ G t,k ( ˇ A (cid:93), [ κ ] t,j,p,k , ˇ G [ κ ] j,p ) , (3.50)so that ˇ γ [0] t,j,k = 0, andˇ ω (cid:93), [ κ ] t = ˇ ω can + Σ ∗ t ω F + ˇ γ t, ,k + · · · + ˇ γ t,j − ,k + ˇ γ [1] t,j,k + · · · + ˇ γ [ κ ] t,j,k + ˇ η ‡ t . (3.51)For each κ (cid:62) B [ κ ] t := δ − j − αt (cid:32) c t ˇ ω m can ∧ (Σ ∗ t ω F ) n (ˇ ω (cid:93), [ κ ] t ) m + n − (cid:33) . (3.52)For convenience, we will say that a t -dependent function on Σ − t ( B (cid:48) × Y ) satisfies condition ( (cid:63) ) if it isequal to δ − j − αt Σ ∗ t (cid:32) f t, + N (cid:88) i =1 f t,i h i (cid:33) + o (1) , (3.53)where the functions f t, , f t,i are smooth and pulled back from B (cid:48) (and smooth in t ), the functions h i are smooth on B (cid:48) × Y with fiberwise average zero (with respect to ( ω F | { z }× Y ) n ), the functions f t, , f t,i converge smoothly to zero, and the o (1) is a term that converges smoothly to zero. Remark 3.12.
Observe that if a t -dependent function satisfies ( (cid:63) ) and it converges locally uniformlyon C m × Y as t → ∞ , then this convergence is actually locally smooth. To see this, first by replaceeach h i by h i − h i we may assume that each h i has fiber average zero. We then apply Proposition 3.1to these (with parameter J = j + 1, and with the empty list of functions H k ), so up to shrinking B (cid:48) wemay assume that the h i are fiberwise L orthonormal (the errors coming from δ − j − αt Σ ∗ t ( rK ) in (3.1)go to zero locally smoothly thanks to Remark 3.4, since δ − t e − t = o (1)). Then, as before, we expandin Taylor series f t,i ( z ) = (cid:80) I a i,I,t z I in C m , 0 (cid:54) i (cid:54) N , where the coefficients a i,I,t go to zero as t → ∞ ,and we split f t,i ( z ) = p t,i ( z ) + r t,i ( z ) where p t,i ( z ) = (cid:80) | I | (cid:54) j a i,I,t z I and r t,i ( z ) = (cid:80) | I | >j a i,I,t z I , so that δ − j − αt Σ ∗ t r t,i → δ − t e − t = o (1). It follows that δ − j − αt Σ ∗ t (cid:16) r t, + (cid:80) Ni =1 r t,i h i (cid:17) → δ − j − αt Σ ∗ t (cid:16) p t, + (cid:80) Ni =1 p t,i h i (cid:17) converges locally uniformly. Taking thefiberwise L inner product against the constant 1 and against each h i shows that δ − j − αt Σ ∗ t p t,i , (cid:54) i (cid:54) N, also converges locally uniformly, and by Lemma 3.9 this convergence is locally smooth. It then followsthat δ − j − αt Σ ∗ t (cid:16) p t, + (cid:80) Ni =1 p t,i h i (cid:17) converges locally smoothly, as desired. ans-Joachim Hein and Valentino Tosatti 33 Lemma 3.13.
Suppose either κ = 0 or κ (cid:62) and we have selected the functions G [ q ] j,p as above for (cid:54) q (cid:54) κ. Then the function B [ κ ] t satisfies ( (cid:63) ) . Furthermore, we have δ j + αt B [ κ ] t = O ( e − α t ) + o (1) from base , (3.54) where the last term is a function from the base which goes to zero smoothly.Proof. For clarity we will split the proof into two steps. For ease of notation, let us write ω (cid:3) t = ω can + e − t ω F + η ‡ t , (3.55)and C t = (cid:18) c t ω m can ∧ ( e − t ω F ) n ( ω (cid:3) t ) m + n − (cid:19) . (3.56)The first step is to show that δ − j − αt Σ ∗ t C t satisfies ( (cid:63) ). Observe that( ω (cid:3) t ) m + n = ( ω can + e − t ω F + η ‡ t ) m + n = (cid:18) m + nn (cid:19) e − nt ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff + m (cid:88) q =1 e − ( n + q ) t (cid:18) m + nn + q (cid:19) ( ω can + e − t ω F, bb + η ‡ t ) m − q ∧ ( ω F, ff + ω F, bf ) n + q =: (cid:18) m + nn (cid:19) e − nt (cid:104) ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff + e − t D t (cid:105) , (3.57)and recalling that c t = (cid:0) m + nn (cid:1) (1 + e − t ˜ c t ) with ˜ c t smoothly convergent, we can write C t as c t e − nt ω m can ∧ ω nF − ( ω (cid:3) t ) m + n ( ω (cid:3) t ) m + n = (1 + e − t ˜ c t ) ω m can ∧ ω nF, ff − ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff − e − t D t ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff + e − t D t = (1 + e − t ˜ c t ) ω m can − ( ω can + e − t ω F, bb + η ‡ t ) m ( ω can + e − t ω F, bb + η ‡ t ) m (cid:32) e − t D t ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff (cid:33) − − e − t D t ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff (cid:32) e − t D t ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff (cid:33) − = e − t ˜ c t + m (cid:88) q =0 m − (cid:88) p =0 m ! p ! q !( m − p − q )! ω p can e − qt ω qF, bb ( η ‡ t ) m − p − q ω m can m (cid:88) q =0 m − (cid:88) p =0 m ! p ! q !( m − p − q )! ω p can e − qt ω qF, bb ( η ‡ t ) m − p − q ω m can − (cid:32) e − t D t ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff (cid:33) − − e − t D t ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff (cid:32) e − t D t ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff (cid:33) − , (3.58)and from this we can see that δ − j − αt Σ ∗ t C t satisfies ( (cid:63) ) as follows. First note that the functions ω p can e − qt ω qF, bb ( η ‡ t ) m − p − q ω m can , (3.59)with q = 0 are pulled back from B (cid:48) , while when q > f t, + (cid:80) Ni =1 f t,i h i with the same notation as above, where the f t, , f t,i converge smoothly to zero at least as O ( e − qt ). An analogous statement holds for e − t D t ( ω can + e − t ω F, bb + η ‡ t ) m ∧ ω nF, ff , (3.60)and so expanding (1 + x ) − using the geometric sum, we thus see that δ − j − αt Σ ∗ t C t satisfies ( (cid:63) ), and that C t = O ( e − t ) + o (1) from base . (3.61)The second step is then to write δ j + αt B [ κ ] t = Σ ∗ t C t − (Σ ∗ t C t + 1) (cid:32) (ˇ ω (cid:93), [ κ ] t ) m + n − (ˇ ω (cid:3) t ) m + n (ˇ ω (cid:93), [ κ ] t ) m + n (cid:33) = Σ ∗ t C t − (Σ ∗ t C t + 1) m + n (cid:88) i =1 (cid:18) m + ni (cid:19) (ˇ γ t, ,k + · · · + ˇ γ t,j − ,k + ˇ γ [1] t,j,k + · · · + ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω (cid:3) t ) m + n − i (ˇ ω (cid:3) t ) m + n m + n (cid:88) i =1 (cid:18) m + ni (cid:19) (ˇ γ t, ,k + · · · + ˇ γ t,j − ,k + ˇ γ [1] t,j,k + · · · + ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω (cid:3) t ) m + n − i (ˇ ω (cid:3) t ) m + n − , (3.62)and for all the terms with ˇ γ t, ,k + · · · + ˇ γ t,j − ,k + ˇ γ [1] t,j,k + · · · + ˇ γ [ κ ] t,j,k we employ Lemma 3.7 which allowsus to write schematically ˇ G t,k ( ˇ A, ˇ G ) = j (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t ( ˇΦ ι,(cid:96) ( ˇ G ) (cid:126) D ι ˇ A ) , (3.63)recalling from (3.40) that we haveˇ γ t,i,k = N i (cid:88) p =1 i∂∂ ˇ G t,k ( ˇ A (cid:93)t,i,p,k , ˇ G i,p ) , ˇ γ [ q ] t,j,k = N [ q ] j (cid:88) p =1 i∂∂ ˇ G t,k ( ˇ A (cid:93), [ q ] t,j,p,k , ˇ G [ q ] j,p ) , (3.64)with the bounds (3.41). Plugging this into (3.62) and arguing as we did above for δ − j − αt Σ ∗ t C t revealsthat B [ κ ] t satisfies ( (cid:63) ) and that (3.54) holds. (cid:3) For our initial list G [1] j,p we would then naively like to take the h i ’s in (3.53) for B [0] t . More precisely,we apply Proposition 3.1 with the functions F i there equal to the h i − h i ’s, and with the H k thereequal to the G i,p , (cid:54) i < j and with parameter J = j + 1. Up to shrinking the ball B (cid:48) we thusobtain our desired list G [1] j,p so that these together with the G i,p , (cid:54) i < j are fiberwise orthonormaland orthogonal to the constants, and the h i ’s lie in the fiberwise linear span of the G i,p , (cid:54) i < j together with the G [1] j,p and the constants. Indeed, the errors coming from δ − j − αt Σ ∗ t ( rK ) in (3.1) go tozero locally smoothly thanks to Remark 3.4, since δ − t e − t = o (1), and so they can be moved into the o (1) term in the expansion (3.53) for B [0] t .3.5.3. The iterative procedure.
Here for a given κ (cid:62) G i,p for 2 (cid:54) i < j and we have have constructed the list G [ q ] j,p for 1 (cid:54) q (cid:54) κ , and hence we have the function B [ κ ] t in (3.52). ans-Joachim Hein and Valentino Tosatti 35 Then we have δ j + αt ( B [ κ ] t − B [ κ − t ) = − ( δ j + αt B [ κ − t + 1) (cid:32) (ˇ ω (cid:93), [ κ ] t ) m + n − (ˇ ω (cid:93), [ κ − t ) m + n (ˇ ω (cid:93), [ κ ] t ) m + n (cid:33) = − ( δ j + αt B [ κ − t + 1) m + n (cid:88) i =1 (cid:18) m + ni (cid:19) (ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω (cid:93), [ κ − t ) m + n − i (ˇ ω (cid:93), [ κ − t ) m + n m + n (cid:88) i =1 (cid:18) m + ni (cid:19) (ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω (cid:93), [ κ − t ) m + n − i (ˇ ω (cid:93), [ κ − t ) m + n − . (3.65)Let us thus first look in details at the term m + n (cid:88) i =1 (cid:18) m + ni (cid:19) (ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω (cid:93), [ κ − t ) m + n − i (ˇ ω (cid:93), [ κ − t ) m + n . (3.66)Thanks to (3.41) we can writeˇ ω (cid:93), [ κ − t = ˇ ω can + (Σ ∗ t ω F ) ff + O ( e − α t ) + o (1) from base , (3.67)(here O ( · ) , o (1) are in the smooth topology on ˇ B Cδ − t ) and so(ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω (cid:93), [ κ − t ) m + n − i (ˇ ω (cid:93), [ κ − t ) m + n = (ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω can + (Σ ∗ t ω F ) ff ) m + n − i (cid:0) m + nn (cid:1) ˇ ω m can ∧ (Σ ∗ t ω F ) n ff (1 + O ( e − α t ) + o (1) from base ) . (3.68)We thus study the terms (ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω can + (Σ ∗ t ω F ) ff ) m + n − i ˇ ω m can ∧ (Σ ∗ t ω F ) n ff , (3.69)with 1 (cid:54) i (cid:54) m + n . To do this, we use again Lemma 3.7 and write schematically (for 1 (cid:54) q (cid:54) κ )ˇ γ [ q ] t,j,k = N [ q ] j (cid:88) p =1 j (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t i∂∂ ( ˇΦ ι,(cid:96) ( ˇ G [ q ] j,p ) (cid:126) D ι ˇ A (cid:93), [ q ] t,j,p,k ) , (3.70)with the key relation ˇΦ , ( ˇ G ) = (∆ ω F | {·}× Y ) − ˇ G, (3.71)proved in (3.33). Decompose (3.70) schematically into the sum of 6 piecesˇ γ [ q ] t,j,k = N [ q ] j (cid:88) p =1 j (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t (cid:26) ( i∂∂ ˇΦ ι,(cid:96) ( ˇ G [ q ] j,p )) ff (cid:126) D ι ˇ A (cid:93), [ q ] t,j,p,k + ( i∂∂ ˇΦ ι,(cid:96) ( ˇ G [ q ] j,p )) bf (cid:126) D ι ˇ A (cid:93), [ q ] t,j,p,k + ( i∂∂ ˇΦ ι,(cid:96) ( ˇ G [ q ] j,p )) bb (cid:126) D ι ˇ A (cid:93), [ q ] t,j,p,k + ( i∂ ˇΦ ι,(cid:96) ( ˇ G [ q ] j,p )) b (cid:126) ∂ D ι ˇ A (cid:93), [ q ] t,j,p,k + ( i∂ ˇΦ ι,(cid:96) ( ˇ G [ q ] j,p )) f (cid:126) ∂ D ι ˇ A (cid:93), [ q ] t,j,p,k + ˇΦ ι,(cid:96) ( ˇ G [ q ] j,p ) (cid:126) i∂∂ D ι ˇ A (cid:93), [ q ] t,j,p,k (cid:27) =: j (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) I [ q ] ι,(cid:96) + · · · + VI [ q ] ι,(cid:96) . (3.72) Then we claim that the term in (3.66) is equal to m + n (cid:88) i =1 (cid:18) m + ni (cid:19) (ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω (cid:93), [ κ − t ) m + n − i (ˇ ω (cid:93), [ κ − t ) m + n = ( m + n ) (cid:0) m + nn (cid:1) I [ κ ]0 , ∧ (ˇ ω can + (Σ ∗ t ω F ) ff ) m + n − ˇ ω m can ∧ (Σ ∗ t ω F ) n ff (1 + O ( e − α t ) + o (1) from base ) + F [ κ ] t = N [ κ ] j (cid:88) p =1 ˇ G [ κ ] j,p ˇ A (cid:93), [ κ ] t,j,p,k (1 + O ( e − α t ) + o (1) from base ) + F [ κ ] t , (3.73)where (cid:107) F [ κ ] t (cid:107) L ∞ ( ˇ B Cδ − t ) (cid:54) Cδ α t (cid:107) I [ κ ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) . (3.74)To prove (3.73), we first show that there is C such that for all t and all R (cid:54) Cδ − t and all 1 (cid:54) q (cid:54) κ we have C − N [ q ] j (cid:88) p =1 (cid:107) ˇ A (cid:93), [ q ] t,j,p,k (cid:107) L ∞ ( B R ) (cid:54) (cid:107) I [ q ]0 , (cid:107) L ∞ ( B R × Y, ˇ g t ) (cid:54) C N [ q ] j (cid:88) p =1 (cid:107) ˇ A (cid:93), [ q ] t,j,p,k (cid:107) L ∞ ( B R ) . (3.75)The second inequality is obvious, as for the first one observe first that for every z ∈ B R the (1 , { i∂∂ ( ˇΦ , ( ˇ G [ q ] j,p ) | { z }× Y ) } p on { z } × Y are R -linearly independent since using (3.71) and taking thefiberwise trace, a nontrivial linear dependence among these would give a nonexistent linear dependenceamong the { ˇ G [ q ] j,p | { z }× Y } p . But then (cid:107) I [ q ]0 , (cid:107) L ∞ ( { z }×× Y, ˇ g t ) = sup { z }× Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) p ˇ A (cid:93), [ q ] t,j,p,k ( z ) i∂∂ ( ˇΦ , ( ˇ G [ q ] j,p ) | { z }× Y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω F | { z }× Y and the RHS is a norm on the finite-dimensional R -vector space of (1 , { z } × Y spanned by { i∂∂ ( ˇΦ , ( ˇ G [ q ] j,p ) | { z }× Y ) } p , which is therefore uniformly equivalent to (cid:80) p | ˇ A (cid:93), [ q ] t,j,p,k ( z ) | , and (3.75) follows.Next, for ( ι, (cid:96) ) (cid:54) = (0 ,
0) we can apply Lemma 3.9 to balls of radius Cδ − t and get (cid:107) I [ q ] ι,(cid:96) (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Ce − ( (cid:96) − ι ) t (cid:88) p (cid:107) D ι ˇ A (cid:93), [ q ] t,j,p,k (cid:107) L ∞ ( B Cδ − t ) (cid:54) Ce − ( (cid:96) − ι ) t δ ιt (cid:88) p (cid:107) ˇ A (cid:93), [ q ] t,j,p,k (cid:107) L ∞ ( B Cδ − t ) (cid:54) Ce − ( (cid:96) − ι ) t δ ιt (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) , (3.76)using also (3.75).Arguing in the same way, and using also that ˇ G [ q ] j,p = Σ ∗ t G [ q ] j,p gains a factor of e − t for each basedifferentiation, we see that for all ι, (cid:96) as in (3.70) we have (cid:107) II [ q ] ι,(cid:96) (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Ce − t e − ( (cid:96) − ι ) t (cid:88) p (cid:107) D ι ˇ A (cid:93), [ q ] t,j,p,k (cid:107) L ∞ ( B Cδ − t ) (cid:54) Ce − ( (cid:96) − ι + ) t δ ιt (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) , (3.77)and similarly (cid:107) III [ q ] ι,(cid:96) (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Ce − ( (cid:96) − ι +1 ) t δ ιt (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) , (3.78) (cid:107) IV [ q ] ι,(cid:96) (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Ce − ( (cid:96) − ι + ) t δ ι +1 t (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) , (3.79) (cid:107) V [ q ] ι,(cid:96) (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Ce − ( (cid:96) − ι ) t δ ι +1 t (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) , (3.80) ans-Joachim Hein and Valentino Tosatti 37 (cid:107) VI [ q ] ι,(cid:96) (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Ce − ( (cid:96) − ι ) t δ ι +2 t (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) , (3.81)and so putting all of these together, and recalling that δ t (cid:29) e − t , we see that (cid:107) • [ q ] ι,(cid:96) (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Cδ t (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) , (3.82)whenever • (cid:54) = I or ( ι, (cid:96) ) (cid:54) = (0 , (cid:107) ˇ γ [ q ] t,j,k − I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Cδ t (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) . (3.83)Also, for all i (cid:62)
2, we can use (3.41) and obtain (cid:107) (ˇ γ [ κ ] t,j,k ) i (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Cδ α t (cid:107) ˇ γ [ κ ] t,j,k (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Cδ α t (cid:107) I [ κ ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) , (3.84)and combining (3.82) and (3.84) with (3.72) easily gives the first equality in (3.73) and (3.74). Thesecond equality in (3.73) then follows from( m + n ) (cid:0) m + nn (cid:1) I [ κ ]0 , ∧ (ˇ ω can + (Σ ∗ t ω F ) ff ) m + n − ˇ ω m can ∧ (Σ ∗ t ω F ) n ff = N [ κ ] j (cid:88) p =1 ˇ G [ κ ] j,p ˇ A (cid:93), [ κ ] t,j,p,k , (3.85)which is a consequence of (3.71).Next, we combining (3.41) with (3.73), (3.74), (3.75) gives (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m + n (cid:88) i =1 (cid:18) m + ni (cid:19) (ˇ γ [ κ ] t,j,k ) i ∧ (ˇ ω (cid:93), [ κ − t ) m + n − i (ˇ ω (cid:93), [ κ − t ) m + n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ˇ B Cδ − t , ˇ g t ) (cid:54) Cδ α t , (3.86)so inserting (3.54), (3.73), (3.74), (3.86) into (3.65) gives B [ κ ] t = B [ κ − t + δ − j − αt N [ κ ] j (cid:88) p =1 ˇ G [ κ ] j,p ˇ A (cid:93), [ κ ] t,j,p,k (1 + o (1) from base ) + δ − j − αt E [ κ ] t + o (1) , (3.87)where the error function E [ κ ] t satisfies the estimate (cid:107) E [ κ ] t (cid:107) L ∞ ( ˇ B Cδ − t ) (cid:54) Cδ α t (cid:107) I [ κ ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) . (3.88)Since by Lemma 3.13 the functions B [ κ ] t and B [ κ − t satisfy ( (cid:63) ), so does the error term δ − j − αt E [ κ ] t (whichis also clear from way it is defined). We thus consider the functions h i that arise in the expression (3.53)for δ − j − αt E [ κ ] t . We then apply Proposition 3.1 with the functions F i there equal to the h i − h i ’s, andwith the H k there equal to the ˇ G i,p , (cid:54) i < j together with the ˇ G [ q ] j,p , (cid:54) q (cid:54) κ , and with parameter J = j + 1. Up to shrinking the ball B (cid:48) again we thus obtain our desired list G [ κ +1] j,p so that thesetogether with the G i,p , (cid:54) i < j and the ˇ G [ q ] j,p , (cid:54) q (cid:54) κ are fiberwise orthonormal and orthogonalto the constants, and the functions h i in the expression (3.53) for δ − j − αt E [ κ ] t lie in the fiberwise linearspan of all these ˇ G ’s and the constants. Again, the errors coming from δ − j − αt Σ ∗ t ( rK ) in (3.1) go to zerolocally smoothly thanks to Remark 3.4, since δ − t e − t = o (1), and so they can be moved into the o (1)term in the expansion (3.53) for δ − j − αt E [ κ ] t . This is the step from κ to κ + 1 in the iterative procedureto select these functions. Iterating (3.87) gives B [ κ ] t = B [0] t + δ − j − αt κ (cid:88) q =1 N [ q ] j (cid:88) p =1 ˇ G [ q ] j,p ˇ A (cid:93), [ q ] t,j,p,k (1 + o (1) from base ) + δ − j − αt κ (cid:88) q =1 E [ q ] t + o (1) , (3.89)with (cid:107) E [ q ] t (cid:107) L ∞ ( ˇ B Cδ − t ) (cid:54) Cδ α t (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B Cδ − t , ˇ g t ) , (3.90)with (3.54) implying in particular the crude estimate (cid:107)B [0] t (cid:107) L ∞ ( ˇ B Cδ − t ) (cid:54) Cδ − j − αt , (3.91)and by construction for each 1 (cid:54) q (cid:54) κ the functions h i in the expression (3.53) for δ − j − αt E [ q ] t lie inthe fiberwise linear span of the ˇ G i,p , (cid:54) i < j and ˇ G [ r ] j,p , (cid:54) r (cid:54) q + 1 together with the constants.3.5.4. Iteration and conclusion.
We now repeat the iterative step that we just described (shrinkingalso B (cid:48) at each step) until step κ := (cid:100) j + αα (cid:101) and then we stop, so the last set of functions which areadded to the list are the G [ κ +1] j,p . Our choice of κ is made so that δ − j − αt δ ( κ +1) α t →
0. The resulting G [ q ] j,p with 1 (cid:54) q (cid:54) κ + 1 are then renamed simply G j,p . These, together with the G i,p , (cid:54) i < j , are theobstruction functions that we seek. It remains to show that the statement of Theorem 3.10 holds withthis choice of obstruction functions. By definition, the quantity in (3.44) equals B [ κ +1] t , which satisfies( (cid:63) ) thanks to Lemma 3.13. From Remark 3.12 we see that if it converges locally uniformly, then itconverges locally smoothly, which is the last claim in Theorem 3.10. We are then left with showingthat if B [ κ +1] t converges locally uniformly, then (3.45) holds.For this, we go back to (3.89) setting κ = κ + 1, and write it as B [ κ +1] t = B [0] t + κ +1 (cid:88) q =1 ( D [ q ] t + ˜ E [ q ] t ) + o (1) , (3.92)where o (1) is in the smooth topology and we have set D [ q ] t = δ − j − αt N [ q ] j (cid:88) p =1 ˇ G [ q ] j,p ˇ A (cid:93), [ q ] t,j,p,k (1 + o (1) from base ) , ˜ E [ q ] t = δ − j − αt E [ q ] t , (3.93)which thanks to (3.75), (3.85) and (3.90) satisfy (cid:107) D [ q ] t (cid:107) L ∞ ( ˇ B R ) ∼ δ − j − αt (cid:107) I [ q ]0 , (cid:107) L ∞ ( ˇ B R , ˇ g t ) , (cid:107) ˜ E [ q ] t (cid:107) L ∞ ( ˇ B R ) (cid:54) Cδ α t (cid:107) D [ q ] t (cid:107) L ∞ ( ˇ B R ) , (3.94)for R (cid:54) Cδ − t . Thanks to our definition of the G i,p , the quantity B [0] t + κ +1 (cid:88) q =1 D [ q ] t + κ (cid:88) q =1 ˜ E [ q ] t , (3.95)is already of the desired form (3.45) (and lies in the fiberwise span of the G i,p , (cid:54) i (cid:54) j ), and so wewill be done if we can show that (cid:107) ˜ E [ κ +1] t (cid:107) L ∞ ( ˇ B R ) = o (1) , (3.96)for any given R (cid:54) Cδ − t . Indeed, we have shown earlier that ˜ E [ κ +1] t satisfies ( (cid:63) ), so (3.96) togetherwith Remark 3.12 would imply that it is o (1) locally smoothly. ans-Joachim Hein and Valentino Tosatti 39 To prove (3.96), we denote by P [ q ] the fiberwise L orthogonal projection onto the span of the ˇ G [ q ] j,p ’s.Since by assumption the B [ κ +1] t converges locally uniformly, so does B [0] t + (cid:80) κ +1 q =1 ( D [ q ] t + ˜ E [ q ] t ) and sodoes P [ q ] applied to it, hence by definition (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P [1] B [0] + D [1] t + P [1] κ +1 (cid:88) q =1 ˜ E [ q ] t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C, (3.97)and for 2 (cid:54) r (cid:54) κ + 1 , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) D [ r ] t + P [ r ] κ +1 (cid:88) q = r − ˜ E [ q ] t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C, (3.98)where we used that by construction P [ r ] B [0] = 0 for r (cid:62) P [ r ] ˜ E [ q ] t = 0 for q < r − h i that arise in the expression (3.53) for ˜ E [ q ] t lie in the fiberwise linear span of theˇ G i,p , (cid:54) i < j together with the ˇ G [ s ] j,p , (cid:54) s (cid:54) q + 1). We then claim that by induction on 1 (cid:54) r (cid:54) κ + 1we have (cid:13)(cid:13)(cid:13) D [ r ] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + Cδ ( r − α t (cid:13)(cid:13)(cid:13) B [0] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) + Cδ α t κ +1 (cid:88) q = r +1 (cid:13)(cid:13)(cid:13) D [ q ] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) , (3.99)for all t sufficiently large. Once this is proved then (3.96) follows by taking r = κ + 1 (cid:13)(cid:13)(cid:13) D [ κ +1] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + Cδ κα t (cid:13)(cid:13)(cid:13) B [0] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) , (3.100)and using (3.94) and (3.91) we obtain (cid:13)(cid:13)(cid:13) ˜ E [ κ +1] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) Cδ α t + Cδ ( κ +1) α t (cid:13)(cid:13)(cid:13) B [0] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) Cδ α t + Cδ ( κ +1) α t δ − j − αt = o (1) , (3.101)as desired.To prove (3.99) we first show the case r = 1. For this, we use (3.97) and (3.91) to bound (cid:13)(cid:13)(cid:13) D [1] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + (cid:107) P [1] B [0] (cid:107) L ∞ ( ˇ B R ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P [1] κ +1 (cid:88) q =1 ˜ E [ q ] t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + C (cid:107)B [0] (cid:107) L ∞ ( ˇ B R ) + C κ +1 (cid:88) q =1 (cid:13)(cid:13)(cid:13) ˜ E [ q ] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + C (cid:107)B [0] (cid:107) L ∞ ( ˇ B R ) + Cδ α t κ +1 (cid:88) q =1 (cid:13)(cid:13)(cid:13) D [ q ] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) , (3.102)and we may assume that Cδ α t (cid:54) so the term with q = 1 in the sum on the RHS can be absorbed bythe LHS, thus proving (3.99) with r = 1. As for the induction step, for r (cid:62) and the induction hypothesis (3.99) to bound (cid:13)(cid:13)(cid:13) D [ r +1] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P [ r +1] κ +1 (cid:88) q = r ˜ E [ q ] t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + C κ +1 (cid:88) q = r (cid:13)(cid:13)(cid:13) ˜ E [ q ] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + Cδ α t κ +1 (cid:88) q = r (cid:13)(cid:13)(cid:13) D [ q ] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + Cδ α t (cid:13)(cid:13)(cid:13) D [ r ] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) + Cδ α t (cid:13)(cid:13)(cid:13) D [ r +1] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) + Cδ α t κ +1 (cid:88) q = r +2 (cid:13)(cid:13)(cid:13) D [ q ] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) (cid:54) C + Cδ rα t (cid:13)(cid:13)(cid:13) B [0] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) + Cδ α t (cid:13)(cid:13)(cid:13) D [ r +1] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) + Cδ α t κ +1 (cid:88) q = r +2 (cid:13)(cid:13)(cid:13) D [ q ] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) , (3.103)and again we may assume that Cδ α t (cid:54) and absorb the term Cδ α t (cid:13)(cid:13)(cid:13) D [ r +1] t (cid:13)(cid:13)(cid:13) L ∞ ( ˇ B R ) on the RHS bythe LHS, thus proving (3.99). This completes the proof of (3.96) and hence also of Theorem 3.10.4. The asymptotic expansion theorem
This section contains the proof of the asymptotic expansion Theorem 4.1, and is the main part ofthe paper.4.1.
Statement of the asymptotic expansion.
Let us first explain the precise assumptions for thetheorem. We work on B × Y , where B ⊂ C m is the unit ball and Y is a closed real 2 n -fold. Theproduct B × Y is equipped with a complex structure J such that pr B is ( J, J C m )-holomorphic, andwith a K¨ahler metric ω X , and such that the fibers { z } × Y are Calabi-Yau n -folds. When later inSection 5 we will use the expansion to prove Theorems A and B, we will have a compact Calabi-Yaumanifold X with fiber space structure f as in Section 1.1, and B is a ball in the base whose closuredoes not meet the critical locus f ( S ), over which f is smoothly trivial and so f − ( B ) is diffeomorphicto B × Y and the complex structure of X defines our complex structure J .As in (1.5) we define a semi-Ricci-flat form ω F on B × Y by ω F = ω X + i∂∂ρ where ρ z = ρ ( z, · ) issuch that ω X | { z }× Y + i∂∂ρ z is Ricci-flat K¨ahler on { z } × Y and (cid:82) { z }× Y ρ z ω nX = 0. We suppose that wehave a K¨ahler metric ω can on B , and define ω (cid:92)t = ω can + e − t ω F , which we assume is a K¨ahler metricfor all t (cid:62)
0. We also assume we have Ricci-flat K¨ahler metrics ω • t , t (cid:62) , on B × Y , of the form ω • t = ω (cid:92)t + i∂∂ψ t which solve the complex Monge-Amp`ere equation( ω • t ) m + n = ( ω (cid:92)t + i∂∂ψ t ) m + n = c t e − nt ω m can ∧ ω nF , (4.1)where c t is a polynomial in e − t of degree at most m with constant coefficient (cid:0) m + nn (cid:1) . We also cruciallyassume that on B × Y we have C − ω (cid:92)t (cid:54) ω • t (cid:54) Cω (cid:92)t , (4.2)and that ψ t → , (4.3)weakly as distributions as t → ∞ . ans-Joachim Hein and Valentino Tosatti 41 We are given 0 (cid:54) j (cid:54) k , and given any point z ∈ B , fix a ball B r ( z ) ⊂ B and apply Theorem 3.10to it. This produces for us a much smaller ball B (cid:48) = B r (cid:48) ( z ) ⊂ B and a list G i,p , (cid:54) i (cid:54) j, (cid:54) p (cid:54) N i of smooth functions on B (cid:48) × Y . For each of these, we define P t,i,p = P t,G i,p , (4.4)as in (3.5). Recall also that we defined an approximate Green’s operator G t,k in § F on B × Y we denote by F ∈ C ∞ ( B ) its fiberwise average with respect to ω F | { z }× Y .Throughout the whole proof we will also fix a family of shrinking product Riemannian metrics g t on B × Y , for example g t = g C m + e − t g Y,z for some fixed z ∈ B . These will only be used to measurenorms and distances, so the particular choice of g t will not matter. Lastly, the H¨older seminorms thatwe use are those defined in § Theorem 4.1.
For all (cid:54) j (cid:54) k , given any z ∈ B we can find a ball B (cid:48) ⊂ B centered at z andfunctions G i,p as above, such that on B (cid:48) × Y we have a decomposition ω • t = ω (cid:92)t + γ t, + γ t, ,k + · · · + γ t,j,k + η t,j,k , (4.5) with the following properties. First of all, for every < α < and every smaller ball B (cid:48)(cid:48) (cid:98) B (cid:48) there is C > such that on B (cid:48)(cid:48) × Y we have for all t (cid:62) : (cid:107) D ι η t,j,k (cid:107) L ∞ ( B (cid:48)(cid:48) × Y,g t ) (cid:54) Ce ι − j − α t for all (cid:54) ι (cid:54) j, (4.6)[ D j η t,j,k ] C α ( B (cid:48)(cid:48) × Y,g t ) (cid:54) C. (4.7) Furthermore, we have γ t, = i∂∂ψ t , γ t, ,k = 0 , γ t,i,k = N i (cid:88) p =1 i∂∂ G t,k ( A t,i,p,k , G i,p ) (2 (cid:54) i (cid:54) j ) , (4.8) where the A t,i,p,k = P t,i,p ( η t,i − ,k ) (2 (cid:54) i (cid:54) j ) , (4.9) are functions from the base, and we have the estimates (cid:107) D ι γ t, (cid:107) L ∞ ( B (cid:48)(cid:48) ,g C m ) = o (1) (0 (cid:54) ι (cid:54) j ) , (4.10)[ D j γ t, ] C α ( B (cid:48)(cid:48) ,g C m ) (cid:54) C, (4.11) (cid:107) D ι A t,i,p,k (cid:107) L ∞ ( B (cid:48)(cid:48) ,g C m ) (cid:54) Ce − ( i +1+ α )(1 − ιj +2+ α ) t (0 (cid:54) ι (cid:54) j + 2 , (cid:54) i (cid:54) j, (cid:54) p (cid:54) N i ) , (4.12) (cid:107) D j +2+ ι A t,i,p,k (cid:107) L ∞ ( B (cid:48)(cid:48) ,g C m ) (cid:54) Ce ι t , (0 (cid:54) ι (cid:54) k, (cid:54) i (cid:54) j, (cid:54) p (cid:54) N i ) , (4.13)sup x,x (cid:48) ∈ B (cid:48)(cid:48) × Y j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − e − ι t (cid:32) | D j +2+ ι A t,i,p,k ( x ) − P x (cid:48) x ( D j +2+ ι A t,i,p,k ( x (cid:48) )) | g t d g t ( x, x (cid:48) ) α (cid:33) (cid:54) C, (4.14) where as usual the supremum is taken over pairs of points that are either horizontally or verticallyjoined. Remark 4.2.
To help understand this statement, morally each piece γ t,i,k should be thought of ashaving ∂∂ -potential of the form e − i +22 t × (fixed function on total space), so γ t,i,k would be then boundedin the shrinking C i ( g t ) but not in the shrinking C i,α ( g t ) for any α >
0. In practice, the potential of γ t,i,k is not quite of this form, but recalling (3.32) we see that its “leading term” is indeed of the form (cid:80) p A t,i,p,k (fixed function) p , and the L ∞ bound for A t,i,p,k in (4.12) just barely falls short of e − i +22 t .Furthermore, as we will see in the proof of Theorem B, the functions A t, ,p,k do satisfy the “optimal” L ∞ bound by e − t . Remark 4.3.
Observe also that when j = k = 0 the statement of theorem is analogous to our earlierwork [19, Thm 1.4], although not exactly identical since a different parallel transport is used there. Inthis very special case, the proof of Theorem 4.1 gives a more streamlined and improved proof of thatresult. Remark 4.4.
When j (cid:54)
1, the decomposition (4.5) simplifies, since γ t, ,k = 0, and the theorem justsays that ω • t − ω (cid:92)t is bounded in a shrinking C j,α -type norm, and goes to zero in shrinking C j . However,already γ t, ,k does not vanish in general (see the proof of Theorem B, and the corresponding discussionin the Introduction), and so this strong statement that holds for j = 0 , , fails for j (cid:62)
2. This is one ofthe main reasons why the statement and proof of Theorem 4.1 are complicated.The proof of Theorem 4.1 occupies the rest of this section (indeed, essentially all of the remainderof the paper), and will be divided into subsections.4.2.
Set-up of an inductive scheme, and initial reductions.
We start with the given point z ∈ B with a ball B (cid:48) ⊂ B centered at z . For a given k = 0 , , , . . . the proof proceeds by induction on j = 0 , , . . . , k . We will treat both the base case and the induction step at once, so, given k , we workat some 0 (cid:54) j (cid:54) k where if j > j − j (and define it for j = 0) and prove (4.6)—(4.14) if j > j = 0.First we give the details on how the decomposition (4.5) is constructed. When j = 0 we simply set γ t, = i∂∂ψ t , η t, ,k = i∂∂ ( ψ t − ψ t ), so that (4.5) holds. For j > j − ω • t = ω (cid:92)t + γ t, + γ t, ,k + · · · + γ t,j − ,k + η t,j − ,k , (4.15)on some ball B (cid:48) centered at z , and we wish to decompose η t,j − ,k = η t,j,k + γ t,j,k on some possiblysmaller ball.As indicated above, when j > B (cid:48) we may assume that we have alreadyselected smooth functions G i,p , (cid:54) i (cid:54) j − , (cid:54) p (cid:54) N i , which are fiberwise L orthonormal, and havefiberwise average zero. When j (cid:62) B (cid:48) gives us alist of functions G j,p , (cid:54) p (cid:54) N j on B (cid:48) × Y with fiberwise average zero and so that the G i,p , (cid:54) i (cid:54) j are all fiberwise L orthonormal, and the conclusion of Theorem 3.10 holds. For ease of notation, wewill rename B (cid:48) to B in all of the following.With these functions, we have the projections P t,i,p = P t,G i,p as in (3.5), and we then define A t,j,p,k = P t,j,p ( η t,j − ,k ) and γ t,j,k = (cid:80) N j p =1 i∂∂ G t,k ( A t,j,p,k , G j,p ), as in (4.8), (4.9), where G t,k was constructed inSection 3.2. Lastly, we define η t,j,k = η t,j − ,k − γ t,j,k , so that (4.5) holds at step j , together with (4.8),(4.9).The goal is thus to prove the estimates (4.6), (4.7), (4.11), (4.12), (4.13) and (4.14) if j >
0, andestimates (4.6), (4.7), (4.10) and (4.11) if j = 0.4.2.1. The bounds that hold thanks to the induction hypothesis.
First, we need to use the inductionhypothesis (4.6) to obtain a uniform bound of the functions A t,i,p,k (2 (cid:54) i (cid:54) j (cid:54) k ), which was definedby (4.9). Note that for any (1 , α on the total space, (cid:107) P t,i,p ( α ) (cid:107) L ∞ ( B ) (cid:54) Ce − t (cid:107) α (cid:107) L ∞ ( B × Y,g t ) . (4.16)Indeed, if we call K = (cid:107) α (cid:107) L ∞ ( B × Y,g t ) then on B × Y we have | α ff | (cid:54) CKe − t , (4.17) | α bf | (cid:54) CKe − t , (4.18) ans-Joachim Hein and Valentino Tosatti 43 | α bb | (cid:54) CK, (4.19)which can be then fed into (3.5) to obtain (4.16).By induction (4.6) (and its analogs for all 2 (cid:54) i (cid:54) j −
1) we know that (cid:107) η t,i − ,k (cid:107) L ∞ ( B × Y,g t ) (cid:54) Ce − i +1 − β t , (4.20)for all β <
1. For the rest of the proof, it suffices to take β = α , say, which we will do from now on.Then (4.16) gives (cid:107) P t,i,p ( η t,i − ,k ) (cid:107) L ∞ ( B ) (cid:54) Ce − i − − α t . (4.21)Putting these together we get (cid:107) A t,i,p,k (cid:107) L ∞ ( B ) (cid:54) Ce − i − − α t , (4.22)for all 2 (cid:54) i (cid:54) j , 1 (cid:54) p (cid:54) N i .4.2.2. Proving (4.6) , (4.10) , (4.12) and (4.13) . The logic now is the following. Suppose first that wehave proved (4.7), (4.11) and (4.14), and use them to quickly establish (4.6), (4.10), (4.12), (4.13) for j >
0, and (4.6), (4.10) for j = 0. And then, and this is the main task, we shall establish that (4.7),(4.11) and (4.14) do hold (this last one is of course vacuous when j = 0 , j = 0, where by assumption we assume that (4.7) and (4.11)hold. Then Theorem 2.9 applies to η t, ,k and it shows that (4.6) follows from (4.7). Next, recall that γ t, = i∂∂ψ t and ψ t → ψ t → γ t, → , (4.23)weakly. On the other hand, the bound (4.2) and the fiber integration argument in [29, p.436] give (cid:107) γ t, (cid:107) L ∞ ( B,g C m ) (cid:54) C. (4.24)Combining this with (4.11) then gives a uniform C α bound for γ t, (up to shrinking B ) and so γ t, → C α (cid:48) for any α (cid:48) < α , which proves (4.10), and completes our discussion of the case when j = 0.We assume then that j >
0. First observe that the exact same arguments show that (4.6) followsfrom (4.7) and Theorem 2.9 applied to η t,j,k , and that (4.10) follows from (4.11) and (4.2), (4.3).The remaining task is then to prove (4.12), (4.13), making use of (4.14) together with (4.22), thatwe have just established holds thanks to induction hypotheses.First, we discuss (4.12). We feed the D j +2 part of (4.14) and the uniform bound (4.22) into theinterpolation inequality in Proposition 2.8 (here R is the radius of B , which is comparable to 1) andobtain j +2 (cid:88) ι =1 ( R − ρ ) ι (cid:107) D ι A t,i,p,k (cid:107) L ∞ ( B ρ ,g C m ) (cid:54) C ( R − ρ ) j +2+ α + Ce − i − − α t , (4.25)and choosing R − ρ = e − i +1+ αj +2+ α t . (4.26)proves (4.12), and it also in particular gives a uniform C j +2 ,α bound on the A t,i,p,k ’s. Note that herewe have only used the D j +2 part of (4.14). It is possible to do slightly better for ι (cid:54) j + 1 by usingthe full strength of (4.14) and some more elaborate interpolations. While similar improvements willin fact be important below, we do not pursue them here because they are more awkward to state anddon’t make enough of a difference to the statement of Theorem 4.1.Finally, we discuss (4.13). From (4.14) we get in particular that[ D j +2+ ι A t,i,p,k ] C α ( B,g C m ) (cid:54) Ce ι t , (cid:54) ι (cid:54) k, (4.27) and doing interpolation between this and | D j +2 A t,i,p,k | = o (1) (which is a slightly suboptimal conse-quence of (4.12)) we get in particular the bound (cid:107) D j +2+ ι A t,i,p,k (cid:107) L ∞ ( B (cid:48) ,g C m ) = o ( e ι t ) , (cid:54) ι (cid:54) k, (4.28)which is even stronger than the statement of (4.13).4.2.3. Setting up the proof of (4.7) , (4.11) and (4.14) . Thanks to the previous section, it remains toestablish the estimates (4.7), (4.11) and (4.14). This is the main task and will occupy almost all therest of the paper.To this effect, suppose we had a uniform bound on the sum of the difference quotients D t,j ( x, x (cid:48) ) := j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − (cid:32) e − ι t | D j +2+ ι A t,i,p,k ( x ) − P x (cid:48) x ( D j +2+ ι A t,i,p,k ( x (cid:48) )) | g t ( x ) d g t ( x, x (cid:48) ) α (cid:33) + | D j γ t, ( x ) − P x (cid:48) x ( D j γ t, )( x (cid:48) )) | g t ( x ) d g t ( x, x (cid:48) ) α + | D j η t,j,k ( x ) − P x (cid:48) x ( D j η t,j,k )( x (cid:48) )) | g t ( x ) d g t ( x, x (cid:48) ) α . (4.29)for all x (cid:54) = x (cid:48) ∈ B (cid:48)(cid:48) × Y which are either horizontally or vertically joined, and all t (cid:62)
0, then it is clearthat the bounds (4.7), (4.11) and (4.14) hold. Observe that for each fixed t we have lim x (cid:48) → x D t,j ( x, x (cid:48) ) =0 since all the objects are smooth.4.3. Set-up of the primary (nonlinear) blowup argument.
Thus, to complete the proof of The-orem 4.1 it suffices to prove a uniform bound for (4.29). We shall assume from now on that B = B w.r.t. the Euclidean metric ω C m , and by replacing B with a slightly smaller ball we will assume withoutloss that all our objects are defined (and satisfy the assumptions of Theorem 4.1) on a concentric ball B σ for some σ >
0. To bound (4.29), define a function µ j,t on B × Y (with usual variables x = ( z, y ))by µ j,t ( x ) = || z | − | j + α sup x (cid:48) =( z (cid:48) ,y (cid:48) ) s.t. | z (cid:48) − z | < || z |− | x (cid:48) and x horizontally or vertically joined D t,j ( x, x (cid:48) ) . (4.30)Then it is clearly sufficient to prove the inequalitymax B × Y µ j,t (cid:54) C. (4.31)If (4.31) is false, then lim sup t →∞ max B × Y µ j,t = ∞ . For simplicity of notation we will assumethat lim t →∞ max B × Y µ j,t = ∞ (usually this will be true only along some sequence t i → ∞ ). Choose x t = ( z t , y t ) ∈ B × Y such that the maximum of µ j,t is achieved at x t , and define λ t by λ j + αt = sup x (cid:48) =( z (cid:48) ,y (cid:48) ) s.t. | z (cid:48) − z t | < || z t |− | x (cid:48) and x t horizontally or vertically joined D t,j ( x t , x (cid:48) ) . (4.32)Let us note for later purposes that after passing to a subsequence, z t → z ∞ ∈ B, y t → y ∞ ∈ Y. (4.33)Now λ t → ∞ since otherwise max B × Y µ j,t would be uniformly bounded. Let us also choose any point x (cid:48) t = ( z (cid:48) t , y (cid:48) t ) with | z (cid:48) t − z t | < || z t |− | realizing the sup in the definition of λ t . Since lim x (cid:48) → x D t,j ( x, x (cid:48) ) =0, we may assume without loss that x (cid:48) t (cid:54) = x t . Consider the diffeomorphismsΨ t : B λ t × Y → B × Y, ( z, y ) = Ψ t (ˆ z, ˆ y ) = ( λ − t ˆ z, ˆ y ) , (4.34) ans-Joachim Hein and Valentino Tosatti 45 pull back any contravariant 2-tensor via Ψ ∗ t , rescale it by λ t and denote the new object with a hat, forexample ˆ ω • t = λ t Ψ ∗ t ω t , etc. Define also ˆ A t,i,p,k = λ t Ψ ∗ t A t,i,p,k , ˆ x t = Ψ − t ( x t ) , ˆ x (cid:48) t = Ψ − t ( x (cid:48) t ) . We thushave ˆ g t = g C m + λ t e − t g Y,z , ˆ ω (cid:92)t = ˆ ω can + λ t e − t Ψ ∗ t ω F , C − ˆ ω (cid:92)t (cid:54) ˆ ω • t (cid:54) C ˆ ω (cid:92)t . (4.35)Also, from (4.24) we get (cid:107) ˆ γ t, (cid:107) L ∞ ( B λt ,g C m ) (cid:54) C, (4.36)and from (4.22) (cid:107) ˆ A t,i,p,k (cid:107) L ∞ ( B λt ) (cid:54) Cλ t e − i − − α t = Cδ t e − i +1 − α t , (4.37)for all 2 (cid:54) i (cid:54) j , 1 (cid:54) p (cid:54) N i , where we define δ t = λ t e − t . (4.38)We will also write ˆ P t,i,p for the stretching of the projection defined by (3.5) and (4.4), namelyˆ P t,i,p ( α ) = n (pr B ) ∗ (cid:16) ˆ G i,p α ∧ Ψ ∗ t ω n − F (cid:17) + δ t tr ˆ ω can (pr B ) ∗ ( ˆ G i,p α ∧ Ψ ∗ t ω nF ) . (4.39)Observe that λ j + αt = D t,j ( x t , x (cid:48) t ) = j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − (cid:32) e − ι t | D j +2+ ι A t,i,p,k ( x ) − P x (cid:48) t x t ( D j +2+ ι A t,i,p,k ( x (cid:48) )) | g t ( x ) d g t ( x t , x (cid:48) t ) α (cid:33) (4.40)+ | D j γ t, ( x t ) − P x (cid:48) t x t ( D j γ t, ( x (cid:48) t )) | g t ( x t ) d g t ( x t , x (cid:48) t ) α + | D j η t,j,k ( x t ) − P x (cid:48) t x t ( D j η t,j,k ( x (cid:48) t )) | g t ( x t ) d g t ( x t , x (cid:48) t ) α (4.41)= j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − (cid:32) e − ι t | D j +2+ ι ˆ A t,i,p,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j +2+ ι ˆ A t,i,p,k (ˆ x (cid:48) t )) | Ψ ∗ t g t (ˆ x t ) d Ψ ∗ t g t (ˆ x t , ˆ x (cid:48) t ) α (cid:33) λ − t (4.42)+ | D j ˆ γ t, (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ γ t, (ˆ x (cid:48) t )) | Ψ ∗ t g t (ˆ x t ) d Ψ ∗ t g t (ˆ x t , ˆ x (cid:48) t ) α λ − t + | D j ˆ η t,j,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ η t,j,k (ˆ x (cid:48) t )) | Ψ ∗ t g t (ˆ x t ) d Ψ ∗ t g t (ˆ x t , ˆ x (cid:48) t ) α λ − t (4.43)= j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − (cid:32) δ ιt | D j +2+ ι ˆ A t,i,p,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j +2+ ι ˆ A t,i,p,k (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) d ˆ g t (ˆ x t , ˆ x (cid:48) t ) α (cid:33) λ j + αt (4.44)+ | D j ˆ γ t, (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ γ t, (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) d ˆ g t (ˆ x t , ˆ x (cid:48) t ) α λ j + αt + | D j ˆ η t,j,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ η t,j,k (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) d ˆ g t (ˆ x t , ˆ x (cid:48) t ) α λ j + αt , (4.45)which implies that j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − (cid:32) δ ιt | D j +2+ ι ˆ A t,i,p,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j +2+ ι ˆ A t,i,p,k (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) d ˆ g t (ˆ x t , ˆ x (cid:48) t ) α (cid:33) + | D j ˆ γ t, (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ γ t, (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) d ˆ g t (ˆ x t , ˆ x (cid:48) t ) α + | D j ˆ η t,j,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ η t,j,k (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) d ˆ g t (ˆ x t , ˆ x (cid:48) t ) α = 1 . (4.46)Now recall that ˆ x (cid:48) t was chosen to maximize the difference quotient of (4.46) among all points ˆ x (cid:48) t =(ˆ z (cid:48) t , ˆ y (cid:48) t ) with | ˆ z (cid:48) t − ˆ z t | < || ˆ z t | − λ t | which are horizontally or vertically joined to ˆ x t . Moreover, the pointˆ x t itself maximizes the quantity µ j,t (Ψ t (ˆ x )) = || ˆ z | − λ t | j + α sup ˆ x (cid:48) =(ˆ z (cid:48) , ˆ y (cid:48) ) s.t. | ˆ z (cid:48) − ˆ z | < || ˆ z |− λ t | ˆ x (cid:48) and ˆ x horizontally or vertically joined ˆ D t,j (ˆ x, ˆ x (cid:48) ) (4.47) among all ˆ x = (ˆ z, ˆ y ) ∈ B λ t × Y , where we used the obvious notationˆ D t,j (ˆ x, ˆ x (cid:48) ) := j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − (cid:32) δ ιt | D j +2+ ι ˆ A t,i,p,k (ˆ x ) − P ˆ x (cid:48) ˆ x ( D j +2+ ι ˆ A t,i,p,k (ˆ x (cid:48) )) | ˆ g t (ˆ x ) d ˆ g t (ˆ x, ˆ x (cid:48) ) α (cid:33) + | D j ˆ γ t, (ˆ x ) − P ˆ x (cid:48) ˆ x ( D j ˆ γ t, )(ˆ x (cid:48) )) | ˆ g t (ˆ x ) d ˆ g t (ˆ x, ˆ x (cid:48) ) α + | D j ˆ η t,j,k (ˆ x ) − P ˆ x (cid:48) ˆ x ( D j ˆ η t,j,k )(ˆ x (cid:48) )) | ˆ g t (ˆ x ) d ˆ g t (ˆ x, ˆ x (cid:48) ) α . (4.48)We observe that || ˆ z t | − λ t | = max µ j + α j,t → ∞ , (4.49)and since by definition | ˆ z t − ˆ z (cid:48) t | (cid:54) || ˆ z t | − λ t | , we also have || ˆ z (cid:48) t | − λ t | (cid:62) || ˆ z t | − λ t | → ∞ . (4.50)This tells us that if we pass to a pointed limit with basepoint ˆ x t , the boundary moves away to infinityand the limit space will be complete. We also learn that for all ˆ x = (ˆ z, ˆ y ) ∈ B λ t × Y ,sup ˆ x (cid:48) =(ˆ z (cid:48) , ˆ y (cid:48) ) s.t. | ˆ z (cid:48) − ˆ z | < || ˆ z |− λ t | ˆ x (cid:48) and ˆ x horizontally or vertically joined ˆ D t,j (ˆ x, ˆ x (cid:48) ) (cid:54) || ˆ z t | − λ t | j + α || ˆ z | − λ t | j + α . (4.51)Using the triangle inequality and (4.49), we deduce in particular that that there exists a C such thatfor all R > t R such that for all t (cid:62) t R ,sup ˆ x, ˆ x (cid:48) ∈ ˆ B R (ˆ z t ) × Y ˆ x (cid:48) and ˆ x horizontally or vertically joined ˆ D t,j (ˆ x, ˆ x (cid:48) ) (cid:54) C, (4.52)where here and in all the following, ˆ B R (ˆ z ) will denote a Euclidean ball in C m (the hat is just a decorationto remind the reader that we are in the hat picture). Furthermore, if the center is one of the two pointsˆ z t and ˆ z (cid:48) t we may omit the center and the Y factor and write simply ˆ B R for short. Obviously (4.52)implies j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − (cid:32) δ ιt [ D j +2+ ι ˆ A t,i,p,k ] C α ( ˆ B R , ˆ g t ) (cid:33) + [ D j ˆ γ t, ] C α ( ˆ B R , ˆ g t ) + [ D j ˆ η t,j,k ] C α ( ˆ B R , ˆ g t ) (cid:54) C, (4.53)for every fixed R .4.4. The non-escaping property.Proposition 4.5.
Assume that δ t (cid:54) C . Then we have the upper bound d ˆ g t (ˆ x t , ˆ x (cid:48) t ) (cid:54) C. (4.54)When δ t → ∞ the proof below does not work, but we will soon show in (4.76) that an even betterbound holds in that case. Proof.
To start, (4.53) gives in particular[ D j +2+ ι ˆ A t,i,p,k ] C α ( ˆ B R , ˆ g t ) (cid:54) Cδ − ιt , − (cid:54) ι (cid:54) k, (4.55)for any given R , while from (4.37) (cid:107) ˆ A t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cδ t e − i +1 − α t , (4.56) ans-Joachim Hein and Valentino Tosatti 47 and we can interpolate between these two, first taking (4.55) with ι = − j (cid:88) (cid:96) =1 ( R − ρ ) (cid:96) (cid:107) D (cid:96) ˆ A t,i,p,k (cid:107) L ∞ ( ˆ B ρ ) (cid:54) C ( R − ρ ) j + α [ D j ˆ A t,i,p,k ] C α ( ˆ B R ) + Cδ t e − i +1 − α t (4.57) (cid:54) C ( R − ρ ) j + α δ t + Cδ t e − i +1 − α t , (4.58)so picking R − ρ ∼ (cid:16) e − i +1 − α t (cid:17) j + α we get (cid:107) D (cid:96) ˆ A t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cδ t (cid:16) e − i +1 − α t (cid:17) − (cid:96)j + α , (cid:54) (cid:96) (cid:54) j, (4.59)for any fixed R . Next, we take (4.55) with 0 < ι + 2 =: (cid:96) and interpolate( R − ρ ) (cid:96) (cid:107) D j + (cid:96) ˆ A t,i,p,k (cid:107) L ∞ ( ˆ B ρ ) (cid:54) C ( R − ρ ) (cid:96) + α [ D j + (cid:96) ˆ A t,i,p,k ] C α ( ˆ B R ) + C (cid:107) D j ˆ A t,i,p,k (cid:107) L ∞ ( ˆ B R ) (cid:54) C ( R − ρ ) (cid:96) + α δ − (cid:96) +2 t + Cδ t (cid:16) e − i +1 − α t (cid:17) αj + α , (4.60)so picking R − ρ ∼ δ (cid:96)(cid:96) + α t (cid:16) e − i +1 − α t (cid:17) α ( j + α )( (cid:96) + α ) (which is small because of our assumption δ t (cid:54) C ) we get (cid:107) D j + (cid:96) ˆ A t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cδ − (cid:96) +2 t (cid:18) δ (cid:96)t (cid:16) e − i +1 − α t (cid:17) αj + α (cid:19) α(cid:96) + α , (cid:54) (cid:96) (cid:54) k + 2 , (4.61)for any fixed R , and so combining (4.59) and (4.61) we see that δ ιt (cid:107) D j +2+ ι ˆ A t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) = o (cid:18) δ ( ι +2) αι +2+ α t (cid:19) = o (1) , − (cid:54) ι (cid:54) k, (4.62)for any fixed R , and applying this to balls centered at ˆ z t and ˆ z (cid:48) t gives in particular (by also invokingSection 2.1.1) j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − δ ιt | D j +2+ ι ˆ A t,i,p,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j +2+ ι ˆ A t,i,p,k (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) = o (1) . (4.63)Next, again thanks to (4.53), for any fixed R we have[ D j ˆ η t,j,k ] C α ( ˆ B R , ˆ g t ) (cid:54) C, (4.64)and Theorem 2.9 gives (in the case when δ t does not go to zero we need to assume here that R issufficiently large, which is of course allowed) (cid:107) D j ˆ η t,j,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cδ αt , (4.65)and applying this to balls centered at ˆ z t and ˆ z (cid:48) t gives in particular | D j ˆ η t,j,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ η t,j,k (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) (cid:54) Cδ αt . (4.66)Similarly, from (4.53) we get [ D j ˆ γ t, ] C α ( ˆ B R , ˆ g t ) (cid:54) C, (4.67)for any fixed R , and interpolating between this and (4.36) easily gives | D j ˆ γ t, (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ γ t, (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) (cid:54) C. (4.68)Then combining (4.46) with (4.63), (4.66) and (4.68) we obtain the desired bound (4.54). (cid:3) We are now in a position to study the possible complete pointed limit spaces of( B λ t × Y, ˆ g t , ˆ x t ) (4.69)as t → ∞ . Modulo translations in the C m factor, we may assume that ˆ x t = (0 , ˆ y t ) ∈ C m × Y . Recallhere that ˆ y t → ˆ y ∞ ∈ Y by (4.33). At this point three cases need to be considered, again up to passingto a subsequence: (1) δ t → ∞ ; (2) δ t → δ ∈ (0 , ∞ ), and without loss of generality δ = 1; and (3) δ t →
0. Observe that thanks to (4.33), it holds in all three cases that the complex structure convergeslocally smoothly to the product complex structure J C m + J Y,z ∞ . Case 1: the blowup is C m + n . In this case we assume that δ t → ∞ .Recall from (4.49) and (4.50) that the two points ˆ z t , ˆ z (cid:48) t satisfy that || ˆ z t | − λ t | → ∞ , || ˆ z (cid:48) t | − λ t | (cid:62) || ˆ z t | − λ t | → ∞ , | ˆ z t − ˆ z (cid:48) t | (cid:54) || ˆ z t | − λ t | . (4.70)Apply the diffeomorphismΞ t : B e t × Y → B λ t × Y, (ˆ z, ˆ y ) = Ξ t (ˇ z, ˇ y ) = ( δ t ˇ z, ˇ y ) , (4.71)and as usual multiply all the pulled back contravariant 2-tensors by δ − t , and denote the new objectsby a check. Denote also ˇ A t,i,p,k = δ − t Ξ ∗ t ˆ A t,i,p,k . Then ˇ g t is uniformly Euclidean, and the points ˇ z t andˇ z (cid:48) t satisfy || ˇ z t | − e t | = δ − t || ˆ z t | − λ t | , || ˇ z (cid:48) t | − e t | = δ − t || ˆ z (cid:48) t | − λ t | , | ˇ z t − ˇ z (cid:48) t | = δ − t | ˆ z t − ˆ z (cid:48) t | . (4.72)Now the balls ˇ B (ˇ z t ) , ˇ B (ˇ z (cid:48) t ) need not be contained in B e t , since we do not have any relation among || ˆ z t | − λ t | and δ t , but they are compactly contained in the larger ball B (1+ σ ) e t for t sufficiently large(cf. the beginning of § Y the metric ˇ ω • t is Ricci-flat and uniformlyEuclidean, so standard local estimates for the complex Monge-Amp`ere equation [19, Proposition 2.3]give us uniform C ∞ estimates for ˇ ω • t on compact subsets of B (1+ σ ) e t × Y , and in particular on ˇ B (ˇ z t ) × Y and ˇ B (ˇ z (cid:48) t ) × Y .Thanks to the definitions (4.8), (4.9) as fiber integrations, we deduce easily from this that thefollowing objects (proved in this order) also have uniform uniform C ∞ estimates on on ˇ B (ˇ z t ) × Y andˇ B (ˇ z (cid:48) t ) × Y : ˇ γ t, , ˇ η t, ,k , ˇ A t, ,p,k , ˇ γ t, ,k , ˇ η t, ,k , . . . , ˇ A t,j,p,k , ˇ γ t,j,k , ˇ η t,j,k . (4.73)Transferring these estimates back to the hat picture we obtain in particular that for ˆ p = ˆ z t , ˆ z (cid:48) t , (cid:107) D j ˆ γ t, (cid:107) L ∞ ( ˆ B δt (ˆ p ) × Y, ˆ g t ) + (cid:107) D j ˆ η t,j,k (cid:107) L ∞ ( ˆ B δt (ˆ p ) × Y, ˆ g t ) + j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − δ ιt (cid:107) D j +2+ ι ˆ A t,i,p,k (cid:107) L ∞ ( ˆ B δt (ˆ p ) × Y, ˆ g t ) (cid:54) Cδ − jt , (4.74)[ D j ˆ γ t, ] C α ( ˆ B δt (ˆ p ) × Y, ˆ g t ) + [ D j ˆ η t,j,k ] C α ( ˆ B δt (ˆ p ) × Y, ˆ g t ) + j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − δ ιt [ D j +2+ ι ˆ A t,i,p,k ] C α ( ˆ B δt (ˆ p ) × Y, ˆ g t ) (cid:54) Cδ − j − αt . (4.75)Using (4.74) and the triangle inequality to bound the numerators in (4.46) (using also the discussionin Section 2.1.1 to bound uniformly the operator norm of P ) gives d ˆ g t (ˆ x t , ˆ x (cid:48) t ) α (cid:54) Cδ − jt , (4.76) ans-Joachim Hein and Valentino Tosatti 49 so the two points ˆ z t and ˆ z (cid:48) t are colliding. Thus ˆ x (cid:48) t belongs to ˆ B δ t (ˆ z t ) × Y for all t large, and so applying(4.75) shows that the quantity in (4.46), which equals 1, is also bounded above by Cδ − j − αt , an obviouscontradiction.4.6. Case 2: the blowup is C m × Y . In this case we have that δ t →
1, without loss of generality.We now have that ˆ g (cid:92)t → g can ( z ∞ ) + g Y,z ∞ =: g P , in C ∞ loc ( C m × Y ) , (4.77)and the complex structure converges to a product. We also know that d ˆ g t (ˆ x t , ˆ x (cid:48) t ) (cid:54) C by (4.54), sopassing to a subsequence we may assume that ˆ x (cid:48) t → ˆ x (cid:48)∞ .Thanks to (4.35), we can apply standard local estimates for the complex Monge-Amp`ere equation[19, Proposition 2.3] on small balls to obtain C ∞ loc ( C m × Y ) bounds for ˆ ω • t . As in Case 1, thanks to thedefinitions as fiber integrations, we deduce easily from this that the following objects (proved in thisorder) also have uniform C ∞ estimates:ˆ γ t, , ˆ η t, ,k , ˆ A t, ,p,k , ˆ γ t, ,k , ˆ η t, ,k , . . . , ˆ A t,j,p,k , ˆ γ t,j,k , ˆ η t,j,k . (4.78)This has many useful consequences. First, going into (4.46) (and using also Remarks 2.6 and 2.7 tocompare H¨older norms), we can see that all the objects appearing there are C ∞ loc ( C m × Y ) bounded,so by estimating the C α difference quotients in (4.46) by C β ones for any β > α , we conclude thatthe two points ˆ x t = (0 , ˆ y t ) and ˆ x (cid:48) t have ˆ g t -distance uniformly bounded away from zero. Second, up topassing to a subsequence, we may assume that all of the above objects converge in C ∞ loc ( C m × Y ). Butwe already know that ˆ γ t, → A t,i,p,k → γ t, and ˆ A t,i,p,k go to zero locally smoothly.Recalling that by definition we haveˆ γ t,i,k = N i (cid:88) p =1 i∂∂ ˆ G t,k ( ˆ A t,i,p,k , ˆ G i,p ) , (4.79)and that Lemma 3.7 writes this asˆ γ t,i,k = i∂∂ N i (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t δ ιt ( ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D ι ˆ A t,i,p,k ) , (4.80)we conclude that ˆ γ t,i,k , (cid:54) i (cid:54) j, also go to zero locally smoothly.The C ∞ loc ( C m × Y ) limit ˆ ω •∞ = ω P + ˆ η ∞ ,j,k of ˆ ω • t = ˆ ω (cid:92)t + ˆ γ t, + ˆ γ t, ,k + · · · + ˆ γ t,j,k + ˆ η t,j,k will be aRicci-flat K¨ahler metric uniformly equivalent to the standard ω P . By the Liouville Theorem from [17](see also [24]), we see that ∇ g P ˆ ω •∞ = 0, and hence ∇ g P ˆ η ∞ ,j,k = 0 . (4.81)Going back to (4.46), we thus obtain a contradiction because the left-hand side of (4.46) manifestlyconverges to zero: the denominators converge to a strictly positive constant, and the numerators all goto zero (thanks to the above C ∞ loc vanishing of the ˆ A t,i,p,k ’s and ˆ γ t, , and (4.81)).4.7. Case 3: the blowup is C m (modulo linear regularity). We finally assume that δ t →
0. Thisis by far the hardest case.First, we can interpolate between the uniform bounds (4.36) and the seminorm bound [ D j ˆ γ t, ] C α ( B R ) (cid:54) C from (4.53), to see that ˆ γ t, has a uniform C j,α loc ( C m ) bound. By Ascoli-Arzel`a, up to passing to asequence t i → ∞ , ˆ γ t, converges in C j,β loc ( C m ) for β < α , but from (4.23) it must converge to zero. Thusin particular this gives (cid:107) D ι ˆ γ t, (cid:107) L ∞ ( ˆ B R , ˆ g t ) = o (1) , (4.82) for all 0 (cid:54) ι (cid:54) j and fixed R , where as usual ˆ B R here can be centered at ˆ z t or ˆ z (cid:48) t . In particular, | D j ˆ γ t, (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ γ t, (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) = o (1) . (4.83)Now the key claim is the following non-colliding estimate: there exists an ε > t itholds that d ˆ g t (ˆ x t , ˆ x (cid:48) t ) (cid:62) ε. (4.84)Assuming (4.84), let us quickly complete the proof of Theorem 4.1. Indeed from (4.46) together with(4.84) we know that j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − δ ιt | D j +2+ ι ˆ A t,i,p,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j +2+ ι ˆ A t,i,p,k (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) + | D j ˆ γ t, (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ γ t, (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) + | D j ˆ η t,j,k (ˆ x t ) − P ˆ x (cid:48) t ˆ x t ( D j ˆ η t,j,k (ˆ x (cid:48) t )) | ˆ g t (ˆ x t ) (cid:62) ε α , (4.85)but from (4.63), (4.66) and (4.83) we see that all terms on the LHS go to zero, which is a contradiction.Thus, the proof of Theorem 4.1 is now complete modulo the crucial linear regularity claim (4.84),which will occupy almost all the rest of the paper.4.8. Set-up of the secondary (linear) blowup argument in Case 3.
If the desired estimate(4.84) was false, then, since ˆ x t (cid:54) = ˆ x (cid:48) t for all t , there would exist a sequence t i → ∞ such that d t i = d ˆ g ti (ˆ x t i , ˆ x (cid:48) t i ) →
0. As usual, we will pretend that d t = d ˆ g t (ˆ x t , ˆ x (cid:48) t ) →
0. Define also a new parameter ε t = d − t δ t , (4.86)and consider the diffeomorphismsΘ t : B d − t λ t × Y → B λ t × Y, (ˆ z, ˆ y ) = Θ t (˜ z, ˜ y ) = ( d t ˜ z, ˜ y ) . (4.87)Pull back all our objects under Θ t , multiply the metrics and 2-forms by d − t , and denote the resultingobjects by the same letters with each hat replaced by a tilde. Define also ˜ A t,i,p,k = d − t Θ ∗ t ˆ A t,i,p,k . Thenfirst of all ˜ g t = g C m + ε t g Y,z , ˜ ω (cid:92)t = ˜ ω can + ε t Θ ∗ t Ψ ∗ t ω F . (4.88)Secondly, thanks to (4.1),(˜ ω • t ) m + n = (˜ ω (cid:92)t + ˜ γ t, + ˜ γ t, ,k + · · · + ˜ γ t,j,k + ˜ η t,j,k ) m + n = c t e ˜ H t (˜ ω (cid:92)t ) m + n , (4.89)where the constants c t converge to (cid:0) m + nn (cid:1) and˜ H t = log ˜ ω m can ∧ ( ε t Θ ∗ t Ψ ∗ t ω F ) n (˜ ω can + ε t Θ ∗ t Ψ ∗ t ω F ) m + n . (4.90) ans-Joachim Hein and Valentino Tosatti 51 Next, from (4.46), (4.53), there is C such that for all R there is t R such that for all t (cid:62) t Rj (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − (cid:32) ε ιt [ D j +2+ ι ˜ A t,i,p,k ] C α ( ˜ B Rd − t (˜ z t ) × Y, ˜ g t ) (cid:33) +[ D j ˜ γ t, ] C α ( ˜ B Rd − t (˜ z t ) × Y, ˜ g t ) + [ D j ˜ η t,j,k ] C α ( ˜ B Rd − t (˜ z t ) × Y, ˜ g t ) (cid:54) Cd j + αt , (4.91) j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − ε ιt | D j +2+ ι ˜ A t,i,p,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j +2+ ι ˜ A t,i,p,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α + | D j ˜ γ t, (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ γ t, (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α + | D j ˜ η t,j,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η t,j,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α = d j + αt , (4.92) d ˜ g t (˜ x t , ˜ x (cid:48) t ) = 1 . (4.93)It is now the time to separate each of our objects into a jet part and a remainder. As it turns out,this separation for ˜ η t,j,k is only needed in one special subcase, and we will deal with this later.First, from now on ˜ B R will always denote ˜ B R (˜ z t ) × Y . This will always include the other blowuppoint ˜ z (cid:48) t provided R >
1. As usual, [ · ] C α ( ˜ B R , ˜ g t ) will denote the shrinking seminorm defined using P andin (2.21), and we will write [ · ] C α base ( ˜ B R , ˜ g t ) for the seminorm where we only consider pairs of points thatare horizontally joined.Let us then discuss the jet subtraction for ˜ A t,i,p,k . Define a polynomial function ˜ A (cid:93)t,i,p,k as the j -jetof ˜ A t,i,p,k at ˜ x t with respect to the standard coordinates on C m , and define˜ A ∗ t,i,p,k = ˜ A t,i,p,k − ˜ A (cid:93)t,i,p,k , (4.94)so that ˜ A ∗ t,i,p,k vanishes to order j at ˜ x t .Define also ˜ ω can = d − t λ t Θ ∗ t Ψ ∗ t ω can where Ψ t ◦ Θ t ( z, y ) = ( d t λ − t z, y ), and for all ι (cid:62) < β < (cid:107) D ι ˜ ω can (cid:107) L ∞ ( ˜ B λtd − t (0) , ˜ g t ) = d ιt λ − ιt (cid:107) D ι ω can (cid:107) L ∞ ( B (0) ,g t ) (cid:54) Cd ιt λ − ιt , [ D ι ˜ ω can ] C β ( ˜ B λtd − t (0) , ˜ g t ) = d ι + βt λ − ι − βt [ D ι ω can ] C β ( B (0) ,g t ) (cid:54) Cd ι + βt λ − ι − βt . (4.95)We again need to perform a jet subtraction to ˜ γ t, = i∂∂ ˜ ψ t , by defining a polynomial function ˜ ψ (cid:93)t asthe ( j + 2)-jet of ˜ ψ t at ˜ x t with respect to the standard coordinates on C m and letting˜ η ‡ t = i∂∂ ˜ ψ (cid:93)t , ˜ η ♦ t = ˜ γ t, − ˜ η ‡ t . (4.96)so that ˜ η ♦ t vanishes to order j at x t .Let us introduce some new notation. Recall that we have defined˜ γ t,i,k = N i (cid:88) p =1 i∂∂ ˜ G t,k ( ˜ A t,i,p,k , ˜ G i,p ) , (4.97)so let us split ˜ A t,i,p,k = ˜ A ∗ t,i,p,k + ˜ A (cid:93)t,i,p,k and define˜ η ◦ t = j (cid:88) i =2 N i (cid:88) p =1 i∂∂ ˜ G t,k ( ˜ A ∗ t,i,p,k , ˜ G i,p ) , ˜ η † t = j (cid:88) i =2 N i (cid:88) p =1 i∂∂ ˜ G t,k ( ˜ A (cid:93)t,i,p,k , ˜ G i,p ) , (4.98)so that we have ˜ ω • t = ˜ ω (cid:92)t + ˜ η ‡ t + ˜ η ♦ t + ˜ η ◦ t + ˜ η † t + ˜ η t,j,k . (4.99)Let us also write ˜ ω (cid:93)t = ˜ ω (cid:92)t + ˜ η † t + ˜ η ‡ t , (4.100) so that ˜ ω • t = ˜ ω (cid:93)t + ˜ η ♦ t + ˜ η ◦ t + ˜ η t,j,k . (4.101)Clearly equations (4.91) and (4.92) hold verbatim with ˜ A t,i,p,k and ˜ γ t, replaced by ˜ A ∗ t,i,p,k and ˜ η ♦ t respectively, i.e. for any fixed R j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − (cid:32) ε ιt [ D j +2+ ι ˜ A ∗ t,i,p,k ] C α ( ˜ B Rd − t , ˜ g t ) (cid:33) + [ D j ˜ η ♦ t ] C α ( ˜ B Rd − t , ˜ g t ) + [ D j ˜ η t,j,k ] C α ( ˜ B Rd − t , ˜ g t ) (cid:54) Cd j + αt , (4.102) j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − ε ιt | D j +2+ ι ˜ A ∗ t,i,p,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j +2+ ι ˜ A ∗ t,i,p,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α + | D j ˜ η ♦ t (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η ♦ t (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α + | D j ˜ η t,j,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η t,j,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α = d j + αt . (4.103)4.9. Estimates on the solution components and on the background data.
The following sec-tion is the technical heart of the paper. Having split up the Monge-Amp`ere equation into background,jets, and “good” parts as above, we now derive precise estimates on the various components, whichwill ultimately allow us to expand and linearize the Monge-Amp`ere equation.In the following sections, the radius R will be any fixed radius, unless otherwise specified.4.9.1. Estimates for ˜ η t,j,k . Recalling (4.53), we can apply Theorem 2.9 to get (cid:107) D ι ˆ η t,j,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cδ j + α − ιt , [ D ι ˆ η t,j,k ] C α ( ˆ B R , ˆ g t ) (cid:54) Cδ j − ιt , (4.104)for 0 (cid:54) ι (cid:54) j , which in the tilde picture becomes d − ιt (cid:107) D ι ˜ η t,j,k (cid:107) L ∞ ( ˜ B Rd − t , ˜ g t ) (cid:54) Cδ j + α − ιt , d − ι − αt [ D ι ˜ η t,j,k ] C α ( ˜ B Rd − t , ˜ g t ) (cid:54) Cδ j − ιt , (4.105)for 0 (cid:54) ι (cid:54) j, or equivalently d − j − αt (cid:107) D ι ˜ η t,j,k (cid:107) L ∞ ( ˜ B Rd − t , ˜ g t ) (cid:54) Cε j + α − ιt , d − j − αt [ D ι ˜ η t,j,k ] C α ( ˜ B Rd − t , ˜ g t ) (cid:54) Cε j − ιt . (4.106)4.9.2. Estimates for ˜ γ t, , ˜ η ‡ t and ˜ η ♦ t . Thanks to (4.82) we have (cid:107) D ι ˆ γ t, (cid:107) L ∞ ( ˆ B R , ˆ g t ) = o (1) , (4.107)for 0 (cid:54) ι (cid:54) j and from (4.53) we get [ D j ˆ γ t, ] C α ( ˆ B R , ˆ g t ) (cid:54) C, (4.108)and so for 0 (cid:54) ι (cid:54) j,d − ιt (cid:107) D ι ˜ γ t, (cid:107) L ∞ ( ˜ B Rd − t , ˜ g t ) = o (1) , d − ι − αt [ D ι ˜ γ t, ] C α ( ˜ B Rd − t , ˜ g t ) = (cid:40) o (1) , (cid:54) ι < j,O (1) , ι = j. (4.109)Since ˜ η ‡ t is annihilated by [ D j · ], it follows from (4.108) that[ D j ˜ η ♦ t ] C α ( ˜ B Rd − t , ˜ g t ) = [ D j ˜ γ t, ] C α ( ˜ B Rd − t , ˜ g t ) (cid:54) Cd j + αt , (4.110)which integrating along segments (starting at ˜ x t where ˜ η ♦ t vanishes to order j ) gives d − ιt (cid:107) D ι ˜ η ♦ t (cid:107) L ∞ ( ˜ B R , ˜ g t ) (cid:54) Cd j + α − ιt R j + α − ι , d − ι − αt [ D ι ˜ η ♦ t ] C α ( ˜ B R , ˜ g t ) (cid:54) Cd j − ιt R j − ι , (4.111) ans-Joachim Hein and Valentino Tosatti 53 for 0 (cid:54) ι (cid:54) j and R (cid:54) Cd − t . In particular, if we assume that ε t (cid:62) C − then we can take R = ˜ Rε t for˜ R (cid:54) Cδ − t and get d − ιt (cid:107) D ι ˜ η ♦ t (cid:107) L ∞ ( ˜ B ˜ Rεt , ˜ g t ) (cid:54) Cδ j + α − ιt ˜ R j + α − ι , d − ι − αt [ D ι ˜ η ♦ t ] C α ( ˜ B ˜ Rεt , ˜ g t ) (cid:54) Cδ j − ιt ˜ R j − ι . (4.112)On the other hand, from its definition as a jet ˜ η ‡ t inherits from (4.109) the bounds d − ιt (cid:107) D ι ˜ η ‡ t (cid:107) L ∞ ( ˜ B R , ˜ g t ) = (cid:40) o (1) , (cid:54) ι < j, , ι (cid:62) j, d − ι − αt [ D ι ˜ η ‡ t ] C α ( ˜ B R , ˜ g t ) = (cid:40) o (1) , (cid:54) ι < j, , ι (cid:62) j. (4.113)for R (cid:54) Cd − t .4.9.3. Estimates for ˜ A ∗ t,i,p,k . Since ˆ A (cid:93)t,i,p,k is a polynomial from the base of degree at most j , it followsfrom (4.53) that [ D j ˆ A ∗ t,i,p,k ] C α ( ˆ B R , ˆ g t ) (cid:54) Cδ t , (4.114)for all fixed R , which integrating along segments (starting at ˆ x t where ˆ A ∗ t,i,p,k vanishes to order j ) gives (cid:107) D ι ˆ A ∗ t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) CR j + α − ι δ t , [ D ι ˆ A ∗ t,i,p,k ] C α ( ˆ B R , ˆ g t ) (cid:54) CR j − ι δ t , (4.115)for 0 (cid:54) ι (cid:54) j , and taking the radius R = Sd t we can translate to the tilde picture d − ι +2 t (cid:107) D ι ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cd j + α − ιt δ t S j + α − ι , d − ι +2 − αt [ D ι ˜ A ∗ t,i,p,k ] C α ( ˜ B S , ˜ g t ) (cid:54) Cd j − ιt δ t S j − ι , (4.116)for 0 (cid:54) ι (cid:54) j and S (cid:54) Cd − t .For derivatives of order higher than j , recall that from (4.53) we have the bounds[ D j + (cid:96) ˆ A ∗ t,i,p,k ] C α ( ˆ B R , ˆ g t ) (cid:54) Cδ − (cid:96)t , (4.117)for 0 (cid:54) (cid:96) (cid:54) k + 2 and R fixed. For 1 (cid:54) (cid:96) (cid:54) k + 2 we then interpolate between this and the bound(4.115) with ι = j as follows:( R − ρ ) (cid:96) (cid:107) D j + (cid:96) ˆ A ∗ t,i,p,k (cid:107) L ∞ ( ˆ B ρ , ˆ g t ) (cid:54) C ( R − ρ ) (cid:96) + α [ D j + (cid:96) ˆ A ∗ t,i,p,k ] C α ( ˆ B R , ˆ g t ) + C (cid:107) D j ˆ A ∗ t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) C ( R − ρ ) (cid:96) + α δ − (cid:96)t + CR α δ t , (4.118)and now assume first that ε (cid:62) C − . Then we choose ρ = ˜ Rδ t with ˜ R >
1, let
A > A α(cid:96) + α A − = ˜ R (cid:96)(cid:96) + α and define R = A ˜ Rδ t , so that R − ρ = ( R α δ (cid:96)t ) (cid:96) + α = δ t ( A ˜ R ) α(cid:96) + α , and so we obtain (cid:107) D j + (cid:96) ˆ A ∗ t,i,p,k (cid:107) L ∞ ( ˆ B ˜ Rδt , ˆ g t ) (cid:54) C ˜ R δ − (cid:96) + αt , (4.119)which in the tilde picture becomes d − j − (cid:96) +2 t (cid:107) D j + (cid:96) ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B ˜ Rεt , ˜ g t ) (cid:54) C ˜ R δ − (cid:96) + αt (4.120)for all 1 (cid:54) (cid:96) (cid:54) k + 2 and ˜ R > R fixed). It will also be useful to rewrite(4.116) and (4.120) as d − j − αt (cid:107) D ι ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cε t S j + α − ι ,d − j − αt (cid:107) D j + (cid:96) ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B ˜ Rεt , ˜ g t ) (cid:54) C ˜ R ε − (cid:96) + αt , (4.121)where 0 (cid:54) ι (cid:54) j, (cid:54) (cid:96) (cid:54) k + 2 and S, ˜ R are fixed.If on the other hand ε t →
0, then in (4.118) we choose R = 2 Sd t with S fixed, and pick ρ = R − ( R α δ (cid:96)t ) (cid:96) + α (cid:62) R = Sd t (provided t sufficiently large, since ( R α δ (cid:96)t ) (cid:96) + α = (2 S ) α(cid:96) + α d t ε (cid:96)(cid:96) + α t ) to obtain (cid:107) D j + (cid:96) ˆ A ∗ t,i,p,k (cid:107) L ∞ ( ˆ B Sdt , ˆ g t ) (cid:54) Cδ − (cid:96)t ( d αt δ (cid:96)t ) α(cid:96) + α S α (cid:96) + α , (4.122) which in the tilde picture becomes d − j − (cid:96) +2 t (cid:107) D j + (cid:96) ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cδ − (cid:96)t ( d αt δ (cid:96)t ) α(cid:96) + α S α (cid:96) + α = Cδ − (cid:96)t d αt ε (cid:96)α(cid:96) + α t S α (cid:96) + α , (4.123)for all 1 (cid:54) (cid:96) (cid:54) k + 2 and fixed S . Again we can rewrite (4.116) and (4.123) as d − j − αt (cid:107) D ι ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cε t S j + α − ι ,d − j − αt (cid:107) D j + (cid:96) ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cε − (cid:96)t ( S α ε (cid:96)t ) α(cid:96) + α , (4.124)where 0 (cid:54) ι (cid:54) j, (cid:54) (cid:96) (cid:54) k + 2 and S fixed.4.9.4. Estimates for ˜ A (cid:93)t,i,p,k . From (4.59) we have (cid:107) D ι ˆ A t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cδ t (cid:16) e − i +1 − α t (cid:17) − ιj + α , (cid:54) ι (cid:54) j, (4.125)for all given R , and since ˆ A (cid:93)t,i,p,k is the j -jet of ˆ A t,i,p,k at ˆ x t then in particular all the coefficients of thepolynomial ˆ A (cid:93)t,i,p,k have size bounded by Cδ t e − i +1 − α αj + α t , and so (cid:107) D ι ˆ A (cid:93)t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) C max(1 , R j − ι ) δ t e − i +1 − α αj + α t , [ D ι ˆ A (cid:93)t,i,p,k ] C β ( ˆ B R , ˆ g t ) (cid:54) C max(1 , R j − ι − β ) δ t e − i +1 − α αj + α t , (4.126)for 0 (cid:54) ι (cid:54) j, < β <
1, and so in the tilde picture d − ι +2 t (cid:107) D ι ˜ A (cid:93)t,i,p,k (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) C max(1 , S j − ι d j − ιt ) δ t e − i +1 − α αj + α t , (4.127) d − ι +2 − βt [ D ι ˜ A (cid:93)t,i,p,k ] C β ( ˜ B S , ˜ g t ) (cid:54) C max(1 , S j − ι − β d j − ι − βt ) δ t e − i +1 − α αj + α t , (4.128)for 0 (cid:54) ι (cid:54) j, < β < , S (cid:54) Cd − t .4.9.5. Estimates for ˜ η ◦ t . By definition, we seek to bound derivatives ofˆ η ◦ t = j (cid:88) i =2 N i (cid:88) p =1 i∂∂ ˆ G t,k ( ˆ A ∗ t,i,p,k , ˆ G i,p ) , (4.129)where from Lemma 3.7 we haveˆ η ◦ t = i∂∂ j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t δ ιt ( ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D ι ˆ A ∗ t,i,p,k ) , (4.130)and we will apply D r to this. We need to use the schematics D r +2 (cid:16) ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D ι ˆ A ∗ t,i,p,k (cid:17) = (cid:88) i + i = r +2 D i ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D i + ι ˆ A ∗ t,i,p,k , (4.131)and estimating (worst-case scenario) (cid:107) D i ˆΦ (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cδ − it , [ D i ˆΦ] C α ( ˆ B R , ˆ g t ) (cid:54) Cδ − i − αt . (4.132)We first assume that ε t (cid:62) C − , so using (4.116) and (4.120) with tilde radius S = ˜ Rε t with ˜ R fixed,we can bound (cid:107) D i ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D i + ι ˆ A ∗ t,i,p,k (cid:107) L ∞ ( ˆ B ˜ Rδt , ˆ g t ) (cid:54) C ˜ R δ − i t δ j +2+ α − i − ιt = C ˜ R δ j + α − r − ιt , (4.133)[ D i ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D i + ι ˆ A ∗ t,i,p,k ] C α ( ˆ B ˜ Rδt , ˆ g t ) (cid:54) C ˜ R δ − i − αt δ j +2+ α − i − ιt + C ˜ R δ − i t δ j +2 − i − ιt (cid:54) C ˜ R δ j − r − ιt , (4.134) ans-Joachim Hein and Valentino Tosatti 55 and combining (4.130), (4.131), (4.133), (4.134) gives d − ιt (cid:107) D ι ˜ η ◦ t (cid:107) L ∞ ( ˜ B ˜ Rεt , ˜ g t ) (cid:54) C ˜ R δ j + α − ιt , d − ι − αt [ D ι ˜ η ◦ t ] C α ( ˜ B ˜ Rεt , ˜ g t ) (cid:54) C ˜ R δ j − ιt , (4.135)for 0 (cid:54) ι (cid:54) j , which we can rewrite as d − j − αt (cid:107) D ι ˜ η ◦ t (cid:107) L ∞ ( ˜ B ˜ Rεt , ˜ g t ) (cid:54) C ˜ R ε j + α − ιt , d − j − αt [ D ι ˜ η ◦ t ] C α ( ˜ B ˜ Rεt , ˜ g t ) (cid:54) C ˜ R ε j − ιt . (4.136)On the other hand when ε t →
0, we shall take only derivatives and difference quotient in the basedirections, however the ∂∂ in (4.129) can still be in all directions, so at most two fiber derivatives canland on ˆΦ. Thus, going back to (4.131) in the term D i ˆΦ ι,(cid:96) ( ˆ G i,p ) there are u fiber derivatives with u (cid:54) D i f = u ), and so we can use (4.116) and (4.123) with tilde radius S tobound δ ιt (cid:107) D i f = u ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D i + ι ˆ A ∗ t,i,p,k (cid:107) L ∞ ( ˆ B Sdt , ˆ g t ) (cid:54) C S δ ιt λ − i + ut δ − ut δ t d j + α − i − ιt , i + ι (cid:54) jd αt δ j − i − ιt ε ( i ι − j ) αi ι − j + α t , i + ι > j (cid:54) C S d j + α − rt , (4.137)since i + u (cid:54) r + 2 and using also u (cid:54)
2, and δ ιt [ D i f = u ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D i + ι ˆ A ∗ t,i,p,k ] C α base ( ˆ B Sdt , ˆ g t ) (cid:54) C S d j + α − rt λ − αt + C S δ ιt λ − i + ut δ − ut (cid:40) δ t d j − i − ιt , i + ι (cid:54) jδ j − i − ιt , i + ι > j (cid:54) C S d j − rt , (4.138)and in particular we obtain d − rt (cid:107) D r b ··· b ˜ η ◦ t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) C S d j + α − rt , d − r − αt [ D r b ··· b ˜ η ◦ t ] C α base ( ˜ B S , ˜ g t ) (cid:54) C S d j − rt , (4.139)for 0 (cid:54) r (cid:54) j and S fixed.It is important to make here the following observation: in (4.137), whenever in the above estimateswe converted a δ t into a d t (using that ε t → λ t or the term e − ( (cid:96) − ι ) t , the actual result is o ( d j + α − rt ) rather than O ( d j + α − rt ). For later use, in the case when r = j ,we need to identify exactly which terms in (4.137) are not a priori o ( d αt ). Inspecting the above bounds,we must have i = u so that the factor of λ − t is absent, and hence i = j + 2 − u , and we must alsohave (cid:96) = ι so that the exponential term is absent. Let us first examine the case when i + ι (cid:54) j . Thismeans that j + 2 − u + ι (cid:54) j i.e. ι (cid:54) u − (cid:54)
0, and so we must have (cid:96) = ι = 0 , u = 2. This term is O ( d αt ) but not a priori smaller, and thanks to (3.33) it equals j (cid:88) i =2 N i (cid:88) p =1 ∂ f ∂ f (∆ Ψ ∗ t ω F | {·}× Y ) − ˆ G i,p D j b ··· b ˆ A ∗ t,i,p,k . (4.140)On the other hand, in the second case when i + ι > j we always get o ( d αt ) thanks to the term ε ( i ι − j ) αi ι − j + α t which goes to zero by assumption. So the conclusion is that in L ∞ ( ˜ B R , ˜ g t ) we have d − j − αt D j b ··· b ˜ η ◦ t = d − j − αt j (cid:88) i =2 N i (cid:88) p =1 ∂ f ∂ f (∆ Ψ ∗ t ω F | {·}× Y ) − ˜ G i,p D j b ··· b ˜ A ∗ t,i,p,k + o (1) . (4.141) Lastly, let us show that when ε → g t with the fixed metric g X wedo get d − ιt (cid:107) D ι ˜ η ◦ t (cid:107) L ∞ ( ˜ B S ,g X ) (cid:54) C S d j + α − ιt , d − ι − αt [ D ι ˜ η ◦ t ] C α ( ˜ B S ,g X ) (cid:54) C S d j − ιt , (4.142)for 0 (cid:54) ι (cid:54) j and fixed S . This is proved similarly to (4.139), using that all derivatives of ˜Φ ι,(cid:96) haveuniformly bounded norm with respect to g X . Briefly, in the tilde picture we have˜ η ◦ t = i∂∂ j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t ε ιt ( ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D ι ˜ A ∗ t,i,p,k ) , (4.143)and we apply D r to this ( r (cid:54) j ), using the schematics D r +2 (cid:16) ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D ι ˜ A ∗ t,i,p,k (cid:17) = (cid:88) i + i = r +2 D i ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D i + ι ˜ A ∗ t,i,p,k , (4.144)we can bound d − rt ε ιt (cid:107) D i ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D i + ι ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B S ,g X ) (cid:54) C S d − i t δ ιt δ t d j + α − i − ιt , i + ι (cid:54) jd αt δ j − i − ιt ε ( i ι − j ) αi ι − j + α t , i + ι > j (cid:54) C S d j + α − rt , (4.145)and d − r − αt ε ιt [ D i ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D i + ι ˜ A ∗ t,i,p,k ] C α ( ˜ B S ,g X ) (cid:54) C S d j − rt + C S d − i t δ ιt (cid:40) δ t d j − i − ιt , i + ι (cid:54) jδ j − i − ιt , i + ι > j (cid:54) C S d j − rt , (4.146)and (4.142) follows.4.9.6. Estimates for ˜ η † t . We seek to boundˆ η † t = j (cid:88) i =2 N i (cid:88) p =1 i∂∂ ˆ G t,k ( ˆ A (cid:93)t,i,p,k , ˆ G i,p ) , (4.147)where from Lemma 3.7 we haveˆ η † t = i∂∂ j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t δ ιt ( ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D ι ˆ A (cid:93)t,i,p,k ) , (4.148)and we will apply D r to this, for any r (cid:62)
0. Recall from (4.126) that (cid:107) D ι ˆ A (cid:93)t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) C max(1 , R j − ι ) δ t e − i +1 − α αj + α t , [ D ι ˆ A (cid:93)t,i,p,k ] C β ( ˆ B R , ˆ g t ) (cid:54) C max(1 , R j − ι − β ) δ t e − i +1 − α αj + α t , (4.149)for 0 (cid:54) ι (cid:54) j, < β < R , while of course derivatives of order > j vanish. We need to usethe schematics D r +2 (cid:16) ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D ι ˆ A (cid:93)t,i,p,k (cid:17) = (cid:88) i + i = r +2 D i ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D i + ι ˆ A (cid:93)t,i,p,k , (4.150)and bound (for 0 < β < R ) (cid:107) D i ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D i + ι ˆ A (cid:93)t,i,p,k (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cδ − i t δ t e − i +1 − α αj + α t (cid:54) Cδ − rt e − i +1 − α αj + α t , (4.151) ans-Joachim Hein and Valentino Tosatti 57 [ D i ˆΦ ι,(cid:96) ( ˆ G i,p ) (cid:126) D i + ι ˆ A (cid:93)t,i,p,k ] C β ( ˆ B R , ˆ g t ) (cid:54) Cδ − i − βt δ t e − i +1 − α αj + α t (cid:54) Cδ − r − βt e − i +1 − α αj + α t . (4.152)Overall, this gives d − rt (cid:107) D r ˜ η † t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cδ − rt e − − α αj + α t , d − r − βt [ D r ˜ η † t ] C β ( ˜ B S , ˜ g t ) (cid:54) Cδ − r − βt e − − α αj + α t , (4.153)for all r (cid:62) , < β < S , while if we take only derivatives in the base directions, then in(4.151) and (4.152) there are at most 2 fiber derivatives landing on ˆΦ (and the C β difference quotientis in the base only), which implies that the negative powers of δ t at the end of (4.151) and (4.152)disappear and we get d − rt (cid:107) D r b ··· b ˜ η † t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Ce − − α αj + α t , d − r − βt [ D r b ··· b ˜ η † t ] C β base ( ˜ B S , ˜ g t ) (cid:54) Ce − − α αj + α t , (4.154)for all r (cid:62) , < β < S (cid:54) Rd − t ( R fixed), and all of these go to zero. On the other hand, ifin (4.151), (4.152) we use the fixed metric g X instead of ˆ g t then all derivatives of ˆΦ ι,(cid:96) are uniformlybounded and we obtain (cid:107) D r ˆ η † t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:54) Cδ t e − − α αj + α t , [ D r ˆ η † t ] C β ( ˆ B R ,g X ) (cid:54) Cδ t e − − α αj + α t , (4.155)for all r (cid:62) , < β < R . Lastly, we will also need a similar estimate using the fixed metric g X but in the tilde picture when ε t → , which says that (cid:107) D r ˜ η † t (cid:107) L ∞ ( ˜ B S ,g X ) (cid:54) Cε t e − − α αj + α t , [ D r ˜ η † t ] C β ( ˜ B S ,g X ) (cid:54) Cε t e − − α αj + α t , (4.156)for r (cid:62) , < β < , fixed S , and again these all go to zero. Briefly, we have˜ η † t = i∂∂ j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t ε ιt ( ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D ι ˜ A (cid:93)t,i,p,k ) , (4.157)and we apply D r to this ( r (cid:54) j ), using the schematics D r +2 (cid:16) ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D ι ˜ A (cid:93)t,i,p,k (cid:17) = (cid:88) i + i = r +2 D i ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D i + ι ˜ A (cid:93)t,i,p,k , (4.158)and using that all derivatives of ˜Φ ι,(cid:96) have uniformly bounded norm with respect to g X we can bound d − rt ε ιt (cid:107) D i ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D i + ι ˜ A (cid:93)t,i,p,k (cid:107) L ∞ ( ˜ B S ,g X ) (cid:54) Cd − i t δ ιt δ t e − i +1 − α αj + α t (cid:54) Cd − rt ε t e − i +1 − α αj + α t , (4.159)and d − r − βt ε ιt [ D i ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D i + ι ˜ A (cid:93)t,i,p,k ] C β ( ˜ B S ,g X ) (cid:54) Cd − i − βt δ ιt δ t e − i +1 − α αj + α t (cid:54) Cd − r − βt ε t e − i +1 − α αj + α t , (4.160)and (4.156) follows.4.9.7. Estimates for ˜ ω (cid:93)t . Since ˜ ω (cid:93)t = ˜ ω • t − ˜ η t,j,k − ˜ η ◦ t − ˜ η ♦ t , we can see, using (4.105), (4.111), (4.135)and the fact that the complex structure has uniformly bounded ˜ g t -norm, that for any given R , ˜ ω (cid:93)t is aK¨ahler form on ˜ B R for all t sufficiently large, with associated metric uniformly equivalent to ˜ g t (thanksto (4.2)).We claim that we have the bounds d − ιt (cid:107) D ι ˜ ω (cid:93)t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cδ − ιt , d − ι − βt [ D ι ˜ ω (cid:93)t ] C β ( ˜ B S , ˜ g t ) (cid:54) Cδ − ι − βt , (4.161)for all ι (cid:62) j ), 0 < β <
1, and fixed S .Indeed, recall that ˜ ω (cid:93)t = ˜ ω can + ε t Θ ∗ t Ψ ∗ t ω F + ˜ η † t + ˜ η ‡ t . (4.162) The term ˜ ω can is bounded by (4.95), the term ˜ η † t is bounded by (4.153), and ˜ η ‡ t by (4.113) (and thisone vanishes when differentiated more than j times). Lastly, for the term ε t Θ ∗ t Ψ ∗ t ω F we have d − ιt (cid:107) D ι ( ε t Θ ∗ t Ψ ∗ t ω F ) (cid:107) L ∞ ( ˜ B S , ˜ g t ) = λ − ιt (cid:107) D ι ( e − t ω F ) (cid:107) L ∞ ( B Sdtλ − t ,g t ) (cid:54) Cδ − ιt , (4.163) d − ι − βt [ D ι ( ε t Θ ∗ t Ψ ∗ t ω F )] C β ( ˜ B S , ˜ g t ) = λ − ι − βt [ D ι ( e − t ω F )] C β ( B Sdtλ − t ,g t ) (cid:54) Cδ − ι − βt , (4.164)by simple “index counting”, for all ι (cid:62) , < β < S (cid:54) Cd − t , and (4.161) now follows.Lastly, we claim that if we move only in the base directions, then we get d − ιt (cid:107) D ι b ··· b ˜ ω (cid:93)t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) (cid:40) O (1) , ι = 0 ,o (1) , ι > , d − ι − βt [ D ι b ··· b ˜ ω (cid:93)t ] C β base ( ˜ B S , ˜ g t ) = o (1) , (4.165)for all ι (cid:62) , < β < S . Indeed this follows by the same argument as above, replacing(4.153) by (4.154), and observing that when we go base-only the bounds in (4.163), (4.164) improvetrivially to d − ιt (cid:107) D ι b ··· b ( ε t Θ ∗ t Ψ ∗ t ω F ) (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cλ − ιt , (4.166) d − ι − βt [ D ι b ··· b ( ε t Θ ∗ t Ψ ∗ t ω F )] C β base ( ˜ B S , ˜ g t ) (cid:54) Cλ − ι − βt , (4.167)for all ι (cid:62) , < β < S (cid:54) Cd − t .4.9.8. Expansion of the Monge-Amp`ere equation.
Using the decomposition (4.101) for the Ricci-flatmetrics ˜ ω • t , we expand the Monge-Amp`ere equation (4.89) astr ˜ ω (cid:93)t (˜ η t,j,k + ˜ η ♦ t + ˜ η ◦ t ) (4.168)+ (cid:88) i + ι + p (cid:62) ( m + n )! i ! ι ! p !( m + n − i − ι − p )! (˜ η t,j,k ) i ∧ (˜ η ♦ t ) ι ∧ (˜ η ◦ t ) p ∧ (˜ ω (cid:93)t ) m + n − i − ι − p (˜ ω (cid:93)t ) m + n = c t e ˜ H t (˜ ω (cid:92)t ) m + n (˜ ω (cid:93)t ) m + n − . For ease of notation, call C iιp = (˜ η t,j,k ) i ∧ (˜ η ♦ t ) ι ∧ (˜ η ◦ t ) p ∧ (˜ ω (cid:93)t ) m + n − i − ι − p (˜ ω (cid:93)t ) m + n . (4.169) Proposition 4.6.
For any fixed R , we have d − j − αt (cid:34) D j b ··· b (cid:32) c t e ˜ H t (˜ ω (cid:92)t ) m + n (˜ ω (cid:93)t ) m + n − (cid:33)(cid:35) C α base ( ˜ B R , ˜ g t ) = o (1) , (4.170) ∀ i + ι + p (cid:62) d − j − αt [ D j b ··· b C iιp ] C α base ( ˜ B R , ˜ g t ) = o (1) , (4.171) for all t sufficiently large. Combining these with (4.168) we obtain d − j − αt [ D j b ··· b (tr ˜ ω (cid:93)t (˜ η t,j,k + ˜ η ♦ t + ˜ η ◦ t ))] C α base ( ˜ B R , ˜ g t ) = o (1) . (4.172) On the other hand, if we assume ε t (cid:62) C − then for every fixed R and (cid:54) a (cid:54) j we have d − j − αt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) D a (cid:32) c t e ˜ H t (˜ ω (cid:92)t ) m + n (˜ ω (cid:93)t ) m + n − (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ˜ B Rεt , ˜ g t ) (cid:54) Cε j + α − at , (4.173) d − j − αt (cid:34) D a (cid:32) c t e ˜ H t (˜ ω (cid:92)t ) m + n (˜ ω (cid:93)t ) m + n − (cid:33)(cid:35) C α ( ˜ B Rεt , ˜ g t ) (cid:54) Cε j − at , (4.174) ∀ i + ι + p (cid:62) d − j − αt (cid:107) D a C iιp (cid:107) L ∞ ( ˜ B Rεt , ˜ g t ) (cid:54) Cδ j + αt ε j + α − at = o (1) , (4.175) ans-Joachim Hein and Valentino Tosatti 59 ∀ i + ι + p (cid:62) d − j − αt [ D a C iιp ] C α ( ˜ B Rεt , ˜ g t ) (cid:54) Cδ j + αt ε j − at = o (1) , (4.176) for t sufficiently large.Proof. First, let us prove (4.170). Using the definition of ˜ H t , we can write e ˜ H t (˜ ω (cid:92)t ) m + n (˜ ω (cid:93)t ) m + n = ˜ ω m can ∧ ( ε t Θ ∗ t Ψ ∗ t ω F ) n (˜ ω (cid:93)t ) m + n . (4.177)To bound the numerator we shall employ (4.95), (4.166) and (4.167). As for the denumerator (˜ ω (cid:93)t ) m + n ,let us temporarily use the shorthand b = (˜ ω (cid:93)t ) m + n , so that we can write schematically (again omittingcombinatorial factors) D r ( b − ) = r (cid:88) (cid:96) =1 (cid:88) j + ··· + j (cid:96) = r D j bb · · · D j (cid:96) bb b − , (4.178)and it then suffices to use (4.165), and (4.170) follows.Estimate (4.171) also follows easily from (4.105), (4.111) and (4.139), together with (4.165) and(4.178).On the other hand, assuming now that ε t (cid:62) C − , estimates (4.175) and (4.176) follow from (4.105),(4.112), (4.135), together with (4.161) and (4.178).Lastly, to prove (4.173), (4.174), using (4.168), (4.175) and (4.176) it suffices to show that d − j − αt (cid:13)(cid:13)(cid:13) D a tr ˜ ω (cid:93)t (˜ η t,j,k + ˜ η ♦ t + ˜ η ◦ t ) (cid:13)(cid:13)(cid:13) L ∞ ( ˜ B Rεt , ˜ g t ) (cid:54) Cε j + α − at , (4.179) d − j − αt (cid:104) D a tr ˜ ω (cid:93)t (˜ η t,j,k + ˜ η ♦ t + ˜ η ◦ t ) (cid:105) C α ( ˜ B Rεt , ˜ g t ) (cid:54) Cε j − at , (4.180)for R fixed and 0 (cid:54) a (cid:54) j , which is also a direct consequence of (4.105), (4.112), (4.135), together with(4.161) and (4.178). (cid:3) We are now in position to derive a contradiction on each of the three possible scenarios (up to passingto a sequence t i → ∞ as usual), according to whether ε t → ∞ , ε t remains bounded away from 0 and ∞ , or ε t → Subcase A: ε t → ∞ . A first complication in this case is that the right hand sides of (4.106) and(4.136) blow up, and thus we do not control d − j − αt (cid:107) D ι ˜ η t,j,k (cid:107) L ∞ ( ˜ B R , ˜ g t ) and d − j − αt (cid:107) D ι ˜ η ◦ t (cid:107) L ∞ ( ˜ B R , ˜ g t ) for 0 (cid:54) ι (cid:54) j , although these do give control on d − j − αt [ D j ˜ η t,j,k ] C α ( ˜ B Rd − t , ˜ g t ) (cid:54) C and d − j − αt [ D j ˜ η ◦ t ] C α ( ˜ B Rεt , ˜ g t ) (cid:54) C for any given R .To fix this, recall that ˜ g t = g C m + ε t g Y,z . Let x m +1 , . . . , x m +2 n be normal coordinates for g Y,z centered at y t . Viewed as a map from Y to R n these depend on t , but we prefer to instead pull backour setup to R n under the inverse map. In this sense we may then assume without loss that (cid:12)(cid:12)(cid:12)(cid:12) ∂ ι ∂ x ι ( g Y,z ( x ) ab − δ ab ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) | x | − min { ,ι } for | x | (cid:54) ι ∈ { , . . . , j + 1 } . (4.181)This is possible thanks to the compactness of Y . Define ˜ x j = ε t x j , so that ˜ x m +1 , . . . , ˜ x m +2 n arenormal coordinates for ε t g Y,z centered at y t . Formally also write ˜ x , . . . , ˜ x m for the standard realcoordinates on C m . Then ˜ x , . . . , ˜ x m +2 n are normal coordinates for ˜ g t centered at ˜ x t with (cid:12)(cid:12)(cid:12)(cid:12) ∂ ι ∂ ˜ x ι (˜ g t (˜ x ) ab − δ ab ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) ε − max { ,ι } t | ˜ x | − min { ,ι } for | ˜ x | (cid:54) ε t and ι ∈ { , . . . , j + 1 } . (4.182) Define then ∂∂ -exact (1 , η (cid:93)t , ˜ η (cid:52) t on ˜ B ε t (˜ x t ) as the j -th order Taylor polynomials of ˜ η t,j,k and˜ η ◦ t at ˜ x t with respect to the coordinate system ˜ x , . . . , ˜ x m +2 n (more precisely, as we did for ˜ γ t, , wetake i∂∂ of the ( j + 2)-jet of the potentials of ˜ η t,j,k and ˜ η ◦ t ). We then define˜ η ∗ t = ˜ η t,j,k − ˜ η (cid:93)t , ˜ η (cid:53) t = ˜ η ◦ t − ˜ η (cid:52) t , (4.183)which are also ∂∂ -exact (1 , B ε t (˜ x t ).4.10.1. Improved estimates after additional jet subtractions.
Proposition 4.7.
For all t sufficiently large we have d − j − αt D ι ˜ η ∗ t (˜ x t ) = 0 , d − j − αt D ι ˜ η (cid:53) t (˜ x t ) = 0 , (cid:54) ι (cid:54) j, (4.184) d − j − αt (cid:107) D ι ˜ η ∗ t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) CS j + α − ι , d − j − αt [ D ι ˜ η ∗ t ] C α ( ˜ B S , ˜ g t ) (cid:54) CS j − ι , S (cid:54) ε t , (cid:54) ι (cid:54) j, (4.185) d − j − αt (cid:107) D ι ˜ η (cid:53) t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) CS j + α − ι , d − j − αt [ D ι ˜ η (cid:53) t ] C α ( ˜ B S , ˜ g t ) (cid:54) CS j − ι S (cid:54) ε t , (cid:54) ι (cid:54) j, (4.186) d − j − αt (cid:32) j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − ε ιt | D j +2+ ι ˜ A ∗ t,i,p,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j +2+ ι ˜ A ∗ t,i,p,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α + | D j ˜ η ♦ t (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η ♦ t (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α + | D j ˜ η ∗ t (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η ∗ t (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α (cid:33) = 1 + Cε α − t . (4.187) Proof.
Equation (4.184) follows from the definition of ˜ η ∗ t and ˜ η (cid:53) t , together with (2.15) that relates D ι with ∇ ι , and using induction on ι .To prove the remaining estimates, the key is the following claim: for all ι (cid:62) , < β < S (cid:54) ε t we have d − ι − αt [ D ι ˜ η (cid:93)t ] C β ( ˜ B S , ˜ g t ) (cid:54) Cε α − t δ j − ιt S − β , (4.188) d − ιt (cid:107) D ι ˜ η (cid:93)t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cδ j − ι + αt , (4.189)and exactly the same bounds for ˜ η (cid:52) t .To prove this claim, note that g Y,z is at bounded distance to g Y,z in C ∞ ( Y ), thus |∇ ι, ˜ g t ˜ g ˜ z,t | ˜ g t (cid:54) C ι ε − ιt on ˜ B R , (4.190)for all ι (cid:62) , R (cid:54) ε t .We can now give the proof of (4.188), (4.189) for ˜ η (cid:93)t , and the exact same proof applies to ˜ η (cid:52) t .We can join ˜ x and ˜ x (cid:48) by concatenating two P -geodesics, one vertical and one horizontal. Bound d − j + ιt times the left-hand side of (4.188) by integrating the D ι +1 -derivative of d − j − αt ˜ η (cid:93)t along these geodesics.Converting D ι +1 into ∇ ι +1 using again (2.15), this allows us to estimate d − j − αt | D ι ˜ η (cid:93)t (˜ x ) − P ˜ x (cid:48) ˜ x ( D ι ˜ η (cid:93)t (˜ x (cid:48) )) | ˜ g t (˜ x ) (4.191) (cid:54) d ˜ g t (˜ x, ˜ x (cid:48) ) (cid:13)(cid:13)(cid:13)(cid:13) d − j − αt (cid:18) ∂∂ ˜ x + Γ ˜ g ˜ z,t (˜ x ) (cid:19) ι +1 + ι − (cid:88) p =0 ∇ ι − − p usual / ˜z A (cid:126) (cid:18) ∂∂ ˜ x + Γ ˜ g ˜ z,t (˜ x ) (cid:19) p (4.192) (cid:88) γ ∈ N d + e | γ | (cid:54) j γ ! ∂ | γ | ˜ η t,j,k ∂ ˜ x γ (˜ x t )(˜ x − ˜ x t ) γ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( B ˜ gt (˜ x t , R )) (4.193)and estimating the big L ∞ norm by Cε j − ι + α − t , as follows. ans-Joachim Hein and Valentino Tosatti 61 (1) Schematically ∇ a Γ = ( ∂ + Γ) a Γ = (cid:80) ∂ a Γ · · · ∂ a (cid:96) Γ, where a + · · · + a (cid:96) + (cid:96) = a + 1 by countingthe total number of ∂ s and Γs in each term of a complete expansion of the left-hand side. Now ∂ b Γ = O ( ε − b − t ) by (4.182) and (4.190), so ∇ a Γ = O ( ε − a − t ).(2) The A -tensor in the tilde picture is bounded by O ( ε − t ), since it is schematically of the sametype as ∂ Γ. By the same reason, ∇ p usual / ˜z A is O ( ε − p − t ).(3) Writing d − j − αt ˜ η t,j,k = η , we have ∂ b η = ( ∇− Γ) b η = (cid:80) ∇ b Γ · · · ∇ b (cid:96) Γ ·∇ c η with b + · · · + b (cid:96) + (cid:96) + c = b . Now using (4.106) we get ( D c η )(˜ x t ) = O ( ε j + α − ct ) for 0 (cid:54) c (cid:54) j (here when we replace ˜ η t,j,k with ˜ η ◦ t wewill use instead (4.136)). Then (2.15) together with Step (2) give ∇ c η = D c η + (cid:80) c − d =0 ∇ d η (cid:126) O ( ε − c + dt ),and so induction on c gives ( ∇ c η )(˜ x t ) = O ( ε j + α − ct ) for 0 (cid:54) c (cid:54) j . We can then use Step (1) above, andfinally get ( ∂ b η )(˜ x t ) = O ( ε j + α − bt ) for 0 (cid:54) b (cid:54) j .(4) We can then estimate( ∂ + Γ) p (( ∂ | γ | η )(˜ x t )(˜ x − ˜ x t ) γ ) = ( ∂ | γ | η )(˜ x t ) (cid:88) ∂ a Γ · · · ∂ a (cid:96) Γ · ∂ b (˜ x − ˜ x t ) γ , (4.194)where a + · · · + a (cid:96) + (cid:96) + b = p again by counting the number of ∂ s and Γs, and b (cid:54) | γ | . Nowrecall that R (cid:54) ε t , so that ∂ b (˜ x − ˜ x t ) γ = O ( ε | γ |− bt ). Since ( ∂ | γ | η )(˜ x t ) = O ( ε j + α −| γ | t ) by Step (3) and ∂ a Γ = O ( ε − a − t ) by (4.182), the quantity in (4.194) can be estimated by O ( ε j + α − pt ).(5) From Steps (2) and (4) we can bound (cid:16) ∇ ι − − p usual / ˜z A (cid:126) ( ∂ + Γ) p (cid:17) (cid:88) γ ( ∂ | γ | η )(˜ x t )(˜ x − ˜ x t ) γ = O ( ε j − ι + α − t ) , (4.195)and so using Step (4) again we obtain the desired bound of Cε j − ι + α − t for the big L ∞ norm. Thisimplies (4.188).To prove (4.189), we write as above (with η = d − j − αt ˜ η t,j,k ) d − j − αt D ι ˜ η (cid:93)t = ( ∂ + Γ) ι + ι − (cid:88) p =0 ∇ ι − − p usual / ˜z A (cid:126) ( ∂ + Γ) p (cid:88) γ ( ∂ | γ | η )(˜ x t )(˜ x − ˜ x t ) γ , (4.196)and using the estimates in Step (4) (with p = ι ) and Step (5) (with ι there replaced by ι −
1) we getthat d − j − αt (cid:107) D ι ˜ η (cid:93)t (cid:107) L ∞ ( ˜ B R , ˜ g t ) (cid:54) Cε j − ι + αt , (4.197)for R (cid:54) ε t , which is (4.189). It is now also clear that exact same proof applies to ˜ η (cid:52) t , simply replacing(4.106) above with the analogous bound (4.136) for ˜ η ◦ t .Next, to prove (4.185) we combine (4.188) with d − j − αt [ D j ˜ η t,j,k ] C α ( ˜ B S , ˜ g t ) (cid:54) C for S (cid:54) ε t (which comesfrom (4.105)) and obtain d − j − αt [ D j ˜ η ∗ t ] C α ( ˜ B S , ˜ g t ) (cid:54) C, (4.198)for S (cid:54) ε t , and the rest of (4.185) follows from this and (4.184) by integrating along segments startingat ˜ x t .We obtain (4.186) in the same way, using the bound d − j − αt [ D j ˜ η ◦ t ] C α ( ˜ B S , ˜ g t ) (cid:54) C for S (cid:54) ε t from(4.135). Also, (4.187) is clear from (4.103) and (4.188). (cid:3) We are aiming to obtain a contradiction by showing that the LHS of (4.187) is o (1). Recall that byconstruction d ˜ g t (˜ x t , ˜ x (cid:48) t ) = 1.4.10.2. The noncancellation property.
The following noncancellation property will be crucial. It isstated and proved in the hat picture, and it will then be transferred to the tilde picture.
Proposition 4.8.
Let ˆ B R := B C m (ˆ z t , R ) × Y . The following inequality holds for all a ∈ N and all R : [ D a ˆ γ t, ] C α ( ˆ B R ) (cid:54) C (cid:32) [ D a b ··· b ˆ η t ] C α base ( ˆ B R ,g X ) + (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ − αt (cid:107) D b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:33) , (4.199) The following inequality holds for all a ∈ N , (cid:54) i (cid:54) j , (cid:54) p (cid:54) N i : [ D a ˆ A t,i,p,k ] C α ( ˆ B R ) (cid:54) Cδ t (cid:18) [ D a b ··· b ˆ η t ] C α base ( ˆ B R , ˆ g t ) (4.200)+ (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ − αt (cid:18) (cid:107) D b ˆ η ‡ t (cid:107) L ∞ ( ˆ B R , ˆ g t ) + δ b − at (cid:107) D b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) (cid:107) L ∞ ( ˆ B R , ˆ g t ) + δ − t (cid:107) D b ˆ η † t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:19)(cid:19) (4.201)+ C i − (cid:88) ι =2 N ι (cid:88) q =1 (cid:32) δ k +2 t [ D a +2 k +2 ˆ A t,ι,q,k ] C α ( ˆ B R ) + (cid:18) Rλ t (cid:19) − α a +2 k +2 (cid:88) b =0 e − (2 k +2) t λ b − a − αt (cid:107) D b ˆ A t,ι,q,k (cid:107) L ∞ ( ˆ B R ) (cid:33) . (4.202) Proof.
We begin by writing ˆ γ t, and ˆ A t,i,p,k as the pushforwards of certain forms on the total space thatare roughly proportional to the whole solution ˆ η t . Next, we discuss how to slide D a into a pushforward.Then we discuss how to slide a H¨older difference quotient into a pushforward. At the end we explainhow to put everything together to get the desired inequalities stated above. Claim 1 : We have the following representations of ˆ γ t, and ˆ A t,i,p,k as pushforwards of ˆ η t :ˆ γ t, = (pr B ) ∗ (ˆ η t ∧ Ψ ∗ t ω nF ) , (4.203)and, writing ˆ P t,i,p for the stretched projection in (4.39),ˆ A t,i,p,k = ˆ P t,i,p (ˆ η t ) + e − (2 k +2) t i − (cid:88) ι =2 N ι (cid:88) q =1 2 k +2 (cid:88) κ =0 ˆΦ κ,i,p,k,ι,q (cid:126) λ κt D κ ˆ A t,ι,q,k , (4.204)where the functions ˆΦ κ,i,p,k,ι,q are from the base. Proof of Claim 1 : We can assume without loss that (cid:82) { ˆ z }× Y Ψ ∗ t ω nF | { ˆ z }× Y = 1 for all ˆ z , so we can writeby definition ˆ ψ t = (pr B ) ∗ (cid:16) ˆ ψ t Ψ ∗ t ω nF (cid:17) , (4.205)and note that (pr B ) ∗ commutes with ∂ and ∂ since pr B is holomorphic, soˆ γ t, = i∂∂ ˆ ψ t = (pr B ) ∗ (ˆ η t ∧ Ψ ∗ t ω nF ) . (4.206)Again by definitionˆ A t,i,p,k = ˆ P t,i,p (ˆ η t,i − ,k ) = ˆ P t,i,p (ˆ η t ) − ˆ P t,i,p (ˆ γ t, ) − i − (cid:88) ι =2 ˆ P t,i,p (ˆ γ t,ι,k ) . (4.207)The term ˆ P t,i,p (ˆ γ t, ) is zero thanks to (3.6). For the remaining terms we invoke Lemma 3.6, transplantedto the hat picture:ˆ P t,i,p (ˆ γ t,ι,k ) = N ι (cid:88) q =1 ˆ P t,i,p ( i∂∂ ˆ G t,k ( ˆ A t,ι,q,k , ˆ G ι,q )) (4.208)= N ι (cid:88) q =1 (cid:32) ˆ A t,ι,q,k (cid:90) { ˆ z }× Y ˆ G i,p ˆ G ι,q ˆ ω nF + e − (2 k +2) t k +2 (cid:88) κ =0 ˆΦ κ,i,p,k ( ˆ G ι,q ) (cid:126) λ κt D κ ˆ A t,ι,q,k (cid:33) . (4.209) ans-Joachim Hein and Valentino Tosatti 63 Here the first term vanishes by orthogonality of the ˆ G i,p ’s, and the rest is exactly what we are claiming. (cid:3) Claim 2 : For any (2 n + 2)-form ξ on B × Y and any a (cid:62) , at any point z ∈ B we have D a (pr B ) ∗ ξ = (pr B ) ∗ (cid:0) ∇ z,a b ··· b ξ (cid:1) . (4.210) Proof of Claim 2 : By definition we have D a (pr B ) ∗ ξ = ∂ a b ··· b (pr B ) ∗ ξ, (4.211)and to evaluate this at a point z ∈ B recall that ∇ z is a product connection and in the base directionsits Christoffel symbols are just zero, and so (4.211) equals (cid:32)(cid:90) { z }× Y ∇ z,a b ··· b (cid:16) ι ∂∂zi ι ∂∂zj ξ (cid:17)(cid:33) dz i ∧ dz j = (cid:32)(cid:90) { z }× Y ι ∂∂zi ι ∂∂zj (cid:0) ∇ z,a b ··· b ξ (cid:1)(cid:33) dz i ∧ dz j = (pr B ) ∗ (cid:0) ∇ z,a b ··· b ξ (cid:1) , (4.212)where in the first equality we used ∇ z b (cid:0) ∂∂z i (cid:1) = 0. (cid:3) Claim 3 : Denote by Φ : { ˆ z } × Y → { ˆ z } × Y the map given by Φ(ˆ z , ˆ y ) = (ˆ z , ˆ y ). Then there is aconstant C such that for any smooth form ξ on the total space we have | ((pr B ) ∗ ξ )(ˆ z ) − ((pr B ) ∗ ξ )(ˆ z ) | (cid:54) C sup ˆ x ∈{ ˆ z }× Y | ξ (ˆ x ) − P Φ(ˆ x )ˆ x ξ (Φ(ˆ x )) | g X (ˆ x ) + Cλ − t ( sup { ˆ z }× Y | ξ | g X ) | ˆ z − ˆ z | . (4.213) Proof of Claim 3 : pick any fixed orthonormal vectors Z , . . . , Z r on the base ( r = (deg ξ ) − n ), anddenote by the same notation their trivial extension to horizontal vector fields on the total space. Then ξ ( Z , . . . , Z r , − , . . . , − ) is a 2 n -form on the total space, hence has at most 2 n fiber components in eachof its indecomposable terms, and only those incomposable terms with exactly 2 n fiber componentssurvive the restriction. Note also note that P -transport commutes with restriction since it preservesthe horizontal-vertical decomposition (cf. Section 2.1.1).Now write the fiberwise restriction of ξ ( Z , . . . , Z r , − , . . . , − ) to any fiber { ˆ z }× Y as a scalar function f (ˆ x ) times the fiberwise Ricci-flat volume form Υ ˆ z := Ψ ∗ t ω nF | { ˆ z }× Y . This now allows us to estimate thecrucial error term: (cid:12)(cid:12)(cid:12)(cid:12) ((pr B ) ∗ ξ )(ˆ z )( Z , . . . , Z r ) − (cid:90) ˆ x ∈{ ˆ z }× Y P Φ(ˆ x )ˆ x [ f (Φ(ˆ x ))Υ ˆ z (Φ(ˆ x ))] (cid:12)(cid:12)(cid:12)(cid:12) (4.214)= (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ˆ x ∈{ ˆ z }× Y f (ˆ x )(Υ ˆ z − (Φ − ) ∗ P Φ(ˆ x )ˆ x Υ ˆ z )(ˆ x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) Cλ − t ( sup { ˆ z }× Y | ξ | g X ) | ˆ z − ˆ z | , (4.215)using here that | Υ ˆ z − (Φ − ) ∗ P Φ(ˆ x )ˆ x Υ ˆ z | (ˆ x ) before stretching has size proportional to the unstretcheddistance | z − z | , which becomes λ − t | ˆ z − ˆ z | after stretching. Having estimated the error term, itremains to bound the main term by (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ˆ x ∈{ ˆ z }× Y ξ (ˆ x )( Z , . . . , Z r , − , . . . , − ) | { ˆ z }× Y − P Φ(ˆ x )ˆ x [ ξ (Φ(ˆ x ))( Z , . . . , Z r , − , . . . , − ) | { ˆ z }× Y ] (cid:12)(cid:12)(cid:12)(cid:12) (4.216) (cid:54) C sup ˆ x ∈{ ˆ z }× Y | ξ (ˆ x ) − P Φ(ˆ x )ˆ x ξ (Φ(ˆ x )) | g X (ˆ x ) , (4.217)which is clear. (cid:3) With these 3 claims we can now complete the proof of Proposition 4.8, starting with (4.199). Com-bining Claims 1–2 gives D a ˆ γ t, = (pr B ) ∗ (cid:0) ∇ z,a b ··· b (ˆ η t ∧ Ψ ∗ t ω nF ) (cid:1) , (4.218) and applying Claim 3 gives[ D a ˆ γ t, ] C α ( ˆ B R ) (cid:54) C [ ∇ z,a b ··· b (ˆ η t ∧ Ψ ∗ t ω nF )] C α base ( ˆ B R ,g X ) + Cλ − t R − α (cid:107)∇ z,a b ··· b (ˆ η t ∧ Ψ ∗ t ω nF ) (cid:107) L ∞ ( ˆ B R ,g X ) , (4.219)and we can use the Leibniz rule to bound[ ∇ z,a b ··· b (ˆ η t ∧ Ψ ∗ t ω nF )] C α base ( ˆ B R ,g X ) (cid:54) C a (cid:88) b =0 (cid:18) [ ∇ z,b b ··· b ˆ η t ] C α base ( ˆ B R ,g X ) (cid:107)∇ z,a − b b ··· b (Ψ ∗ t ω nF ) (cid:107) L ∞ ( ˆ B R ,g X ) + (cid:107)∇ z,b b ··· b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) [ ∇ z,a − b b ··· b (Ψ ∗ t ω nF )] C α base ( ˆ B R ,g X ) (cid:19) (cid:54) C a (cid:88) b =0 (cid:18) λ b − at [ ∇ z,b b ··· b ˆ η t ] C α base ( ˆ B R ,g X ) + λ b − a − t R − α (cid:107)∇ z,b b ··· b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:19) (cid:54) C [ ∇ z,a b ··· b ˆ η t ] C α base ( ˆ B R ,g X ) + C (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ b − a − αt (cid:107)∇ z,b b ··· b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) , (4.220)where in the last line for b < a we have used[ ∇ z,b b ··· b ˆ η t ] C α base ( ˆ B R ,g X ) (cid:54) R − α (cid:107)∇ z,b +1 b ··· b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) , (4.221)as in Remark 2.5. Similarly, we bound the last term in (4.219) by Cλ − t R − α (cid:107)∇ z,a b ··· b (ˆ η t ∧ Ψ ∗ t ω nF ) (cid:107) L ∞ ( ˆ B R ,g X ) (cid:54) Cλ − t R − α a (cid:88) b =0 λ b − at (cid:107)∇ z,b b ··· b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) , (4.222)and combining (4.219), (4.220) and (4.222) gives[ D a ˆ γ t, ] C α ( ˆ B R ) (cid:54) C [ ∇ z,a b ··· b ˆ η t ] C α base ( ˆ B R ,g X ) + C (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ b − a − αt (cid:107)∇ z,b b ··· b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) . (4.223)To complete the proof of (4.199) we need to convert the ∇ z,p s into D p s. For this we use (2.15), whichfor all c (cid:62) D c b ··· b ˆ η t = ∇ z,c b ··· b ˆ η t + c − (cid:88) p =0 ( ∇ z,p ˆ η t (cid:126) ∇ c − − p usual / ˜ z ˆ A ) b ··· b , (4.224)where ˆ A denotes the A tensor in the hat picture. We have the very crude estimate (cid:107)∇ a ˆ A (cid:107) L ∞ ( ˆ B R ,g X ) (cid:54) Cλ − t , [ ∇ a ˆ A ] C α base ( ˆ B R ,g X ) (cid:54) CR − α λ − t , (4.225)for any a (cid:62)
0, coming from the fact that at least one of the indices of ˆ A must be in the base direction(otherwise it is zero) and thanks to the stretching Ψ t this gives a factor of λ − t . Using (4.224) and(4.225) and arguing as in (4.220), (4.222) we obtain[ ∇ z,a b ··· b ˆ η t ] C α base ( ˆ B R ,g X ) (cid:54) [ D a b ··· b ˆ η t ] C α base ( ˆ B R ,g X ) + C (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ − αt (cid:107) D b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) , (4.226)and similarly (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ b − a − αt (cid:107)∇ z,b b ··· b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:54) C (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ − αt (cid:107) D b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) , (4.227)and so (4.199) follows from (4.223), (4.226) and (4.227). ans-Joachim Hein and Valentino Tosatti 65 We then move on to the proof of (4.200)–(4.202). Thanks to (4.204), we have[ D a ˆ A t,i,p,k ] C α ( ˆ B R ) = [ D a ˆ P t,i,p (ˆ η t )] C α ( ˆ B R ) + e − (2 k +2) t i − (cid:88) ι =2 N ι (cid:88) q =1 2 k +2 (cid:88) κ =0 λ κt (cid:104) D a (cid:16) ˆΦ κ,i,p,k,ι,q (cid:126) D κ ˆ A t,ι,q,k (cid:17)(cid:105) C α ( ˆ B R ) , (4.228)and let us first discuss the first term on the RHS. Recall thatˆ P t,i,p (ˆ η t ) = n (pr B ) ∗ ( ˆ G i,p ˆ η t ∧ Ψ ∗ t ω n − F ) + δ t tr ˆ ω can (pr B ) ∗ ( ˆ G i,p ˆ η t ∧ Ψ ∗ t ω nF ) , (4.229)and the estimate δ t [ D a tr ˆ ω can (pr B ) ∗ ( ˆ G i,p ˆ η t ∧ Ψ ∗ t ω nF )] C α ( ˆ B R ) (cid:54) Cδ t (cid:32) [ D a b ··· b ˆ η t ] C α base ( ˆ B R ,g X ) + (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ − αt (cid:107) D b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:33) , (4.230)is proved exactly like in the proof of (4.199). We then use the decompositionˆ η t = ˆ η ‡ t + ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k + ˆ η † t , (4.231)and bound trivially (cid:107) D b ˆ η t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:54) (cid:107) D b ˆ η ‡ t (cid:107) L ∞ ( ˆ B R , ˆ g t ) + (cid:107) D b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) (cid:107) L ∞ ( ˆ B R , ˆ g t ) + δ − t (cid:107) D b ˆ η † t (cid:107) L ∞ ( ˆ B R ,g X ) . (4.232)On the other hand, to bound [ D a (pr B ) ∗ ( ˆ G i,p ˆ η t ∧ Ψ ∗ t ω n − F )] C α ( ˆ B R ) , (4.233)we can argue as in Claim 2 and get D a (pr B ) ∗ ( ˆ G i,p ˆ η t ∧ Ψ ∗ t ω n − F ) = (pr B ) ∗ (cid:16) ∇ z,a b ··· b ( ˆ G i,p ˆ η t ∧ Ψ ∗ t ω n − F ) (cid:17) , (4.234)while notice that now since the quantities inside the pushforwards are 2 n -forms, only the ff -componentsof ˆ η t appear. This means that if we bring in the decomposition (4.231) then the term ˆ η ‡ t disappearsfrom (4.234) since it is pulled back from the base. We then apply Claim 3 and we bound (4.233) by C [ ∇ z,a b ··· b ( ˆ G i,p (ˆ η † t + ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) ff ∧ Ψ ∗ t ω n − F )] C α base ( ˆ B R ,g X ) + Cλ − t R − α (cid:107)∇ z,a b ··· b ( ˆ G i,p (ˆ η † t + ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) ff ∧ Ψ ∗ t ω n − F ) (cid:107) L ∞ ( ˆ B R ,g X ) , (4.235)which as in (4.220) and (4.222) is bounded by C [( ∇ z,a b ··· b ˆ η † t + ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) ff ] C α base ( ˆ B R ,g X ) + C (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ b − a − αt (cid:107)∇ z,b b ··· b (ˆ η † t + ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) ff (cid:107) L ∞ ( ˆ B R ,g X ) (4.236) (cid:54) C [( ∇ z,a b ··· b ˆ η † t ) ff ] C α base ( ˆ B R ,g X ) + C (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ b − a − αt (cid:107) ( ∇ z,b b ··· b ˆ η † t ) ff (cid:107) L ∞ ( ˆ B R ,g X ) (4.237)+ C [( ∇ z,a b ··· b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k )) ff ] C α base ( ˆ B R ,g X ) + C (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ b − a − αt (cid:107) ( ∇ z,b b ··· b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k )) ff (cid:107) L ∞ ( ˆ B R ,g X ) . (4.238)As before, for any (1 , α on B × Y we use the conversion( D c b ··· b α ) ff = ( ∇ z,c b ··· b α ) ff + c − (cid:88) p =0 ( ∇ z,p α (cid:126) ∇ c − − p usual / ˜ z ˆ A ) b ··· bff , (4.239) which can be iterated( ∇ z,c b ··· b α ) ff = ( D c b ··· b α ) ff + c − (cid:88) p =0 (cid:98) p (cid:99) (cid:88) r =0 p − (cid:88) p =0 p − (cid:88) p =0 · · · p r − − (cid:88) p r =0 (cid:18) D p r α (cid:126) ∇ c − − p ˆ A (cid:126) ∇ p − − p ˆ A (cid:126) ∇ p − − p ˆ A · · · ∇ p r − − − p r ˆ A (cid:19) b ··· bff . (4.240)For the terms in (4.237), we apply the conversion (4.240) with α = ˆ η † t and we just bound the ˆ A tensorterms by (4.225) and obtain the bound C [( D a b ··· b ˆ η † t ) ff ] C α base ( ˆ B R ,g X ) + C (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ − αt (cid:107) D b ˆ η † t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:54) Cδ t (cid:32) [( D a b ··· b ˆ η † t ) ff ] C α base ( ˆ B R , ˆ g t ) + δ − t (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ − αt (cid:107) D b ˆ η † t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:33) . (4.241)As for the terms in (4.238), we first bound them by Cδ t (cid:32) [( ∇ z,a b ··· b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k )) ff ] C α base ( ˆ B R , ˆ g t ) + (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ b − a − αt (cid:107) ( ∇ z,b b ··· b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k )) ff (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:33) , (4.242)and apply the conversion (4.240) with α = ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k , together with the bounds (cid:107)∇ a ˆ A (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cλ − t δ − a − t , [ ∇ a ˆ A ] C α base ( ˆ B R , ˆ g t ) (cid:54) CR − α λ − t δ − a − t , (4.243)for any a (cid:62) R (as in item (2) in the proof of Proposition 4.7), which imply for example (cid:107)∇ c − − p ˆ A (cid:126) ∇ p − − p ˆ A (cid:126) ∇ p − − p ˆ A · · · ∇ p r − − − p r ˆ A (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:54) Cδ − c + p r t , (4.244)and bound (4.242) by Cδ t (cid:32) [( D a b ··· b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k )) ff ] C α base ( ˆ B R , ˆ g t ) + (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ − αt δ b − at (cid:107) D b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) (cid:107) L ∞ ( ˆ B R , ˆ g t ) (cid:33) , (4.245)so combining (4.241) and (4.245) (and ( D a b ··· b ˆ η ‡ t ) ff = 0) we deduce that (4.233) is bounded by Cδ t (cid:18) [ D a b ··· b ˆ η t ] C α base ( ˆ B R , ˆ g t ) + (cid:18) Rλ t (cid:19) − α a (cid:88) b =0 λ − αt (cid:16) δ b − at (cid:107) D b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) (cid:107) L ∞ ( ˆ B R , ˆ g t ) + δ − t (cid:107) D b ˆ η † t (cid:107) L ∞ ( ˆ B R ,g X ) (cid:17) (cid:19) . (4.246)This together with (4.230) and (4.232) gives us the desired bound for the first term on the RHS of(4.228). Lastly, we deal with the second term on the RHS of (4.228). Using the Leibniz rule we canbound it by e − (2 k +2) t i − (cid:88) ι =2 N ι (cid:88) q =1 2 k +2 (cid:88) κ =0 a (cid:88) b =0 λ κt (cid:18) [ D a − b ˆΦ κ,i,p,k,ι,q ] C α ( ˆ B R ) (cid:107) D b + κ ˆ A t,ι,q,k (cid:107) L ∞ ( ˆ B R ) + (cid:107) D a − b ˆΦ κ,i,p,k,ι,q (cid:107) L ∞ ( ˆ B R ) [ D b + κ ˆ A t,ι,q,k ] C α ( ˆ B R ) (cid:19) , (4.247) ans-Joachim Hein and Valentino Tosatti 67 and we can bound λ κt [ D a − b ˆΦ κ,i,p,k,ι,q ] C α ( ˆ B R ) (cid:54) λ κt R − α (cid:107) D a − b +1 ˆΦ κ,i,p,k,ι,q (cid:107) L ∞ ( ˆ B R ) (cid:54) CR − α λ b + κ − a − t = C (cid:18) Rλ t (cid:19) − α λ b + κ − a − αt , (4.248)and for b + κ < a + 2 k + 2 λ κt (cid:107) D a − b ˆΦ κ,i,p,k,ι,q (cid:107) L ∞ ( ˆ B R ) [ D b + κ ˆ A t,ι,q,k ] C α ( ˆ B R ) (cid:54) Cλ b + κ − at R − α (cid:107) D b + κ +1 ˆ A t,ι,q,k (cid:107) L ∞ ( ˆ B R ) = C (cid:18) Rλ t (cid:19) − α λ b + κ +1 − a − αt (cid:107) D b + κ +1 ˆ A t,ι,q,k (cid:107) L ∞ ( ˆ B R ) , (4.249)and so (4.247) is bounded by Ce − (2 k +2) t i − (cid:88) ι =2 N ι (cid:88) q =1 (cid:18) λ k +2 t [ D a +2 k +2 ˆ A t,ι,q,k ] C α ( ˆ B R ) + (cid:18) Rλ t (cid:19) − α a +2 k +2 (cid:88) c =0 λ c − a − αt (cid:107) D c ˆ A t,ι,q,k (cid:107) L ∞ ( ˆ B R ) (cid:19) , (4.250)which is exactly (4.202). (cid:3) Preliminary estimate on ˜ η ♦ t . First, we prove the following estimate that will be useful to dealwith the second piece on the LHS of (4.187): for any fixed R we have for all t large d − j − αt [ D j ˜ η ♦ t ] C α ( ˜ B R , ˜ g t ) (cid:54) Cd − j − αt [ D j (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C α ( ˜ B R , ˜ g t ) + o (1) . (4.251)To prove this, we start by recalling that˜ η t = ˜ η ♦ t + ˜ η ‡ t + ˜ η † t + ˜ η (cid:52) t + ˜ η (cid:53) t + ˜ η ∗ t + ˜ η (cid:93)t , (4.252)and by taking (4.199) on ˆ B δ t (which translates to ˜ B ε t in the tilde picture) we bound the term (0 (cid:54) b (cid:54) j ) (cid:107) D b ˆ η t (cid:107) L ∞ ( ˆ B δt ,g X ) (cid:54) C (cid:107) D b (ˆ η t − ˆ η † t ) (cid:107) L ∞ ( ˆ B δt , ˆ g t ) + (cid:107) D b ˆ η † t (cid:107) L ∞ ( ˆ B δt ,g X ) (cid:54) C (cid:107) D b (ˆ η ♦ t + ˆ η ‡ t + ˆ η (cid:52) t + ˆ η (cid:53) t + ˆ η ∗ t + ˆ η (cid:93)t ) (cid:107) L ∞ ( ˆ B δt , ˆ g t ) + Cδ t e − − α αj + α t = o (1) , (4.253)using (4.111), (4.113), (4.185), (4.186), (4.189) and (4.155) to bound all the pieces. Using this, we cantransfer (4.199) to the tilde picture to get d − j − αt [ D j ˜ η ♦ t ] C α ( ˜ B S , ˜ g t ) = d − j − αt [ D j ˜ γ t, ] C α ( ˜ B S , ˜ g t ) (cid:54) Cd − j − αt [ D j b ··· b ˜ η t ] C α base ( ˜ B S , ˜ g t ) + o (1) , (4.254)for S (cid:54) ε t . Using (4.113), (4.154), we get d − j − αt [ D j b ··· b (˜ η ‡ t + ˜ η † t )] C α base ( ˜ B εt , ˜ g t ) = o (1) , (4.255)while (4.188) implies in particular that d − j − αt [ D j (˜ η (cid:53) t + ˜ η (cid:93)t )] C α ( ˜ B R , ˜ g t ) = o (1) , (4.256)and combining (4.252), (4.254), (4.255) and (4.256) completes the proof of (4.251). Killing the contribution from ˜ A t,i,p,k : the main claim (4.267) . Our next goal is to show thatthe first piece of (4.187) goes to zero, namely d − j − αt j (cid:88) i =2 N i (cid:88) p =1 j +2+2 k (cid:88) a = j ε a − j − t | D a ˜ A t,i,p,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D a ˜ A t,i,p,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α = o (1) , (4.257)and to do this we will prove the more precise estimate d − j − αt j (cid:88) i =2 N i (cid:88) p =1 ε a − j − t [ D a ˜ A t,i,p,k ] C α ( ˜ B S , ˜ g t ) = (cid:40) O (1) if S = O ( ε t ) ,o (1) if S = O (1) , (4.258)for all a (cid:62) j , which clearly implies (4.257), where somewhat abusively in the rest of this section thenotation O ( ε t ) for a radius will mean a radius bounded above by R ε t where R < § B ε t ).To start, we wish to employ (4.200)–(4.202) with radius R = d t S, S (cid:54) R ε t (so ( R/λ t ) − α (cid:54) Ce − (1 − α ) t ). To do this, we first use (4.105), (4.111), (4.113), (4.135) and (4.155) to bound the term e − (1 − α ) t a (cid:88) b =0 λ − αt (cid:16) (cid:107) D b ˆ η ‡ t (cid:107) L ∞ ( ˆ B dtS , ˆ g t ) + δ b − at (cid:107) D b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) (cid:107) L ∞ ( ˆ B dtS , ˆ g t ) + δ − t (cid:107) D b ˆ η † t (cid:107) L ∞ ( ˆ B dtS ,g X ) (cid:17) (cid:54) Ce − (1 − α ) t a (cid:88) b =0 λ − αt (cid:16) o (1) + δ b − at (cid:107) D b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) (cid:107) L ∞ ( ˆ B dtS , ˆ g t ) + Ce − − α αj + α t (cid:17) (cid:54) o (1) + Ce − (1 − α ) t λ − αt δ j + α − at + Ce − (1 − α ) t a (cid:88) b = j +1 λ − αt δ b − at (cid:107) D b (ˆ η ♦ t + ˆ η ◦ t + ˆ η t,j,k ) (cid:107) L ∞ ( ˆ B dtS , ˆ g t ) (cid:54) o (1) + Ce − (1 − α ) t λ − αt δ j + α − at + Ce − (1 − α ) t a (cid:88) b = j +1 λ − αt δ b − at (cid:107) D b (ˆ η ♦ t + ˆ η ∗ t + ˆ η (cid:53) t ) (cid:107) L ∞ ( ˆ B dtS , ˆ g t ) , (4.259)where in the last line we used the splitting ˆ η ◦ t + ˆ η t,j,k = ˆ η (cid:52) t + ˆ η (cid:53) t + ˆ η ∗ t + ˆ η (cid:93)t and the estimates (4.189)for ˆ η (cid:52) t and ˆ η (cid:93)t .Using this, we can transfer (4.200)–(4.202) to the tilde picture and multiplying it by δ a − j − t we thenget for all a (cid:62) j, S (cid:54) R ε t ,d − j − αt ε a − j − t [ D a ˜ A t,i,p,k ] C α ( ˜ B S , ˜ g t ) (cid:54) Cε a − jt d − j − αt [ D a b ··· b ˜ η t ] C α base ( ˜ B S , ˜ g t ) + Ce − (1 − α ) t a (cid:88) b = j +1 λ − αt δ b − jt d − bt (cid:107) D b (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) (cid:107) L ∞ ( ˜ B S , ˜ g t ) + o (1)+ C i − (cid:88) (cid:96) =2 N (cid:96) (cid:88) q =1 (cid:32) d − j − αt ε k + a − jt [ D a +2 k +2 ˜ A t,(cid:96),q,k ] C α ( ˜ B S , ˜ g t ) + a +2 k +2 (cid:88) b =0 d − j − αt ε a − j − t d a − b + αt e − (2 k +2+1 − α ) t λ b − a − αt (cid:107) D b ˜ A t,(cid:96),q,k (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:33) , (4.260)where the constants depend on a , so the goal (4.258) is to show that all terms on the RHS of (4.260)are O (1) for S = O ( ε t ) and are o (1) for S = O (1), for all a (cid:62) j . The first useful observation is thatthanks to (4.113), (4.154) and (4.188) for all a (cid:62) j we have ε a − jt d − j − αt [ D a b ··· b ˜ η t ] C α base ( ˜ B O ( εt ) , ˜ g t ) = ε a − jt d − j − αt [ D a b ··· b (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C α base ( ˜ B O ( εt ) , ˜ g t ) + O (1) , (4.261) ans-Joachim Hein and Valentino Tosatti 69 as well as ε a − jt d − j − αt [ D a b ··· b ˜ η t ] C α base ( ˜ B S , ˜ g t ) = ε a − jt d − j − αt [ D a b ··· b (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C α base ( ˜ B S , ˜ g t ) + o (1) , (4.262)for all fixed S . We thus proceed to bound the first term on the RHS of (4.261) and (4.262), and theidea is to use the Schauder estimates in Proposition 2.13. In order to do that we need to check thatafter applying the diffeomorphismsΠ t : B e t × Y → B d − t λ t × Y, (˜ z, ˜ y ) = Π t (ˇ z, ˇ y ) = ( ε t ˇ z, ˇ y ) , (4.263)then on Π − t ( ˜ B ε t × Y ) = ˇ B × Y the metrics ˇ g t = ε − t Π ∗ t ˜ g t and ˇ ω (cid:93)t = ε − t Π ∗ t ˜ ω (cid:93)t are smoothly convergent.This is obvious for ˇ g t and for ˇ ω (cid:93)t we haveˇ ω (cid:93)t = ˇ ω can + Π ∗ t Θ ∗ t Ψ ∗ t ω F + ˇ η † t + ˇ η ‡ t , (4.264)where ˇ ω can + Π ∗ t Θ ∗ t Ψ ∗ t ω F is smoothly convergent, while ˇ η † t + ˇ η ‡ t goes smoothly to zero thanks to (4.113)and (4.153). Thus Proposition 2.13 applies, and from now on we fix R to be the value provided bythat proposition. Given two radii 0 < ρ < R (cid:54) R ε t and α (cid:54) β <
1, let ˜ R = ρ + ( R − ρ ) apply theproposition to bound ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B ρ , ˜ g t ) (cid:54) Cε a − jt d − j − αt [ D a tr ˜ ω (cid:93)t (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B ˜ R , ˜ g t ) + Cε a − jt d − j − αt ( R − ρ ) − a − β (cid:107) ˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t (cid:107) L ∞ ( ˜ B ˜ R , ˜ g t ) (cid:54) Cε a − jt d − j − αt [ D a tr ˜ ω (cid:93)t (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B ˜ R , ˜ g t ) + Cε a − jt ˜ R j + α ( R − ρ ) − a − β (cid:54) Cε a − jt d − j − αt [ D a tr ˜ ω (cid:93)t (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B ˜ R , ˜ g t ) + Cε a + αt ( R − ρ ) − a − β , (4.265)where for the L ∞ term we used the estimates from (4.111), (4.185) and (4.186) which together give d − j − αt | D ι (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) | L ∞ ( ˜ B S , ˜ g t ) (cid:54) CS j + α − ι , d − j − αt [ D ι (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C α ( ˜ B S , ˜ g t ) (cid:54) CS j − ι , (4.266)for 0 (cid:54) ι (cid:54) j and S (cid:54) ε t . The idea is to choose β = α in order to prove the first case in (4.258), and β > α to prove the second case.The main claim is then that for all a (cid:62) j , 0 < ρ < R (cid:54) R ε t and α (cid:54) β <
1, letting ˜ R = ρ + ( R − ρ ),we have the bound Cε a − jt d − j − αt [ D a tr ˜ ω (cid:93)t (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B ˜ R , ˜ g t ) (cid:54) ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + C a (cid:88) r =0 ε α − β + rt ( R − ρ ) − r + C a − (cid:88) r =0 ε α + rt ( R − ρ ) − r − β , (4.267)where the constants depend on a, β , which combined with (4.265) gives ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B ρ , ˜ g t ) (cid:54) ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) C a (cid:88) r =0 ε α − β + rt ( R − ρ ) − r + C a (cid:88) r =0 ε α + rt ( R − ρ ) − r − β (4.268)for all 0 < ρ < R (cid:54) R ε t , and then a standard iteration [19, Lemma 3.4] gives that ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B ρ , ˜ g t ) (cid:54) C a (cid:88) r =0 ε α − β + rt ( R − ρ ) − r + C a (cid:88) r =0 ε α + rt ( R − ρ ) − r − β , (4.269) for all 0 < ρ < R (cid:54) R ε t and all a (cid:62) j . Picking R ∼ ρ ∼ ε t this gives ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B O ( εt ) , ˜ g t ) (cid:54) Cε α − βt . (4.270)When β = α , we insert this into (4.261) and get ε a − jt d − j − αt [ D a b ··· b ˜ η t ] C α base ( ˜ B O ( εt ) , ˜ g t ) = O (1) , (4.271)which bounds one of the terms in (4.260). On the other hand, when β > α , then the bound in (4.270)implies in particular an o (1) bound for the C β seminorm on ˜ B S for any fixed S , and hence also an o (1)bound for the C α seminorm on ˜ B S , and inserting this into (4.262) gives ε a − jt d − j − αt [ D a b ··· b ˜ η t ] C α base ( ˜ B S , ˜ g t ) = o (1) . (4.272)To prove the claim (4.267), we will argue by induction on a (cid:62) j . Note that once (4.267) is provedfor some value of a , then the above argument shows that (4.270) also holds for this value of a , andusing Proposition 2.8 and (2.47) to interpolate between this and the L ∞ bound in (4.266) (on balls ofradius comparable to ε t ) gives ε b − j − αt d − j − αt | D b (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) | L ∞ ( ˜ B O ( εt ) , ˜ g t ) (cid:54) C, ε b − j + β − αt d − j − αt [ D b (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B O ( εt ) , ˜ g t ) (cid:54) C, (4.273)for 0 (cid:54) b (cid:54) a and α (cid:54) β <
1. These bounds (4.273) for smaller values of a are all that we will usefrom the induction hypothesis. Observe also that, thanks to (4.266), these bounds (4.273) are alreadyknown to hold in the base case of the induction when a = j and β = α , but at this point they have notyet been established when a = j and β > α .Now, to prove claim (4.267) we need to convert [ D a tr ˜ ω (cid:93)t (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β into [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β with a small coefficient in front, and this is where the PDE (4.168) comes in, which in the tilde picturecan be written (ignoring combinatorial factors) as d − j − αt tr ˜ ω (cid:93)t (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) = d − j − αt (cid:32) c t ˜ ω m can ∧ (Θ ∗ t Ψ ∗ t ω F ) n (˜ ω (cid:93)t ) m + n − (cid:33) − d − j − αt tr ˜ ω (cid:93)t (˜ η (cid:93)t + ˜ η (cid:52) t ) (4.274) − d − j − αt (cid:88) i + p + (cid:96) (cid:62) (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) i ∧ (˜ η (cid:93)t ) p ∧ (˜ η (cid:52) t ) (cid:96) ∧ (˜ ω (cid:93)t ) m + n − i − p − (cid:96) (˜ ω (cid:93)t ) m + n . We then multiply (4.274) by ε a − jt , a (cid:62) j, and we need to see what bounds are available for the pieceson the RHS of (4.274).4.10.5. The piece of (4.274) with d − j − αt tr ˜ ω (cid:93)t (˜ η (cid:93)t + ˜ η (cid:52) t ) . First, from (4.188) and (4.189) we have d − j − αt [ D ι ˜ η (cid:93)t ] C β ( ˜ B S , ˜ g t ) (cid:54) Cε α − j − ιt S − β , d − j − αt (cid:107) D ι ˜ η (cid:93)t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cε j − ι + αt , (4.275) d − j − αt [ D ι ˜ η (cid:52) t ] C β ( ˜ B S , ˜ g t ) (cid:54) Cε α − j − ιt S − β , d − j − αt (cid:107) D ι ˜ η (cid:52) t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cε j − ι + αt , (4.276)for ι (cid:62) S (cid:54) ε t , while (4.161) gives (cid:107) D ι ˜ ω (cid:93)t (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cε − ιt , [ D ι ˜ ω (cid:93)t ] C β ( ˜ B S , ˜ g t ) (cid:54) Cε − ι − βt , (4.277)for ι (cid:62) S (cid:54) ε t . Using (4.275), (4.276) and (4.277) we conclude that for all a (cid:62) > β (cid:62) α we have ε a − jt d − j − αt | D a tr ˜ ω (cid:93)t (˜ η (cid:93)t + ˜ η (cid:52) t ) | L ∞ ( ˜ B εt , ˜ g t ) (cid:54) Cε αt , ε a − jt d − j − αt [ D a tr ˜ ω (cid:93)t (˜ η (cid:93)t + ˜ η (cid:52) t )] C β ( ˜ B εt , ˜ g t ) (cid:54) Cε α − βt . (4.278) ans-Joachim Hein and Valentino Tosatti 71 The piece of (4.274) with the nonlinearities.
This is the term N := Cε a − jt d − j − αt (cid:88) i + p + (cid:96) (cid:62) (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) i ∧ (˜ η (cid:93)t ) p ∧ (˜ η (cid:52) t ) (cid:96) ∧ (˜ ω (cid:93)t ) m + n − i − p − (cid:96) (˜ ω (cid:93)t ) m + n . (4.279)For any a (cid:62)
0, we take D a of (4.279) and expand it schematically as C (cid:88) i + p + (cid:96) (cid:62) µ + ν + φ + ψ + σ = a ( D µ ( ε a − jt d − j − αt (˜ η ∗ t +˜ η ♦ t +˜ η (cid:53) t ) i ))( D ν ((˜ η (cid:93)t ) p ))( D φ ((˜ η (cid:52) t ) (cid:96) ))( D ψ ((˜ ω (cid:93)t ) m + n − i − p − (cid:96) ))( D σ (((˜ ω (cid:93)t ) m + n ) − )) , (4.280)To bound this, first observe that from (4.275), (4.276), (4.277) we can bound for all ν, φ, ψ (cid:62) D ν ˜ η (cid:93)t ] C β ( ˜ B εt , ˜ g t ) (cid:54) Cd j + αt ε α − β + j − νt = Cδ j + αt ε − β − νt , | D ν ˜ η (cid:93)t | L ∞ ( ˜ B εt , ˜ g t ) (cid:54) Cδ j + αt ε − νt , (4.281)[ D φ ˜ η (cid:52) t ] C β ( ˜ B εt , ˜ g t ) (cid:54) Cδ j + αt ε − β − φt , | D φ ˜ η (cid:52) t | L ∞ ( ˜ B εt , ˜ g t ) (cid:54) Cδ j + αt ε − φt , (4.282)[ D ψ ˜ ω (cid:93)t ] C β ( ˜ B εt , ˜ g t ) (cid:54) Cε − β − ψt , | D ψ ˜ ω (cid:93)t | L ∞ ( ˜ B εt , ˜ g t ) (cid:54) Cε − ψt , (4.283)and similarly for the reciprocal of (˜ ω (cid:93)t ) m + n . As for ˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t , we claim that for α (cid:54) β < (cid:54) µ < a we have | D µ (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) | L ∞ ( ˜ B O ( εt ) , ˜ g t ) (cid:54) Cδ j + αt ε − µt , [ D µ (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B O ( εt ) , ˜ g t ) (cid:54) Cδ j + αt ε − β − µt , (4.284)and indeed in the base case of the induction a = j these are simple consequences of (4.266) plusinterpolation (Proposition 2.8 and (2.47)), while for a > j these are given by the bounds (4.273) whichhold by induction.First we will consider the case when in (4.280) we have a (cid:54) j . Using (4.266), (4.281), (4.282),(4.283), we obtain the bounds | D a N | L ∞ ( ˜ B εt , ˜ g t ) (cid:54) Cε αt δ j + αt , [ D a N ] C α ( ˜ B εt , ˜ g t ) (cid:54) Cδ j + αt . (4.285)Next, we take a (cid:62) j in (4.280) and we take [ D a N ] C β ( ˜ B ˜ R , ˜ g t ) , where ˜ R = O ( ε t ) is as above, and1 > β (cid:62) α . The seminorm will distribute onto all the terms in the obvious sense. Let us first considerany term with a D a derivative of (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) on which the difference quotient lands, which is thenmultiplied by at least one other decorated ˜ η piece (since i + p + (cid:96) (cid:62)
2) which is o (1) by (4.281), (4.282),(4.284), and thus the sum of all such terms is bounded above by14 ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) . (4.286)Next, consider any term where (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) i is completely undifferentiated. These terms are allbounded by Cε α − βt by (4.281), (4.282), (4.283), (4.284).Lastly, consider any term which is not already covered, i.e. which contains the factor ε a − jt d − j − αt (˜ η ∗ t +˜ η ♦ t + ˜ η (cid:53) t ) differentiated some number of times, either of the form r with 0 < r (cid:54) a , or of the form r + β with 0 (cid:54) r < a . If there is more than one factor of (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) which is differentiated, we selecthere the one with the largest number of derivatives.Each such term is thus the product of two parts, the first part being what we just discussed, andthe second part consisting of a total of a + β − r (resp. a − r ) derivatives landing on the product of theremaining terms (and these terms contain at least one other decorated ˜ η piece since i + p + (cid:96) (cid:62) η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )term inside this piece will be differentiated strictly less than a times), the second piece can be boundedby o ( ε − a − β + rt ) (resp. o ( ε − a + rt )). As for the first piece, assume first that it contains r derivatives, and interpolate with Proposition2.8 from radius ˜ R = ρ + ( R − ρ ) to radius Rε a − jt d − j − αt (cid:107) D r (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) (cid:107) L ∞ ( ˜ B ˜ R , ˜ g t ) (cid:54) Cε a − jt d − j − αt ( R − ρ ) a + β − r [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + Cε a − jt d − j − αt ( R − ρ ) − r (cid:107) ˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t (cid:107) L ∞ ( ˜ B R , ˜ g t ) (cid:54) Cε a − jt d − j − αt ε a + β − rt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + Cε a − jt d − j − αt ( R − ρ ) − r δ j + αt , (4.287)so that in this case the sum of all such terms overall can be bounded by o (1) Cε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + o (1) ε − a − β + rt Cε a − jt d − j − αt ( R − ρ ) − r δ j + αt (cid:54) ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + o (1) ε α − β + rt ( R − ρ ) − r . (4.288)In the second case when the first piece contains r + β derivatives, we interpolate similarly using (2.47) ε a − jt d − j − αt [ D r (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B ˜ R , ˜ g t ) (cid:54) Cε a − jt d − j − αt ( R − ρ ) a − r [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + Cε a − jt d − j − αt ( R − ρ ) − r − β (cid:107) ˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t (cid:107) L ∞ ( ˜ B R , ˜ g t ) (cid:54) Cε a − jt d − j − αt ε a − rt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + Cε a − jt d − j − αt ( R − ρ ) − r − β δ j + αt , (4.289)so that in this case the sum of all such terms overall can be bounded by o (1) Cε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + o (1) ε − a + rt Cε a − jt d − j − αt ( R − ρ ) − r − β δ j + αt (cid:54) ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + o (1) ε α + rt ( R − ρ ) − r − β . (4.290)We have thus covered all possible terms, and the conclusion is that for all a (cid:62) j, > β (cid:62) α we have[ D a N ] C β ( ˜ B ˜ R , ˜ g t ) (cid:54) ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B R , ˜ g t ) + Cε α − βt + o (1) a (cid:88) r =1 ε α − β + rt ( R − ρ ) − r + o (1) a − (cid:88) r =0 ε α + rt ( R − ρ ) − r − β . (4.291)4.10.7. The piece of (4.274) with c t ˜ ω m can ∧ (Θ ∗ t Ψ ∗ t ω F ) n (˜ ω (cid:93)t ) m + n − . The claim about this is that for all a (cid:62) > β (cid:62) α and all fixed R we have ε a − jt d − j − αt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D a (cid:32) c t ˜ ω m can ∧ (Θ ∗ t Ψ ∗ t ω F ) n (˜ ω (cid:93)t ) m + n − (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ( ˜ B Rεt × Y, ˜ g t ) (cid:54) Cε αt , (4.292) ε a − jt d − j − αt (cid:34) D a (cid:32) c t ˜ ω m can ∧ (Θ ∗ t Ψ ∗ t ω F ) n (˜ ω (cid:93)t ) m + n − (cid:33)(cid:35) C β ( ˜ B Rεt × Y, ˜ g t ) (cid:54) Cε α − βt , where as usual the constants depend on a, β , and note that here we are working on the product ofa ball in the base (centered at ˜ z t ) times the whole fiber Y (not just the small coordinate chart from § a (cid:54) j and β = α follows immediately from (4.173), (4.174).Next, to prove (4.292) for a (cid:62) j and 1 > β (cid:62) α , we apply the diffeomorphismΠ t : B e t × Y → B d − t λ t × Y, (˜ z, ˜ y ) = Π t (ˇ z, ˇ y ) = ( ε t ˇ z, ˇ y ) , (4.293) ans-Joachim Hein and Valentino Tosatti 73 and as usual multiply all the pulled back contravariant 2-tensors by ε − t , and denote the new objectsby a check. This way Π − t ( ˜ B Rε t × Y ) = ˇ B R × Y , the metric ˇ g t is uniformly Euclidean, and we have bydefinition Ψ t ◦ Θ t ◦ Π t = Σ t , where Σ t was defined in (3.9).Transferring to the check picture gives δ − j − αt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D a (cid:32) c t ˇ ω m can ∧ (Σ ∗ t ω F ) n (ˇ ω (cid:93)t ) m + n − (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ( ˇ B R × Y, ˇ g t ) = ε − αt ε a − jt d − j − αt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D a (cid:32) c t ˜ ω m can ∧ (Θ ∗ t Ψ ∗ t ω F ) n (˜ ω (cid:93)t ) m + n − (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ( ˜ B Rεt × Y, ˜ g t ) , (4.294) δ − j − αt (cid:34) D a (cid:32) c t ˇ ω m can ∧ (Σ ∗ t ω F ) n (ˇ ω (cid:93)t ) m + n − (cid:33)(cid:35) C β ( ˇ B R × Y, ˇ g t ) = ε β − αt ε a − jt d − j − αt (cid:34) D a (cid:32) c t ˜ ω m can ∧ (Θ ∗ t Ψ ∗ t ω F ) n (˜ ω (cid:93)t ) m + n − (cid:33)(cid:35) C β ( ˜ B Rεt × Y, ˜ g t ) . (4.295)It then follows from (4.292) with a (cid:54) j, β = α that δ − j − αt (cid:18) c t ˇ ω m can ∧ (Σ ∗ t ω F ) n (ˇ ω (cid:93)t ) m + n − (cid:19) is locally uniformlybounded in C j,α for the (essentially fixed) metric ˇ g t , so by Ascoli-Arzel`a up to passing to sequence itconverges locally uniformly on C m × Y . We can thus apply the last statement in Theorem 3.10 whichimplies that δ − j − αt (cid:18) c t ˇ ω m can ∧ (Σ ∗ t ω F ) n (ˇ ω (cid:93)t ) m + n − (cid:19) converges locally smoothly, so in particular all of its derivativesare locally uniformly bounded on ˇ B R . Thus the LHS of (4.294) and (4.295) with are uniformly boundedfor all a, β and R , and this completes the proof of (4.292).4.10.8. Completion of the proof of the main claim (4.267) . Combining the discussion of the 3 pieces insections 4.10.5, 4.10.6 and 4.10.7, namely combining (4.278), (4.291) and (4.292) with (4.274), completesthe proof of (4.267) and hence of (4.271) and (4.272). This means that the first term on the RHS of(4.260) is O (1) when we take S = O ( ε t ) and o (1) when we take S = O (1). We discuss the other 3terms on the RHS of (4.260) separately.4.10.9. The second term of (4.260) . This is the term e − (1 − α ) t a (cid:88) b = j +1 λ − αt δ b − jt d − bt (cid:107) D b (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) (cid:107) L ∞ ( ˜ B O ( εt ) , ˜ g t ) , (4.296)where a (cid:62) j . To prove that this is term is o (1), first recall that from (4.270) we have in particular ε a − jt d − j − αt [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C α ( ˜ B O ( εt ) , ˜ g t ) (cid:54) C, (4.297)and we then interpolate between this and the L ∞ bound in (4.266) using Proposition 2.8( R − ρ ) b (cid:107) D b (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) (cid:107) L ∞ ( ˜ B ρ , ˜ g t ) (cid:54) C ( R − ρ ) a + α [ D a (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C α ( ˜ B R , ˜ g t ) + C (cid:107) ˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t (cid:107) L ∞ ( ˜ B R , ˜ g t ) (cid:54) C ( R − ρ ) a + α d j + αt ε j − at + Cd j + αt R j + α , (4.298)and taking R ∼ ρ ∼ ε t we get (cid:107) D b (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) (cid:107) L ∞ ( ˜ B O ( εt ) , ˜ g t ) (cid:54) Cε j − b + αt d j + αt , (4.299) and so each term in the sum in (4.296) is bounded by Ce − (1 − α ) t δ b − jt λ − αt d − bt ε j − b + αt d j + αt = Ce − (1 − α ) t δ αt λ − αt = o (1) , (4.300)which establishes that e − (1 − α ) t a (cid:88) b = j +1 λ − αt δ b − jt d − bt (cid:107) D b (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t ) (cid:107) L ∞ ( ˜ B O ( εt ) , ˜ g t ) = o (1) , (4.301)for all a (cid:62) j, as desired.4.10.10. The third term of (4.260) . This term is of the same nature as the LHS of (4.260), but with (cid:96) < i so well-suited for an inductive argument on i , which appears below.4.10.11. The fourth term of (4.260) and putting it all together.
To recap what we have achieved sofar, combining (4.260) with the bounds (4.271) and (4.272) for the first term on the RHS and those in(4.301) for the second term, we obtain for a (cid:62) jd − j − αt ε a − j − t [ D a ˜ A t,i,p,k ] C α ( ˜ B S , ˜ g t ) (cid:54) (cid:40) O (1) if S = O ( ε t ) ,o (1) if S = O (1) , + C i − (cid:88) (cid:96) =2 N (cid:96) (cid:88) q =1 (cid:32) d − j − αt ε k + a − jt [ D a +2 k +2 ˜ A t,(cid:96),q,k ] C α ( ˜ B S , ˜ g t ) + a +2 k +2 (cid:88) b =0 d − j − αt ε a − j − t d a − b + αt e − (2 k +2+1 − α ) t λ b − a − αt (cid:107) D b ˜ A t,(cid:96),q,k (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:33) , (4.302)and we now use this to prove (4.258) by showing by induction on 2 (cid:54) i (cid:54) j that for all a (cid:62) j we have d − j − αt N i (cid:88) p =1 ε a − j − t [ D a ˜ A t,i,p,k ] C α ( ˜ B S , ˜ g t ) = (cid:40) O (1) if S = O ( ε t ) ,o (1) if S = O (1) . (4.303)The base of the induction i = 2 is exactly (4.302). For the induction step, we assume that (4.303)holds up to i −
1, then the second line of (4.302) can also be bounded by O (1) if S = O ( ε t ) and by o (1) if S = O (1). As for the third line of (4.302), first consider the terms with 0 (cid:54) b (cid:54) j . For these,we use directly (4.59) which gives in particular d − bt (cid:107) D b ˜ A t,(cid:96),q,k (cid:107) L ∞ ( ˜ B εt , ˜ g t ) = o ( ε t ) and so we can boundthese terms by d − j − αt ε a − j − t d a + αt e − (2 k +2+1 − α ) t λ b − a − αt ε t o (1) = δ a − jt e − (2 k +2+1 − α ) t λ b − a − αt o (1) = o (1) , (4.304)since b (cid:54) j (cid:54) a . Next, look at the terms with j < b (cid:54) a + 2 k + 2. For these we use interpolationbetween d − jt (cid:107) D j ˜ A t,(cid:96),q,k (cid:107) L ∞ ( ˜ B εt , ˜ g t ) = o ( ε t ) , (4.305)which is provided by (4.59), and d − j − αt ε b − j − t [ D b ˜ A t,(cid:96),q,k ] C α ( ˜ B O ( εt ) , ˜ g t ) = O (1) , (4.306)for b (cid:62) j and (cid:96) (cid:54) i −
1, which comes from the induction hypothesis (4.303) (only the first case, when S = O ( ε t )). Interpolating gives, for j < b (cid:54) a + 2 k + 2,( R − ρ ) b (cid:107) D b ˜ A t,(cid:96),q,k (cid:107) L ∞ ( ˜ B ρ , ˜ g t ) (cid:54) C ( R − ρ ) b + α [ D b ˜ A t,(cid:96),q,k ] C α ( ˜ B R , ˜ g t ) + C ( R − ρ ) j (cid:107) D j ˜ A t,(cid:96),q,k (cid:107) L ∞ ( ˜ B R , ˜ g t ) (cid:54) C ( R − ρ ) b + α d j + αt ε j − bt + C ( R − ρ ) j d jt ε t o (1) , (4.307) ans-Joachim Hein and Valentino Tosatti 75 and taking R ∼ ρ ∼ ε t we get (cid:107) D b ˜ A t,(cid:96),q,k (cid:107) L ∞ ( ˜ B O ( εt ) , ˜ g t ) (cid:54) Cε j + α − bt d j + αt + Cε j − bt d jt , (4.308)and so the terms in the third line of (4.302) with b > j can be bounded by d − j − αt ε a − j − t d a − b + αt e − (2 k +2+1 − α ) t λ b − a − αt ε j + α − bt d j + αt + d − j − αt ε a − j − t d a − b + αt e − (2 k +2+1 − α ) t λ b − a − αt ε j − bt d jt = δ a − b + αt e − (2 k +2+1 − α ) t λ b − a − αt + δ a − bt e − (2 k +2+1 − α ) t λ b − a − αt = e − ( a − b + α +2 k +2+1 − α ) t + e − ( a − b +2 k +2+1 − α ) t λ − αt = o (1) , (4.309)since b (cid:54) a + 2 k + 2. A moment’s thought reveals that this completes the inductive proof of (4.303)(both cases), and hence this also completes the proof of (4.258) and of (4.257).4.10.12. Killing the contribution from ˜ η ♦ t . We claim that for each fixed S we have d − j − αt [ D j ˜ η ♦ t ] C α ( ˜ B S , ˜ g t ) = o (1) , (4.310)which in particular shows that another piece of (4.187) goes to zero. This is now straightforward: take(4.270) with a = j and β > αd − j − αt [ D j (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C β ( ˜ B O ( εt ) , ˜ g t ) = o (1) , (4.311)and hence the same o (1) bound for the C β seminorm on ˜ B S , and hence also for the C α seminorm on˜ B S , namely d − j − αt [ D j (˜ η ∗ t + ˜ η ♦ t + ˜ η (cid:53) t )] C α ( ˜ B S , ˜ g t ) = o (1) , (4.312)Plugging this into (4.251) then proves (4.310).4.10.13. Killing the contribution from ˜ η ∗ t . Lastly, in order to obtain a contradiction we first claim thatfor each fixed S we have d − j − αt [ D j ˜ η ◦ t ] C α ( ˜ B S , ˜ g t ) = o (1) . (4.313)To see this, recall that from Lemma 3.7 we have˜ η ◦ t = i∂∂ j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t ε ιt ( ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D ι ˜ A ∗ t,i,p,k ) , (4.314)and the bounds from (4.116), (4.117), (4.120) d − ι +2 t (cid:107) D ι ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B εt , ˜ g t ) (cid:54) Cδ j +2+ α − ιt , d − ι +2 − αt [ D ι ˜ A ∗ t,i,p,k ] C α ( ˜ B εt , ˜ g t ) (cid:54) Cδ j +2 − ιt , (4.315)for 0 (cid:54) ι (cid:54) j + 2 + 2 k , together with (4.258) which gives d − ι +2 − αt [ D ι ˜ A ∗ t,i,p,k ] C α ( ˜ B S , ˜ g t ) = o ( δ j +2 − ιt ) . (4.316)for j (cid:54) ι (cid:54) j + 2 + 2 k . Also, simply taking (4.116) with S fixed gives d − ι +2 − αt [ D ι ˜ A ∗ t,i,p,k ] C α ( ˜ B S , ˜ g t ) (cid:54) Cd j − ιt δ t S j − ι = Cδ j +2 − ιt ε − j + ιt S j − ι = o ( δ j +2 − ιt ) , (4.317)for 0 (cid:54) ι < j . On the other hand, for the functions ˜Φ := ˜Φ ι,(cid:96) ( ˜ G i,p ) we have the estimates (cid:107) D ι ˜Φ (cid:107) L ∞ ( ˜ B εt , ˜ g t ) (cid:54) Cε − ιt , (4.318)[ D ι ˜Φ] C α ( ˜ B S , ˜ g t ) (cid:54) CS − α | D ι +1 ˜Φ | L ∞ ( ˜ B S , ˜ g t ) (cid:54) CS − α ε − ι − t = o ( ε − ι − αt ) , (4.319) for all ι (cid:62)
0, 0 < α < S fixed. Now we can use all the estimates (4.315), (4.316), (4.317), (4.318)and (4.319), to prove (4.313) by arguing like in the proof of (4.135) as follows: write schematically D j +2 (cid:16) ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D ι ˜ A ∗ t,i,p,k (cid:17) = (cid:88) i + i = j +2 D i ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D i + ι ˜ A ∗ t,i,p,k , (4.320)and estimate d − j − αt ε ιt [ D i ˜Φ ι,(cid:96) ( ˜ G i,p ) (cid:126) D i + ι ˜ A ∗ t,i,p,k ] C α ( ˜ B S , ˜ g t ) (cid:54) Cd − j − αt ε ιt o ( ε − i − αt ) d i + ι − t δ j +2+ α − i − ιt + Cd − j − αt ε ιt ε − i t d i + ι − αt o ( δ j +2 − i − ιt )= o (1) , (4.321)and combining (4.314), (4.320) and (4.321) proves (4.313).Now that (4.313) is established, recall that (4.188) gives d − j − αt [ D j ˜ η (cid:52) t ] C α ( ˜ B S , ˜ g t ) (cid:54) Cε α − t S − α , (4.322)and combining these two gives d − j − αt [ D j ˜ η (cid:53) t ] C α ( ˜ B S , ˜ g t ) = o (1) . (4.323)But recall now (4.310) and (4.312), which together give d − j − αt [ D j (˜ η ∗ t + ˜ η (cid:53) t )] C α ( ˜ B S , ˜ g t ) = o (1) . (4.324)Combining (4.323) and (4.324) gives d − j − αt [ D j ˜ η ∗ t ] C α ( ˜ B S , ˜ g t ) = o (1) , (4.325)which finally together with (4.257) and (4.310) gives an immediate contradiction to (4.187).4.11. Subcase B: ε t → (without loss). Thanks to (4.106), (4.111) and (4.135) together withRemark 2.6 (using also Remark 2.7 in order to compare the mildly varying C j,α loc topologies) we are nowable to say that d − j − αt ˜ η t,j,k , d − j − αt ˜ η ♦ t and d − j − αt ˜ η ◦ t converge in the topology of C j,β loc ( C m × Y ) for every β < α to limiting 2-forms ˜ η ∞ ,j,k , ˜ η ♦∞ and ˜ η ◦∞ in C j,α loc ( C m × Y ), which are O ( r j + α ) at infinity, whichare weakly closed (as a locally uniform limit of smooth closed forms) and of type (1 ,
1) with respectto J C m + J Y,z ∞ . We may clearly assume that ˜ g (cid:93)t → g can (0) + g Y,z ∞ locally smoothly, where g can (0) is aconstant K¨ahler metric on C m , which we will rename simply g C m . As in [19, Proposition 3.11], all ofthese limiting forms are i∂∂ -exact on C m × Y .Thanks to (4.121), the functions d − j − αt ˜ A ∗ t,i,p,k converge in C j +2+2 k,β loc ( C m × Y ) to limiting functions˜ A ∗∞ ,i,p,k ( z ) from C m , while ˜ G i,p converge locally smoothly to functions ˜ G ∞ ,i,p ( y ) pulled back from Y ,and since ˜ η ◦ t = j (cid:88) i =2 N i (cid:88) p =1 i∂∂ ˜ G t,k ( ˜ A ∗ t,i,p,k , ˜ G i,p ) (4.326)it follows that d − j − αt ˜ η ◦ t → ˜ η ◦∞ := j (cid:88) i =2 N i (cid:88) p =1 i∂∂ ˜ G ∞ ,k ( ˜ A ∗∞ ,i,p,k , ˜ G ∞ ,i,p ) , (4.327)at least in the topology of C ,β loc ( C m × Y ), where˜ G ∞ ,k ( A, G ) = k (cid:88) (cid:96) =0 ( − (cid:96) (∆ C m ) (cid:96) A · (∆ Y ) − (cid:96) − G, (4.328)thanks to (3.18). In fact we will not use this explicit formula, but only the obvious fact that ˜ G ∞ ,k ( A, G )is bilinear, and in particular it vanishes when A ≡ ans-Joachim Hein and Valentino Tosatti 77 Now passing to the limit in (4.172) implies that we can writetr ω C m + ω Y,z ∞ (˜ η ∞ ,j,k + ˜ η ♦∞ + ˜ η ◦∞ ) = K ( z ) + (cid:88) q K q ( z ) H q ( y ) , (4.329)where K ( z ) , K q ( z ) are polynomials of degree at most j , and H q ( y ) are functions pulled back from thefiber Y with mean value zero. Furthermore, thanks to (4.168) and (4.175), we see that the C limitof d − j − αt (cid:32) c t ˜ ω m can ∧ ( ε t Θ ∗ t Ψ ∗ t ω F ) n (˜ ω (cid:93)t ) m + n − (cid:33) , (4.330)exists and is equal to the limit of d − j − αt tr ˜ ω (cid:93)t (˜ η t,j,k + ˜ η ♦ t + ˜ η ◦ t ), which is given by (4.329). We canthen apply Theorem 3.10 and see that the functions H q ( y ) in (4.329) lie in the span of the functions˜ G ∞ ,i,p , (cid:54) i (cid:54) j, (cid:54) p (cid:54) N i .Let us now go back to the definition in (4.9) d − j − αt ˜ A t,i,p,k = d − j − αt ˜ P t,i,p (˜ η t,i − ,k ) (4.331)where recall that˜ P t,i,p ( α ) = n (pr B ) ∗ ( ˜ G i,p α ∧ Θ ∗ t Ψ ∗ t ω n − F ) + ε t tr ˜ ω can (pr B ) ∗ ( ˜ G i,p α ∧ Θ ∗ t Ψ ∗ t ω nF ) . (4.332)By definition, we can write˜ η t,i − ,k − (˜ η t,j,k + ˜ η ♦ t + ˜ η ◦ t ) = ˜ η † t + ˜ η ‡ t − ˜ γ t, − i − (cid:88) q =2 ˜ γ t,q,k , (4.333)and since ˜ P t,i,p annihilates any (1 ,
1) form from the base by (3.6), we have d − j − αt ˜ A t,i,p,k = d − j − αt ˜ P t,i,p (˜ η t,j,k + ˜ η ♦ t + ˜ η ◦ t ) + d − j − αt ˜ P t,i,p (˜ η † t ) − d − j − αt i − (cid:88) q =2 ˜ P t,i,p (˜ γ t,q,k ) . (4.334)Now recall that ˜ η † t = j (cid:88) a =2 N a (cid:88) b =1 i∂∂ ˜ G t,k ( ˜ A (cid:93)t,a,b,k , ˜ G a,b ) , (4.335)˜ γ t,q,k = N q (cid:88) c =1 i∂∂ ˜ G t,k ( ˜ A t,q,c,k , ˜ G q,c ) , (4.336)we can employ Lemma 3.6 and get j (cid:88) a =2 N a (cid:88) b =1 ˜ P t,i,p ( i∂∂ ( ˜ G t,k ( ˜ A (cid:93)t,a,b,k , ˜ G a,b ))) = j (cid:88) a =2 N a (cid:88) b =1 ˜ A (cid:93)t,a,b,k (cid:90) { z }× Y ˜ G i,p ˜ G a,b Θ ∗ t Ψ ∗ t ω nF + j (cid:88) a =2 N a (cid:88) b =1 j (cid:88) ι =0 e − (2 k +2 − ι ) t ˜Φ ι,i,p,k ( ˜ G a,b ) (cid:126) D ι ˜ A (cid:93)t,a,b,k = ˜ A (cid:93)t,i,p,k + j (cid:88) a =2 N a (cid:88) b =1 j (cid:88) ι =0 e − (2 k +2 − ι ) t ˜Φ ι,i,p,k ( ˜ G a,b ) (cid:126) D ι ˜ A (cid:93)t,a,b,k , (4.337)using here crucially that ˜ A (cid:93)t,a,b,k is a polynomial of degree at most j , and that the ˜ G i,p are fiberwiseorthonormal. From (4.127) we can in particular crudely bound d − j − αt e − (2 k +2 − ι ) t (cid:13)(cid:13)(cid:13) ˜Φ ι,i,p,k ( ˜ G a,b ) (cid:126) D ι ˜ A (cid:93)t,a,b,k (cid:13)(cid:13)(cid:13) L ∞ ( ˜ B S , ˜ g t ) (cid:54) Cd − j − αt e − (2 k +2 − j ) t = o (1) , (4.338) for any fixed S , since by assumption j (cid:54) k . On the other hand, since q < i , we have N q (cid:88) c =1 ˜ P t,i,p ( i∂∂ ( ˜ G t,k ( ˜ A t,q,c,k , ˜ G q,c ))) = N q (cid:88) c =1 2 k +2 (cid:88) ι =0 e − (2 k +2 − ι ) t ˜Φ ι,i,p,k ( ˜ G q,c ) (cid:126) D ι ˜ A t,q,c,k = N q (cid:88) c =1 j (cid:88) ι =0 e − (2 k +2 − ι ) t ˜Φ ι,i,p,k ( ˜ G q,c ) (cid:126) D ι ˜ A (cid:93)t,q,c,k + N q (cid:88) c =1 2 k +2 (cid:88) ι =0 e − (2 k +2 − ι ) t ˜Φ ι,i,p,k ( ˜ G q,c ) (cid:126) D ι ˜ A ∗ t,q,c,k , (4.339)and we can argue exactly as above for the terms with ˜ A (cid:93)t,q,c,k , while for the terms with ˜ A ∗ t,q,c,k we use(4.121) which gives d − j − αt (cid:107) D ι ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B S , ˜ g t ) (cid:54) C for all 0 (cid:54) ι (cid:54) j + 2 k and fixed S , and we seethat d − j − αt N q (cid:88) c =1 ˜ P t,i,p ( i∂∂ ( ˜ G t,k ( ˜ A t,q,c,k , ˜ G q,c ))) = d − j − αt N q (cid:88) c =1 ˜Φ k +2 ,i,p,k ( ˜ G q,c ) (cid:126) D k +2 ˜ A ∗ t,q,c,k + o (1) , (4.340)locally uniformly, and combining (4.334), (4.337), (4.338) and (4.340) we obtain d − j − αt ˜ A t,i,p,k = d − j − αt ˜ P t,i,p (˜ η t,j,k + ˜ η ♦ t + ˜ η ◦ t ) + d − j − αt ˜ A (cid:93)t,i,p,k − d − j − αt i − (cid:88) q =2 N q (cid:88) c =1 ˜Φ k +2 ,i,p,k ( ˜ G q,c ) (cid:126) D k +2 ˜ A ∗ t,q,c,k + o (1) . (4.341)We now want to pass (4.341) to the limit as t → ∞ (in the C topology say). As mentionedearlier, we have d − j − αt ˜ A ∗ t,i,p,k → ˜ A ∗∞ ,i,p,k , d − j − αt ˜ η ◦ t → ˜ η ◦∞ , d − j − αt ˜ η ♦ t → ˜ η ♦∞ , d − j − αt (˜ η t,j,k ) → ˜ η ∞ ,j,k and˜ G i,p → ˜ G ∞ ,i,p , so that d − j − αt ˜ P t,i,p (˜ η t,j,k + ˜ η ♦ t + ˜ η ◦ t ) converges to (cid:90) { z ∞ }× Y ˜ G ∞ ,i,p tr ω Y,z ∞ (˜ η ∞ ,j,k + ˜ η ♦∞ + ˜ η ◦∞ ) ω nY,z ∞ + tr ω C m (pr B ) ∗ ( ˜ G ∞ ,i,p (˜ η ∞ ,j,k + ˜ η ♦∞ + ˜ η ◦∞ ) ∧ ω nY,z ∞ )= (cid:90) { z ∞ }× Y ˜ G ∞ ,i,p tr ω Y,z ∞ (˜ η ∞ ,j,k + ˜ η ♦∞ + ˜ η ◦∞ ) ω nY,z ∞ + (cid:90) { z ∞ }× Y ˜ G ∞ ,i,p tr ω C m (˜ η ∞ ,j,k + ˜ η ♦∞ + ˜ η ◦∞ ) ω nY,z ∞ = (cid:90) { z ∞ }× Y ˜ G ∞ ,i,p tr ω C m + ω Y,z ∞ (˜ η ∞ ,j,k + ˜ η ♦∞ + ˜ η ◦∞ ) ω nY,z ∞ , (4.342)and so (4.341) limits to˜ A ∗∞ ,i,p,k = (cid:90) { z ∞ }× Y ˜ G ∞ ,i,p tr ω C m + ω Y,z ∞ (˜ η ∞ ,j,k + ˜ η ♦∞ + ˜ η ◦∞ ) ω nY,z ∞ − i − (cid:88) q =2 N q (cid:88) c =1 ˜Φ k +2 ,i,p,k ( ˜ G ∞ ,q,c ) (cid:126) D k +2 ˜ A ∗∞ ,q,c,k . (4.343)We can then plug in (4.329) and we see that˜ A ∗∞ ,i,p,k = Q i,p ( z ) − i − (cid:88) q =2 N q (cid:88) c =1 ˜Φ k +2 ,i,p,k ( ˜ G ∞ ,q,c ) (cid:126) D k +2 ˜ A ∗∞ ,q,c,k , (4.344)where Q i,p ( z ) is a polynomial on C m of degree at most j , and using this we can show by induction on i that ˜ A ∗∞ ,i,p,k = 0 for all i, p . Indeed, in the base case of the induction i = 2 the last term in (4.344)is not present, and so ˜ A ∗∞ , ,p,k is a polynomial of degree at most j , but since it also has vanishing j -jet ans-Joachim Hein and Valentino Tosatti 79 at origin (recall that we have translated the C m factor so that ˜ x t = (0 , ˜ y t )), it must be identically zero.The induction step is then exactly the same. Recalling (4.328) (or simply the remark after it), it thenfollows that ˜ η ◦∞ = 0.It also follows that the contribution d − j − αt j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − ε ιt | D j +2+ ι ˜ A ∗ t,i,p,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j +2+ ι ˜ A ∗ t,i,p,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α (4.345)to (4.103) goes to zero.Going back to (4.343) this means that (cid:90) { z ∞ }× Y ˜ G ∞ ,i,p tr ω C m + ω Y,z ∞ (˜ η ∞ ,j,k + ˜ η ♦∞ ) ω nY,z ∞ = 0 . (4.346)Next, as mentioned earlier we can write ˜ η ∞ ,j,k = i∂∂ ˜ ϕ ∞ with ˜ ϕ ∞ = O ( r j +2+ α ) and ˜ ϕ ∞ = 0, and˜ η ♦∞ = i∂∂ ˜ A ∞ , where ˜ A ∞ , is a function from the base which is O ( r j +2+ α ).We go back to (4.329), which we can write astr ω C m + ω Y,z ∞ (˜ η ∞ ,j,k + ˜ η ♦∞ ) = ∆ ω C m + ω Y,z ∞ ( ϕ ∞ + ˜ A ∞ , ) = K ( z ) + (cid:88) q K q ( z ) H q ( y ) , (4.347)and letting ∆ − C m K ( z ) denote any degree j + 2 polynomial on C m with Laplacian equal to K ( z ), whichclearly exists, we can write∆ ω C m + ω Y,z ∞ ( ϕ ∞ + ˜ A ∞ , − ∆ − C m K ) = (cid:88) j K j ( z ) H j ( y ) . (4.348)We thus see that the function on the LHS belongs to the fiberwise span of the ˜ G ∞ ,i,p ’s, and since by(4.346) it is also fiberwise L orthogonal to such span, we conclude that∆ ω C m + ω Y,z ∞ ( ϕ ∞ + ˜ A ∞ , − ∆ − C m K ) = 0 . (4.349)So the function ϕ ∞ + ˜ A ∞ , − ∆ − C m K is actually a harmonic function on C m × Y , O ( r j +2+ α ) byconstruction. Thanks to [19, Proposition 3.12] this implies that ϕ ∞ + ˜ A ∞ , − ∆ − C m K is a harmonicpolynomial of degree at most j + 2 on C m . Absorbing this polynomial into ∆ − C m K ( z ) (which byconstruction was only unique modulo harmonic polynomials anyway) we obtain the identity˜ ϕ ∞ + ˜ A ∞ , ( z ) = ∆ − C m K ( z ) . (4.350)Taking the fiber average of this identity immediately tells us that ˜ A ∞ , is a polynomial of degree atmost j + 2. This implies that the contribution d − j − αt | D j ˜ η ♦ t (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η ♦ t (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α (4.351)to (4.103) also goes to zero.Then going back to (4.350) we see also that ˜ ϕ ∞ is equal to the pullback of a polynomial from thebase, but since it also has fiberwise average zero, we conclude that ˜ ϕ ∞ = 0, and hence ˜ η ∞ ,j,k = 0 . Thisimplies that the contribution d − j − αt | D j ˜ η t,j,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η t,j,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) d ˜ g t (˜ x t , ˜ x (cid:48) t ) α (4.352)to (4.103) also goes to zero, thus giving a contradiction. Subcase C: ε t → . It follows from (4.106), (4.111), using also Remarks 2.6 and 2.7, that d − j − αt ˜ η t,j,k and d − j − αt ˜ η ♦ t converge in the topology of C j,β loc ( C m × Y ) for every β < α to limiting 2-forms˜ η ∞ ,j,k and ˜ η ♦∞ in C j,α loc ( C m × Y ), which are weakly closed (as a locally uniform limit of smooth closedforms) and of type (1 ,
1) with respect to J C m + J Y,z ∞ . We have ˜ g (cid:93)t → g can (0) locally smoothly thanks to(4.113), (4.156) and the fact that clearly ε t Θ ∗ t Ψ ∗ t ω F → g can (0) is a constantK¨ahler metric on C m , which we will rename simply g C m . Similarly, we can pass d − j − αt ˜ η ◦ t to a limit ˜ η ◦∞ since (4.142) give us a uniform C j,α loc bound (with respect to a fixed metric rather than ˜ g t ).Since ˜ η ♦ t is the pullback of a form from the base, the same is true for ˜ η ♦∞ , which by (4.102) is O ( r j + α )at infinity, and it is also ∂∂ -exact since it is weakly closed on C m . From (4.93) and (4.103) we see that d − j − αt (cid:32) j (cid:88) i =2 N i (cid:88) p =1 2 k (cid:88) ι = − ε ιt | D j +2+ ι ˜ A ∗ t,i,p,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j +2+ ι ˜ A ∗ t,i,p,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) + | D j ˜ η ♦ t (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η ♦ t (˜ x (cid:48) t )) | ˜ g t (˜ x t ) + | D j ˜ η t,j,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η t,j,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) (cid:33) = 1 . (4.353)However, recall that thanks to (4.106) we have for any fixed Rd − j − αt (cid:107) D ι ˜ η t,j,k (cid:107) L ∞ ( ˜ B Rd − t , ˜ g t ) → , (cid:54) ι (cid:54) j, (4.354)and so the contribution of d − j − αt ˜ η t,j,k to (4.353) is negligible. Since ˜ ω (cid:93)t → ω can (0) locally smoothly, italso follows that the contribution of d − j − αt ˜ η t,j,k to (4.172) is negligible too, which implies that d − j − αt [ D j b ··· b (tr ˜ ω (cid:93)t (˜ η ◦ t + ˜ η ♦ t ))] C α base ( ˜ B R , ˜ g t ) = o (1) , (4.355)for all R fixed.Furthermore, thanks to (4.124), the contributions of all the terms of the sum in (4.353) with ι > − d − j − αt (cid:32) j (cid:88) i =2 N i (cid:88) p =1 ε − t | D j ˜ A ∗ t,i,p,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ A ∗ t,i,p,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) + | D j ˜ η ♦ t (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ η ♦ t (˜ x (cid:48) t )) | ˜ g t (˜ x t ) (cid:33) = 1 + o (1) . (4.356)We can write d − j − αt D j b ··· b (tr ˜ ω (cid:93)t (˜ η ◦ t + ˜ η ♦ t )) = d − j − αt D j b ··· b (tr ˜ ω (cid:93)t ˜ η ♦ t ) + ( m + n ) d − j − αt D j b ··· b ˜ η ◦ t ∧ (˜ ω (cid:93)t ) m + n − (˜ ω (cid:93)t ) m + n . (4.357)For the first term we claim that d − j − αt D j b ··· b (tr ˜ ω (cid:93)t ˜ η ♦ t ) = d − j − αt D j b ··· b (tr ˜ ω can ˜ η ♦ t ) + o (1) , (4.358)uniformly on ˜ B R , which follows by observing that tr ˜ ω (cid:93)t ˜ η ♦ t = tr (˜ ω (cid:93)t ) bb ˜ η ♦ t since ˜ η ♦ t is pulled back from thebase, and using the bounds (4.111) for d − j − αt ˜ η ♦ t together with the aforementioned fact that (˜ ω (cid:93)t ) bb − ˜ ω can → D j b ··· b (tr ˜ ω can ˜ η ♦ t ) is the pullback of a function from the base.Ignoring combinatorial constants, we then schematically expand (4.357) as d − j − αt D j b ··· b (tr ˜ ω can ˜ η ♦ t ) + d − j − αt (cid:88) p + q + r = j ( D p b ··· b ˜ η ◦ t ) D q b ··· b ((˜ ω (cid:93)t ) m + n − ) D r b ··· b (((˜ ω (cid:93)t ) m + n ) − ) + o (1) , (4.359) ans-Joachim Hein and Valentino Tosatti 81 and from (4.139), (4.165) and (4.178) we see that (4.359) equals d − j − αt (cid:32) D j b ··· b (tr ˜ ω can ˜ η ♦ t ) + ( D j b ··· b ˜ η ◦ t ) ∧ (˜ ω (cid:93)t ) m + n − (˜ ω (cid:93)t ) m + n (cid:33) + o (1) , (4.360)uniformly on ˜ B R . For the second term, we use (4.141) which gives d − j − αt D j b ··· b ˜ η ◦ t = d − j − αt j (cid:88) i =2 N i (cid:88) p =1 ∂ f ∂ f (∆ Θ ∗ t Ψ ∗ t ω F | {·}× Y ) − ˜ G i,p D j b ··· b ˜ A ∗ t,i,p,k + o (1) , (4.361)uniformly on ˜ B R , and so d − j − αt D j b ··· b (tr ˜ ω (cid:93)t (˜ η ◦ t + ˜ η ♦ t ))= d − j − αt D j b ··· b (tr ˜ ω can ˜ η ♦ t ) + j (cid:88) i =2 N i (cid:88) p =1 ( D j b ··· b ˜ A ∗ t,i,p,k )tr ˜ ω (cid:93)t ∂ f ∂ f (∆ Θ ∗ t Ψ ∗ t ω F | {·}× Y ) − ˜ G i,p + o (1)= d − j − αt D j b ··· b (tr ˜ ω can ˜ η ♦ t ) + ε − t j (cid:88) i =2 N i (cid:88) p =1 ( D j b ··· b ˜ A ∗ t,i,p,k ) ˜ G i,p + o (1) , (4.362)uniformly on ˜ B R . Here we used that d − j − αt (cid:107) D j b ··· b ˜ A ∗ t,i,p,k (cid:107) L ∞ ( ˜ B R , ˜ g t ) (cid:54) Cε t by (4.116) and (cid:107) ˜ η † t (cid:107) L ∞ ( ˜ B R , ˜ g t ) = o (1) by (4.153), so that we can exchange tr ˜ ω (cid:93)t ∂ f ∂ f with tr ε t Θ ∗ t Ψ ∗ t ω F ∂ f ∂ f with only an o (1) error.Now (4.355) says that the quantity in (4.362) is asymptotically independent of the base directionson any fixed ball ˜ B R . Taking the fiberwise average of (4.362) thus shows that d − j − αt D j (tr ˜ ω can ˜ η ♦ t ) isapproaching a constant locally uniformly, and in the limit we obtain ∇ j,g m C tr g C m ˜ η ♦∞ = (const) . (4.363)On the other hand, taking the fiberwise L inner product of (4.362) with each of the ˜ G (cid:96),q ’s shows that ε − t d − j − αt D j ˜ A ∗ t,i,p,k is also approaching a constant locally uniformly (for all 2 (cid:54) i (cid:54) j, (cid:54) p (cid:54) N i ), soin particular j (cid:88) i =2 N i (cid:88) p =1 ε − t d − j − αt | D j ˜ A ∗ t,i,p,k (˜ x t ) − P ˜ x (cid:48) t ˜ x t ( D j ˜ A ∗ t,i,p,k (˜ x (cid:48) t )) | ˜ g t (˜ x t ) → , (4.364)so the contribution from D j ˜ A ∗ t,i,p,k to (4.356) goes to zero too, which shows that ˜ η ♦∞ is not annihilatedby [ ∇ j,g m C · ] C α .Thus ˜ η ♦∞ satisfies |∇ j,g C m ˜ η ♦∞ | = O ( r α ), it has a global ∂∂ -potential of class C j +2 ,α loc ( C m ), and satisfies(4.363), hence it can be written as ˜ η ♦∞ = i∂∂ϕ for some smooth function ϕ on C m with ∇ j,g m C ∆ g C m ϕ = (const) . (4.365)It follows that ϕ = (cid:96) + h where (cid:96) is a real polynomial of degree (cid:54) j + 2 on C m and h is a harmonicfunction on C m with | i∂∂h | = O ( r j + α ) , and Liouville’s Theorem shows that the coefficients of i∂∂h arepolynomials of degree at most j , thus contradicting the fact that ˜ η ♦∞ is not annihilated by [ ∇ j,g m C · ] C α .This finally concludes the proof of (4.84), and hence of Theorem 4.1.5. Proof of the main theorems
For the sake of brevity, in this section all norms and seminorms will be taken on an arbitrary ball B (cid:48) (cid:98) B which we allow to shrink slightly whenever interpolation is used, and the generic uniformconstant C is allowed to depend on d ( B (cid:48) , ∂B ). Proof of Theorem A.
Theorem A follows quite easily from Theorem 4.1, as follows. Recall that on X we have the Ricci-flat K¨ahler metrics ω • t = f ∗ ω B + e − t ω X + i∂∂ϕ t , which satisfy (1.1)( ω • t ) m + n = ˜ c t e − nt ω m + nX , ˜ c t = e nt (cid:82) X ( f ∗ ω B + e − t ω X ) m + n (cid:82) X ω m + nX , sup X ϕ t = 0 . (5.1)On X \ S we let ω F = ω X + i∂∂ρ be the semi-Ricci-flat form defined in (1.5), and we define a smoothfunction G on X \ S by e G = ω m + nX f ∗ ω mB ∧ ω nF . (5.2)One then easily checks [28, 29] that G is pulled back from the base, where it equals e G = f ∗ ( ω m + nX ) ω mB (cid:82) X b ω nX , (5.3)where of course (cid:82) X b ω nX is independent of b . Furthermore, e G it is integrable on B \ f ( S ) (an is even in L p ( ω mB ) for some p >
1) and satisfies (cid:82) B \ f ( S ) e G ω nB = (cid:82) X ω m + nX (cid:82) Xb ω nX . We can then solve the Monge-Amp`ereequation [28, 29] ω m can = ( ω B + i∂∂ψ ∞ ) m = (cid:82) B ω mB (cid:82) X b ω nX (cid:82) X ω m + nX e G ω mB , (5.4)with ψ ∞ smooth on X \ S (and globally continuous, which we will not need) and on X \ S define ω (cid:92)t = f ∗ ω can + e − t ω F and ψ t = ϕ t − ψ ∞ − e − t ρ , so that we have ω • t = ω (cid:92)t + i∂∂ψ t . Thus, combining(5.1) and (5.4), we see that on X \ S we have( ω • t ) m + n = ( ω (cid:92)t + i∂∂ψ t ) m + n = c t e − nt ω m can ∧ ω nF , (5.5)where c t = e nt (cid:82) X ( f ∗ ω B + e − t ω X ) m + n (cid:82) B ω mB (cid:82) X b ω nX , (5.6)which is indeed equal to a polynomial in e − t of degree at most m with constant coefficient (cid:0) m + nn (cid:1) . It iseasy to see that given any K (cid:98) X \ S , there is t K such that ω (cid:92)t is a K¨ahler metric on K for all t (cid:62) t K ,uniformly equivalent to f ∗ ω B + e − t ω X .To prove Theorem A we can assume that we are given an arbitrary coordinate unit ball compactlycontained in B \ f ( S ), over which f is C ∞ trivial. As usual, we simply denote by B this ball, and itspreimage is B × Y equipped with a complex structure J as in Theorem 4.1. Thanks to [29] we knowthat on B × Y we have C − ω (cid:92)t (cid:54) ω • t (cid:54) Cω (cid:92)t , (5.7)for all t (cid:62) ω (cid:92)t is K¨ahler for all t (cid:62) i∂∂ψ t → ψ t → L by standard psh functions theory, since ϕ t is normalized by sup X ϕ t = 0).We are thus in good shape to apply Theorem 4.1. To prove that ω • t is locally uniformly bounded in C k of a fixed metric (where k (cid:62) j = k in Theorem 4.1, so that up to shrinking B we have the decomposition (4.5) ω • t = ω (cid:92)t + γ t, + γ t, ,k + · · · + γ t,k,k + η t,k,k , (5.8)where ω (cid:92)t is clearly smoothly bounded, γ t, has uniform C k,α bounds by (4.10) and (4.11), η t,k,k haseven shrinking uniform C k,α bounds by (4.6) and (4.7) (hence a standard uniform C k,α bound thanks ans-Joachim Hein and Valentino Tosatti 83 to Remark 2.6), and the γ t,i,k have uniform C k bounds by the following argument: from the definition(4.8) and (3.32) we can write for 2 (cid:54) i (cid:54) kγ t,i,k = i∂∂ N i (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t Φ ι,(cid:96) ( G i,p ) (cid:126) D ι A t,i,p,k , (5.9)and so schematically for 0 (cid:54) r (cid:54) k D r γ t,i,k = (cid:88) i + i = r +2 N i (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t D i Φ ι,(cid:96) ( G i,p ) (cid:126) D i + ι A t,i,p,k , (5.10)and using a fixed metric g X we have clearly | D i Φ ι,(cid:96) ( G i,p ) | g X (cid:54) C , while from (4.12) we see that | D i + ι A t,i,p,k | = o (1) when i + ι (cid:54) k + 2 and from (4.28) | D i + ι A t,i,p,k | = o ( e ( i + ι − k − t ) when k + 2
From Theorem 4.1 applied with j = 2 and k (cid:62) ω • t = ω can + e − t ω F + γ t, + γ t, ,k + η t, ,k , (5.12)where we have that γ t, = i∂∂ψ t is uniformly bounded in C ,α and small in C ,β for 0 < β < α thanksto (4.10) and (4.11), and γ t, ,k has the schematic structure γ t, ,k = i∂∂ N (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t Φ ι,(cid:96) ( G ,p ) (cid:126) D ι A t, ,p,k , (5.13)where A t, ,p,k are functions from the base which thanks to (4.12) and (4.28) satisfy | D i A t, ,p,k | (cid:54) (cid:40) Ce − (3+ α ) ( − i α ) t , (cid:54) i (cid:54) ,o ( e − (4 − i ) t ) , (cid:54) i (cid:54) k. (5.14)Interestingly, we will also need another interpolation, as in (4.59), interpolating between | A t, ,p,k | (cid:54) Ce − (3+ α ) t from (4.12) and [ D A t, ,p,k ] C α (cid:54) Ce − t from (4.14) gives | D i A t, ,p,k | (cid:54) Ce − t (cid:16) e − − α t (cid:17) − i α , (cid:54) i (cid:54) . (5.15)Our goal is to clarify the structure of the term γ t, ,k . This will take us some work, and the very firststep is the claim that arguing as in (5.11) and using (5.14) and (5.15) we will have | γ t, ,k | g X = o ( e − t ) . (5.16)To prove the claim (5.16), we argue as in (5.10) and bound | γ t, ,k | g X (cid:54) C (cid:88) i + i =2 N (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t | D i + ι A t, ,p,k | , (5.17)and we bound the RHS of (5.17) by o ( e − t ) by considering the possible values of i + ι ∈ { , . . . , k + 2 } :if i + ι = 0 , , | D i + ι A t, ,p,k | = o ( e − t ), so good. If i + ι = 3 , necessarily ι (cid:62) (cid:96) (cid:62)
1, while (5.14) in particular gives | D i + ι A t, ,p,k | = o (1) so the RHS of(5.17) is again o ( e − t ). And if i + ι (cid:62) o (1) (cid:88) i + i =2 N (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t e ( i + ι − t (cid:54) o ( e − t ) k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t = o ( e − t ) , (5.18)using i (cid:54)
2, which concludes the proof of (5.16).On the other hand, thanks to (4.6) and (4.7), together with interpolation (Proposition 2.8 and(2.47)), η t, ,k satisfies | D i η t, ,k | g t (cid:54) Ce i − − α t , [ D i η t, ,k ] C β ( g t ) (cid:54) Ce i − β − α t , (cid:54) i (cid:54) , < β (cid:54) α, (5.19)so in particular | η t, ,k | g X = o ( e − t ) , | ( η t, ,k ) ff | g X = o ( e − t ) . (5.20)Next, we seek better estimates than (5.16) for the ff components of γ t, ,k . We claim that we have( γ t, ,k ) ff = N (cid:88) p =1 A t, ,p,k ∂ f ∂ f (∆ ω F | {·}× Y ) − G ,p , + o ( e − t ) , (5.21)where the o ( e − t ) is in L ∞ loc ( g X ). Indeed from (5.9) we can write( γ t, ,k ) ff = N (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t ∂ f ∂ f Φ ι,(cid:96) ( G ,p ) (cid:126) D ι A t, ,p,k , (5.22)and we can estimate each term as follows. For ι (cid:62) | D ι A t, ,p,k | = o ( e − (4 − ι ) t ) from (5.14), andso e − (cid:96)t | ∂ f ∂ f Φ ι,(cid:96) ( G ,p ) (cid:126) D ι A t, ,p,k | g X (cid:54) o (1) e − t e − ( (cid:96) − ι ) t = o ( e − t ) , (5.23)since (cid:96) (cid:62) ι . For ι = 3, we have (cid:96) (cid:62) | D ι A t, ,p,k | = o (1) from (5.14), so the term is again o ( e − t ).For ι = 1 ,
2, we have (cid:96) (cid:62) | D ι A t, ,p,k | = o ( e − t ) from (5.15), so the term is again o ( e − t ). Andfor ι = 0, let us first look at the terms with (cid:96) (cid:62)
1. For these, we have | A t, ,p,k | = O ( e − (3+ α ) t ), andso when multiplied by e − (cid:96)t , (cid:96) (cid:62)
1, these terms are indeed o ( e − t ). So we are only left with the termswhere ι = (cid:96) = 0 which equal N (cid:88) p =1 A t, ,p,k ∂ f ∂ f (∆ ω F | {·}× Y ) − G ,p , (5.24)since Φ , ( G ) = (∆ ω F | {·}× Y ) − G by (3.33), thus proving (5.21). In particular, using the bound (5.14)in (5.21) gives | ( γ t, ,k ) ff | g X (cid:54) Ce − α t . (5.25)Our next goal is to prove that | A t, ,p,k | (cid:54) Ce − t , (5.26)which improves upon (4.12). To see this, recall from (5.5) that c t e − nt ω m can ∧ ω nF = ( ω • t ) m + n = ( ω can + i∂∂ψ t + e − t ω F + γ t, ,k + η t, ,k ) m + n , (5.27)with c t = (cid:0) m + nn (cid:1) + O ( e − t ). Multiply this by e nt ( m + nn ) and using the above estimates (5.16), (5.20), (5.25)(which imply that | e t γ t, ,k + e t η t, ,k | g X = o (1) and | ( e t γ t, ,k + e t η t, ,k ) ff | g X = o ( e − t )) and using also ans-Joachim Hein and Valentino Tosatti 85 that i∂∂ψ t is small in C by (4.11), we can expand it as c t (cid:0) m + nn (cid:1) ω m can ∧ ω nF = ( ω can + i∂∂ψ t ) m ∧ ( ω F + e t γ t, ,k + e t η t, ,k ) n ff + mn + 1 e − t ( ω can + i∂∂ψ t ) m − ∧ ( ω F + e t γ t, ,k + e t η t, ,k ) n +1 + O ( e − t )= ( ω can + i∂∂ψ t ) m ∧ ω nF + ne t ( ω can + i∂∂ψ t ) m ( ω F ) n − ff ( γ t, ,k ) ff + mn + 1 e − t ( ω can + i∂∂ψ t ) m − ∧ ω n +1 F + o ( e − t ) , (5.28)where the error terms o ( e − t ) are in L ∞ loc ( g X ). Define a function S on B × Y by mn + 1 ω m − ∧ ω n +1 F = S ω m can ∧ ω nF , (5.29)so that (using again that i∂∂ψ t is o (1)) (5.28) gives( ω can + i∂∂ψ t ) m ∧ ( ω F ) n ff (cid:16) e t tr ω F | {·}× Y ( γ t, ,k ) ff + e − t S (cid:17) = c t (cid:0) m + nn (cid:1) ω m can ∧ ( ω F ) n ff + o ( e − t ) . (5.30)Divide both sides by ω m can ∧ ( ω F ) n ff , obtaining an equality of functions, which we can decompose intoits fiber average ( ω can + i∂∂ψ t ) m ω m can (cid:0) e − t S (cid:1) = c t (cid:0) m + nn (cid:1) + o ( e − t ) , (5.31)(using here that γ t, ,k is ∂∂ -exact so its fiberwise trace has zero integral) and what is left after sub-tracting the fiberwise average e − t ( S − S ) + e t tr ω F | {·}× Y ( γ t, ,k ) ff = o ( e − t ) . (5.32)Next, taking the fiberwise trace of (5.21) with respect to the fiberwise restriction of ω F givestr ω F | {·}× Y ( γ t, ,k ) ff = N (cid:88) p =1 A t, ,p,k G ,p + o ( e − t ) . (5.33)Subtituting (5.33) into (5.32) and integrating against G ,p on any fiber { z } × Y gives A t, ,p,k = − e − t (cid:90) { z }× Y S G ,p ω nF + o ( e − t ) , (5.34)which is valid as usual in L ∞ , and the desired (5.26) follows.The next step is to use the improved bound in (5.26) to obtain better bounds for γ t, ,k and itsderivatives, as follows. Interpolating between (5.26) and [ D A t, ,p,k ] C α (cid:54) C from (4.14) as in (4.59)gives | D i A t, ,p,k | (cid:54) Ce − − i + γ t , (cid:54) i (cid:54) , (5.35)where γ > α . We then interpolate again as in (4.28)between [ D ι A t, ,p,k ] C α (cid:54) Ce ι t from (4.14) (1 (cid:54) ι (cid:54) k ) and | D A t, ,p,k | (cid:54) Ce − γ t from (5.35) and wecan improve this to | D i A t, ,p,k | (cid:54) Ce − − i + γ t , (cid:54) i (cid:54) k, (5.36)up to shrinking γ . Similarly, for 0 < β < α using interpolation we can bound[ D i A t, ,p,k ] C β (cid:54) Ce − − i − β + γ t , (cid:54) i (cid:54) k. (5.37) Then (5.36) and (5.37) together with (5.26) can be used to bound γ t, ,k much as we did to derive (5.11).Namely, for 0 (cid:54) r (cid:54)
2, expand schematically D r γ t, ,k = (cid:88) i + i = r +2 N (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t D i Φ ι,(cid:96) ( G ,p ) (cid:126) D i + ι A t, ,p,k , (5.38)and using | D i Φ ι,(cid:96) ( G i,p ) | g t (cid:54) Ce i t and (5.26), (5.36) we can bound | D r γ t, ,k | g t (cid:54) C (cid:88) i + i = r +2 2 k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t e ( i + i + ι − t = e ( r − t k (cid:88) ι =0 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − ( (cid:96) − ι ) t (cid:54) Ce ( r − t , (5.39)i.e. | D i γ t, ,k | g t (cid:54) Ce − − i t , (cid:54) i (cid:54) . (5.40)Similarly, we can bound the H¨older C β ( g t ) seminorm of (5.38) for 0 < β < α by using the estimate[ D i Φ ι,(cid:96) ( G i,p )] C β ( g t ) (cid:54) Ce ( i + β ) t together with (5.26), (5.36), (5.37) and get[ D i γ t, ,k ] C β ( g t ) (cid:54) Ce − − i − β t , (cid:54) i (cid:54) . (5.41)These can be refined as follows. Using again (3.33) we have γ t, ,k = i∂∂ N (cid:88) p =1 A t, ,p,k (∆ ω F | {·}× Y ) − G ,p + i∂∂ N (cid:88) p =1 2 k (cid:88) ι =0 k (cid:88) (cid:96) =max(1 , (cid:100) ι (cid:101) ) e − (cid:96)t Φ ι,(cid:96) ( G ,p ) (cid:126) D ι A t, ,p,k , (5.42)and writing γ (cid:93)t, ,k := γ t, ,k − i∂∂ N (cid:88) p =1 A t, ,p,k (∆ ω F | {·}× Y ) − G ,p , (5.43)then we can expand schematically D r γ (cid:93)t, ,k (0 (cid:54) r (cid:54)
2) as above by (cid:88) i + i = r +2 N (cid:88) p =1 2 k (cid:88) ι =1 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t D i Φ ι,(cid:96) ( G ,p ) (cid:126) D i + ι A t, ,p,k + (cid:88) i + i = r +2 N (cid:88) p =1 k (cid:88) (cid:96) =1 e − (cid:96)t D i Φ ,(cid:96) ( G ,p ) (cid:126) D i A t, ,p,k , (5.44)and using (5.26) and (5.36) we can bound the g t -norm of this as in (5.39) by Ce − γ t (cid:88) i + i = r +2 2 k (cid:88) ι =1 k (cid:88) (cid:96) = (cid:100) ι (cid:101) e − (cid:96)t e ( i + i + ι − t + C (cid:88) i + i = r +2 k (cid:88) (cid:96) =1 e − (cid:96)t e ( i + i − t (cid:54) Ce ( r − − γ ) t , (5.45)i.e. | D i γ (cid:93)t, ,k | g t (cid:54) Ce − − i + γ t , (cid:54) i (cid:54) . (5.46)Similarly, we can bound the H¨older C β ( g t ) seminorm of (5.44) for 0 < β < α by using the estimate[ D i Φ ι,(cid:96) ( G i,p )] C β ( g t ) (cid:54) Ce ( i + β ) t together with (5.26), (5.36), (5.37) and get[ D i γ (cid:93)t, ,k ] C β ( g t ) (cid:54) Ce − − i − β + γ t , (cid:54) i (cid:54) , (5.47)and (5.46), (5.47) are o (1) provided we pick 0 < β < γ , or in other words γ (cid:93)t, ,k is o (1) in C ,β ( g t ) for0 < β < γ .As for the first term on the RHS of (5.42) we can write it as i∂∂ N (cid:88) p =1 A t, ,p,k (∆ ω F | {·}× Y ) − G ,p = i∂∂ (∆ ω F | {·}× Y ) − N (cid:88) p =1 A t, ,p,k G ,p − N (cid:88) p =1 A t, ,p,k G ,p , (5.48) ans-Joachim Hein and Valentino Tosatti 87 and we would like to inject here (5.32) and (5.33), but in order to do this, we need to clarify the natureof the error terms there, which so far have been proven to be o ( e − t ) in L ∞ ( g X ). The claim is that sucherror terms are in fact also o (1) in C ,β ( g t ) for 0 < β < γ . For the error in (5.33), this follows rightaway from (5.46), (5.47). As for the error in (5.32), let us go back to (5.27), and divide it by (cid:0) m + nn (cid:1) e − nt to get c t (cid:0) m + nn (cid:1) ω m can ∧ ω nF = ( ω can + i∂∂ψ t ) m ∧ (cid:0) ω F + e t γ t, ,k + e t η t, ,k (cid:1) n ff + m (cid:88) j =1 m ! n !( m − j )!( n + j )! (cid:0) ω can + i∂∂ψ t (cid:1) m − j ∧ (cid:0) ω F + e t γ t, ,k + e t η t, ,k (cid:1) n ∧ (cid:0) e − t ω F + γ t, ,k + η t, ,k (cid:1) j = ( ω can + i∂∂ψ t ) m ∧ (cid:0) ( ω F ) n ff + ne t ( ω F ) n − ff ( γ t, ,k ) ff (cid:1) + ne t ( ω can + i∂∂ψ t ) m ∧ ( ω F ) n − ff ( η t, ,k ) ff + n (cid:88) k =2 (cid:18) nk (cid:19) ( ω can + i∂∂ψ t ) m ∧ ( ω F ) n − k ff ( e t γ t, ,k + e t η t, ,k ) k ff + mn + 1 e − t (cid:0) ω can + i∂∂ψ t (cid:1) m − ∧ ω n +1 F + mn + 1 n (cid:88) p =1 (cid:18) np (cid:19) e − t (cid:0) ω can + i∂∂ψ t (cid:1) m − ∧ ω n − p +1 F ∧ ( e t γ t, ,k + e t η t, ,k ) p + mn + 1 (cid:0) ω can + i∂∂ψ t (cid:1) m − ∧ (cid:0) ω F + e t γ t, ,k + e t η t, ,k (cid:1) n ∧ ( γ t, ,k + η t, ,k )+ m (cid:88) j =2 m ! n !( m − j )!( n + j )! (cid:0) ω can + i∂∂ψ t (cid:1) m − j ∧ (cid:0) ω F + e t γ t, ,k + e t η t, ,k (cid:1) n ∧ (cid:0) e − t ω F + γ t, ,k + η t, ,k (cid:1) j , (5.49)and so the error in (5.32) consists precisely of the terms in lines − , − , − , − o (1) in C ,β ( g t ) for 0 < β < γ . To do this,first recall that i∂∂ψ t is o (1) in C ,β from (4.10) and (4.11). We will need the following “hybrid”norm | D j T | hyb of any contravariant tensor T , obtained by using g X for the components of T and g t for the norm of the vectors in the derivatives D j . We will also use the corresponding hybrid seminorm[ D j T ] C α hyb where g t is used also to measure the distances in the H¨older difference quotient. In particularwhen j = 0 the hybrid norm just equals the plain g X tensor norm, while on scalar functions the hybrid(semi)norm is equal to the g t (semi)norm, so to estimate the C ,β ( g t )-norm of various lines of (5.49)(divided of course by the product volume form ω m can ∧ ω nF ) we can just D -differentiate and estimateeach term by its hybrid (semi)norm. The hybrid norm trivially satisfies | · | g X (cid:54) C | · | hyb (cid:54) C | · | g t , andsimilarly for the seminorm, and we have the following trivial estimates | D i ω F | hyb (cid:54) Ce i t , [ D i ω F ] C β hyb (cid:54) Ce i + β t , (cid:54) i (cid:54) , < β < α. (5.50)Thanks to (5.19) we have in particular | D i η t, ,k | hyb (cid:54) Ce i − − α t , [ D i η t, ,k ] C β hyb (cid:54) Ce i − β − α t , (cid:54) i (cid:54) , < β < α, (5.51)and we claim that similarly | D i γ t, ,k | hyb (cid:54) Ce i − − γ t , [ D i γ t, ,k ] C β hyb (cid:54) Ce i − β − γ t , (cid:54) i (cid:54) , < β < α, (5.52)which follows by examining the proof of (5.40), (5.41): using the hybrid norm, we drop the factor of e i t/ on the RHS of (5.39) and thus we gain a factor of e − t/ over (5.40) except when i = 0, but in this case i (cid:62) A t, ,p,k is differentiated (at least twice) and so in this case we gain a factor of e − γt/ over (5.40) thanks to (5.36). The same discussion applies to (5.41), using (5.37), and (5.52) follows.We can summarize these as | D i ( ω F + e t γ t, ,k + e t η t, ,k ) | hyb (cid:54) Ce i t , | D i ( e t γ t, ,k + e t η t, ,k ) | hyb (cid:54) Ce i − γ t , (cid:54) i (cid:54) , (5.53)[ D i ( ω F + e t γ t, ,k + e t η t, ,k )] C β hyb (cid:54) Ce i + β t , [ D i ( e t γ t, ,k + e t η t, ,k )] C β hyb (cid:54) Ce i + β − γ t , (cid:54) i (cid:54) , (5.54)while in the fiber directions from (5.19), (5.40) and (5.41) we obtain the better bounds | D i ( e t γ t, ,k ) ff | hyb (cid:54) Ce i − t , | D i ( e t η t, ,k ) ff | hyb (cid:54) Ce i − − α t , (cid:54) i (cid:54) , (5.55)[ D i ( e t γ t, ,k ) ff ] C β hyb (cid:54) Ce i + β − t , [ D i ( e t η t, ,k ) ff ] C β hyb (cid:54) Ce i + β − − α t , (cid:54) i (cid:54) , (5.56)for 0 < β < α , and using these we can easily see that the terms in lines − , − , − , − o (1) in C ,β ( g t ) provided that 0 < β < γ . Indeed, consider the case when [ D · ] C β hyb is applied tothese terms. Then line − Ce − t e β t , lines − − Ce − t e β t e − γ t , andline − Ce β t e − α t + Ce β t e − t , and all of these go to zero. And when a lower numberof derivatives is applied, the estimates are even better.Combining these discussions of (5.32) and (5.33), we conclude that N (cid:88) p =1 A t, ,p,k G ,p = − e − t ( S − S ) + (Err) , (5.57)where (Err) is o (1) in C ,β ( g t ) for 0 < β < γ . Since i∂∂ψ t is o (1) in C ,β , it can be eliminated from(5.57) without changing the error term. On the other hand, from (5.42) and the discussion that followsit, together with (5.48), we see that γ t, ,k = i∂∂ (∆ ω F | {·}× Y ) − N (cid:88) p =1 A t, ,p,k G ,p − N (cid:88) p =1 A t, ,p,k G ,p + (Err) , (5.58)where (Err) is of the same kind.When substituting (5.57) into (5.58), we need to understand the regularity of i∂∂ (∆ ω F | {·}× Y ) − (Err)where (Err) are functions that are o (1) in C ,β ( g t ). When we restrict purely to a fiber { z } × Y , theoperator (∆ ω F | {·}× Y ) − increases regularity by 2 derivatives in H¨older spaces, by standard Schaudertheory, and so (∆ ω F | {·}× Y ) − (Err) is o (1) in C ,β ( g t ) when we look only in the fiber directions. It followsthat the fiber-fiber components of i∂∂ (∆ ω F | {·}× Y ) − (Err) are o (1) in C ,β ( g t ) when we restrict to anyfiber. Thus from (5.57) and (5.58) we obtain that γ t, ,k = − e − t i∂∂ (∆ ω F | {·}× Y ) − ( S − S ) + (err) , (5.59)where the (1 , o (1) in C ,β ( g t ). In addition,(err) is also O (1) in C ( g t ) because γ t, ,k is O (1) in C ( g t ) thanks to (5.40), while the first term on theRHS of (5.59) also clearly O (1) in C ( g t ). More precisely, we obtain | D i (err) | g t (cid:54) Ce − − i t , (cid:54) i (cid:54) . (5.60)Finally, we wish to have a more explicit understanding of the term S − S . For this, we use the workof Schumacher and collaborators [1, 25]. For 1 (cid:54) µ (cid:54) m we let W µ be the unique (1 ,
0) smooth vectorfield on B × Y which is the ω F -horizontal lift of the standard coordinate vector field ∂ w µ on C m , i.e.it satisfies (pr B ) ∗ W µ = ∂ w µ and ω F ( W µ , T , X z ) = 0 for all z ∈ B (where as usual X z = { z } × Y ), cf.[1, § ∂ operator to W µ gives A µ = ∂ X z W µ , (5.61) ans-Joachim Hein and Valentino Tosatti 89 where A µ is the unique T , X z -valued (0 , X z harmonic with respect to the Ricci-flat metric ω F | X z that represents the Kodaira-Spencer class κ ( ∂ w µ ) ∈ H ( X z , T , X z ). We now claim that wehave the identity S − S = (∆ ω F | {·}× Y ) − (cid:16) g µ ¯ ν can ( (cid:104) A µ , A ¯ ν (cid:105) − (cid:104) A µ , A ¯ ν (cid:105) ) (cid:17) , (5.62)where (cid:104)· , ·(cid:105) is the fiberwise Ricci-flat inner product. This identity is proved as follows: first, a classicalcomputation of Semmes [26] shows that our “geodesic curvature” quantity S defined in (5.29) is equalto g µ ¯ ν can ω F ( W µ , W ν ) , (5.63)and then (5.62) follows from this together with the fiberwise Laplacian formula of Schumacher (see [1,Lemma 4.1]) valid on the fiber X z ∆ ω F | Xz (cid:20) ω F ( W µ , W ν ) (cid:21) = ω WP ( ∂ w µ , ∂ w ν ) | z − (cid:104) A µ , A ν (cid:105) , (5.64)where ω WP (cid:62) B , so the term ω WP ( ∂ w µ , ∂ w ν ) | z is constant on X z .In conclusion, we have the decomposition ω • t = f ∗ ω can + e − t ω F + γ t, − e − t ∂∂ (∆ ω F | {·}× Y ) − (∆ ω F | {·}× Y ) − ( g µ ¯ ν can ( (cid:104) A µ , A ¯ ν (cid:105)−(cid:104) A µ , A ¯ ν (cid:105) ))+ η t, ,k +(err) , (5.65)where (err) is O (1) in C ( g t ), its fiber-fiber components are fiberwise o (1) in C ,β ( g t ), and it satisfies(5.60). But the term η t, ,k satisfies even better estimates than these thanks to (5.19), and so it can beabsorbed in (err), while the term γ t, is pulled back from the base and by (4.10) and (4.11) it is o (1)in C ,β .To complete the proof of Theorem B, we set error = γ t, , error = (err) and we thus need to verifythat our estimates for (err) imply the stated bounds (1.7). For simplicity let us write (err)= η , which is a ∂∂ -exact (1 , D j η to ∇ z,j η (at an arbitrary point ( z, y )). By definition these are equal for j = 0 , , while for j = 2we have D η = ∇ z, η + A (cid:126) η by (2.13), and | A | g t (cid:54) Ce t (since by definition A • fff = 0).On the other hand, from (5.60) we know that | D j η | g t (cid:54) Ce − − j t , (cid:54) j (cid:54) , (5.66)and so | η | g t (cid:54) Ce − t , |∇ z η | g t (cid:54) Ce − t , (5.67)and |∇ z, η | g t (cid:54) | D η | g t + C | A | g t | η | g t (cid:54) C + Ce t e − t (cid:54) C. (5.68)We then work in local product coordinates, and convert ∇ z η = ∂η + Γ z (cid:126) η, (5.69)with | Γ z | g t (cid:54) Ce t along the fiber over z (since Γ z are the Christoffel symbols of the product metric g z,t ) and so | ∂η | g t (cid:54) Ce − t , (5.70)and also ∇ z, η = ∂ η + ∂ Γ z (cid:126) η + Γ z (cid:126) ∂η + Γ z (cid:126) Γ z (cid:126) η, (5.71)and using | ∂ Γ z | g t (cid:54) Ce t (since ∂ f (Γ z ) bff = 0) we obtain | ∂ η | g t (cid:54) C. (5.72)Combining (5.67), (5.70) and (5.72) gives (in a fixed metric) | ( ∂ j η ) { (cid:96) }| (cid:54) Ce − − (cid:96) + j t , (cid:54) j (cid:54) , (5.73) which for (cid:96) < j + 2 agrees with the statement of (1.7). To also obtain the slightly better estimate for (cid:96) = j + 2 stated in (1.7), recall that we have proved earlier that[ ∇ z, ff ( η ff | { z }× Y )] C β ( { z }× Y,g t ) = o (1) , (5.74)for some β >
0. Writing η = i∂∂ϕ with ϕ = 0, and applying Lemma 2.11 to ( { z } × Y, g
Y,z ) as in (2.61),in the proof of Theorem 2.9, we can bound for 0 (cid:54) j (cid:54) , sup { z }× Y |∇ z,j f ··· f i∂∂ϕ | { z }× Y | g Y,z (cid:54) C [ ∇ z, f ··· f i∂∂ϕ | { z }× Y ] C β ( { z }× Y,g
Y,z ) (cid:54) Ce − β t [ ∇ z, ff ( η ff | { z }× Y )] C β ( { z }× Y,g t ) (cid:54) Ce − β t , (5.75)which means that |∇ z,j f ··· f ( η ff | { z }× Y ) | g t (cid:54) Ce − − j + β t , (cid:54) j (cid:54) , (5.76)and using (5.76) together with (5.69), (5.71) and | Γ z | g t (cid:54) Ce t , | ∂ Γ z | g t (cid:54) Ce t gives | ( ∂ j ( η ff | { z }× Y ) | g t (cid:54) Ce − − j + β t , (cid:54) j (cid:54) , (5.77)and so using a fixed metric | ( ∂ j ( η ff | { z }× Y ) | (cid:54) Ce − ( β ) t , (cid:54) j (cid:54) , (5.78)which is the improvement over (5.73) for (cid:96) = j + 2 claimed in (1.7). (cid:3) Remark 5.1.
The remaining terms in lines − , − o (1) of C ( g t ) (which the otherlines are, as they are even o (1) in C ,β ( g t ) for 0 < β < γ ), are easily seen to be O (1) in C ( g t ) usingthe same estimates as above with the hybrid norm. This shows that( ω • t ) m + n = c t e − nt ω m can ∧ ω nF (1 + f t ) , (5.79)where (locally as usual) the functions f t are o (1) in L ∞ and O (1) in C ( g t ). References [1] M. Braun, Y.-J. Choi, G. Schumacher,
K¨ahler forms for families of Calabi-Yau manifolds , Publ. Res. Inst. Math.Sci. (2020), no. 1, 1–13.[2] J. Cao, H. Guenancia, M. P˘aun, Variation of singular K¨ahler-Einstein metrics: Kodaira dimension zero. With anappendix by Valentino Tosatti , to appear in J. Eur. Math. Soc.[3] G. Chen, X.X. Chen,
Gravitational instantons with faster than quadratic curvature decay (III) , to appear in Math.Ann.[4] G. Chen, J. Viaclovsky, R. Zhang,
Collapsing Ricci-flat metrics on elliptic K3 surfaces , Comm. Anal. Geom. (2020), no. 8, 2019–2133.[5] V. Datar, A. Jacob, Y. Zhang, Adiabatic limits of anti-self-dual connections on collapsed K surfaces , to appear inJ. Differential Geom.[6] J.-P. Demailly, N. Pali, Degenerate complex Monge-Amp`ere equations over compact K¨ahler manifolds , Internat. J.Math. (2010), no. 3, 357–405.[7] P. Eyssidieux, V. Guedj, A. Zeriahi, A priori L ∞ -estimates for degenerate complex Monge-Amp`ere equations , Int.Math. Res. Not. IMRN 2008, Art. ID rnn 070, 8 pp.[8] J. Fine, Constant scalar curvature K¨ahler metrics on fibred complex surfaces , J. Differential Geom. (2004), no. 3,397–432.[9] J. Fine, Fibrations with constant scalar curvature K¨ahler metrics and the CM-line bundle , Math. Res. Lett. (2007),no. 2, 239–247.[10] W. Fischer, H. Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten , Nachr. Akad. Wiss.G¨ottingen Math.-Phys. Kl. II (1965), 89–94.[11] D. Gilbarg, N.S. Trudinger,
Elliptic partial differential equations of second order , Classics in Mathematics, Springer-Verlag, Berlin, 2001.[12] B. Greene, A. Shapere, C. Vafa, S.-T. Yau,
Stringy cosmic strings and noncompact Calabi-Yau manifolds , NuclearPhys. B (1990), no. 1, 1–36. ans-Joachim Hein and Valentino Tosatti 91 [13] M. Gross, V. Tosatti, Y. Zhang,
Collapsing of abelian fibered Calabi-Yau manifolds , Duke Math. J. (2013), no.3, 517–551.[14] M. Gross, V. Tosatti, Y. Zhang,
Geometry of twisted K¨ahler-Einstein metrics and collapsing , Comm. Math. Phys. (2020), no. 3, 1401–1438.[15] M. Gross, P.M.H. Wilson,
Large complex structure limits of K surfaces , J. Differential Geom. (2000), no. 3,475–546.[16] H.-J. Hein, Gravitational instantons from rational elliptic surfaces , J. Amer. Math. Soc. (2012), 355–393.[17] H.-J. Hein, A Liouville theorem for the complex Monge-Amp`ere equation on product manifolds , Comm. Pure Appl.Math. (2019), no. 1, 122–135.[18] H.-J. Hein, V. Tosatti, Remarks on the collapsing of torus fibered Calabi-Yau manifolds , Bull. Lond. Math. Soc. (2015), no. 6, 1021–1027.[19] H.-J. Hein, V. Tosatti, Higher-order estimates for collapsing Calabi-Yau metrics , Camb. J. Math. (2020), no. 4,683–773.[20] W. Jian, Y. Shi, A “boundedness implies convergence” principle and its applications to collapsing estimates in K¨ahlergeometry , Nonlinear Anal. (2021), 112255.[21] W. Jian, Y. Shi,
Global higher-order estimates for collapsing Calabi-Yau metrics on elliptic K surfaces , to appearin J. Geom. Anal.[22] S. Ko(cid:32)lodziej, The complex Monge-Amp`ere equation , Acta Math. (1998), no. 1, 69–117.[23] Y. Li,
A gluing construction of collapsing Calabi-Yau metrics on K fibred -folds , Geom. Funct. Anal. (2019),no. 4, 1002–1047.[24] C. Li, J. Li, X. Zhang, A mean value formula and a Liouville theorem for the complex Monge-Amp`ere equation , Int.Math. Res. Not. IMRN 2020, no. 3, 853–867.[25] G. Schumacher,
Positivity of relative canonical bundles and applications , Invent. Math. (2012), no. 1, 1–56.Erratum (2013), no. 1, 253–255.[26] S. Semmes,
Complex Monge-Amp`ere and symplectic manifolds , Amer. J. Math. (1992), no. 3, 495–550.[27] J. Song, G. Tian,
The K¨ahler-Ricci flow on surfaces of positive Kodaira dimension , Invent. Math. (2007), no. 3,609–653.[28] J. Song, G. Tian,
Canonical measures and K¨ahler-Ricci flow , J. Amer. Math. Soc. (2012), no. 2, 303–353.[29] V. Tosatti, Adiabatic limits of Ricci-flat K¨ahler metrics , J. Differential Geom. (2010), no. 2, 427–453.[30] V. Tosatti, Degenerations of Calabi-Yau metrics , in
Geometry and Physics in Cracow,
Acta Phys. Polon. B Proc.Suppl. (2011), no. 3, 495–505.[31] V. Tosatti, Calabi-Yau manifolds and their degenerations , Ann. N.Y. Acad. Sci. (2012), 8–13.[32] V. Tosatti,
Collapsing Calabi-Yau manifolds , Surveys in Differential Geometry (2018), 305–337, InternationalPress, 2020.[33] V. Tosatti, B. Weinkove, X. Yang, The K¨ahler-Ricci flow, Ricci-flat metrics and collapsing limits , Amer. J. Math. (2018), no. 3, 653–698.[34] V. Tosatti, Y. Zhang,
Triviality of fibered Calabi-Yau manifolds without singular fibers , Math. Res. Lett. (2014),no. 4, 905–918.[35] V. Tosatti, Y. Zhang, Infinite time singularities of the K¨ahler-Ricci flow , Geom. Topol. (2015), no. 5, 2925–2948.[36] V. Tosatti, Y. Zhang, Collapsing hyperk¨ahler manifolds , Ann. Sci. ´Ec. Norm. Sup´er. (2020), no.3, 751–786.[37] S.-T. Yau, On the Ricci curvature of a compact K¨ahler manifold and the complex Monge-Amp`ere equation, I , Comm.Pure Appl. Math. (1978), 339–411. Mathematisches Institut, WWU M¨unster, 48149 M¨unster, GermanyDepartment of Mathematics, Fordham University, Bronx, NY 10458, USA
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