Collapsing Calabi-Yau fibrations and uniform diameter bounds
aa r X i v : . [ m a t h . DG ] J a n Collapsing Calabi-Yau fibrations and uniformdiameter bounds
Yang LiJanuary 26, 2021
Abstract
As a sequel to [19], we study Calabi-Yau metrics collapsing along aholomorphic fibration over a Riemann surface. Assuming at worst canon-ical singular fibres, we prove a uniform diameter bound for all fibres inthe suitable rescaling. This has consequences on the geometry around thesingular fibres.
The present paper studies the adiabatic limiting behaviour of Ricci flat K¨ahlermetrics on a Calabi-Yau manifold under the degeneration of the K¨ahler class.The basic setting is:
Setting 1.1.
Let ( X, ω X ) be an n -dimensional projective manifold with nowherevanishing holomorphic volume form Ω, normalised to ∫ X i n Ω ∧ Ω =
1. Let π ∶ X → Y be a holomorphic fibration onto a Riemann surface, with connectedfibres denoted by X y for y ∈ Y , and without loss of generality ∫ X y ω n − X =
1, and ∫ Y ω Y =
1. The singular fibres lie over the discriminant locus S ⊂ Y , and π is asubmersion over Y ∖ S . We assume the singular fibres are normal and have atworst canonical singularities . Let ω Y be a K¨ahler metric on Y , and let ˜ ω t bethe Calabi-Yau metrics on X in the class of ω t = tω X + π ∗ ω Y , for 0 < t ≪ Example 1.2.
The most elementary examples are projective Calabi-Yau man-ifolds with Lefschetz fibrations over P , for n ≥
3. The basic non-example is aK3 surface with an elliptic fibration, such that the singular fibres are of type I .The wider question of collapsing Calabi-Yau metrics is intensely investi-gated by Tosatti and collaborators [25][26][27][13][14][17]. Most of these worksconcentrate only on what happens away from the singular fibres. The author’sprevious work [19] recognized the importance of the uniform fibre diameterbound for the geometry near the singular fibres. This meansdiam ( X y , t − ˜ ω t ) ≤ C. (1)1ith constants independent of the fibre X y and the collapsing parameter t . Moreprecisely, we mean that any two points on X y can be joined by some path in X ( not necessarily contained in X y ) whose t − ˜ ω t -length is uniformly bounded.The central result in [19] (modulo some technical generalizations) is essentially Theorem 1.3.
In setting 1.1, we assume the uniform diameter bound (1). Fixa singular fibre X and a point P ∈ X , and let Z be a pointed Gromov-Hausdorffsubsequential limit of ( X, t − ˜ ω t , P ) . Assuming in addition that any holomorphicvector field on the regular part of X vanishes, then Z is isometric to ¯ X × C with the product metric, where we equip C with the Euclidean metric, and ¯ X stands for the metric completion of the singular Calabi-Yau metric on X reg inthe class [ ω X ] . A detailed review of the main steps of [19] will be given in section 2 (partlybecause some intermediate conclusions are useful, and partly for technical gen-eralizations). It was also observed in [19] that in some special cases the uniformfibre diameter bound can be implied by a conjectural H¨older bound on theK¨ahler potential uniformly on the fibres, and the main evidence in [19] is a non-trivial diameter bound for nodal K3 fibres. While this H¨older bound strategy hasrecently found a number of interesting applications (eg. [8][11]), the conjectureremains hitherto unresolved, due to the difficulty of complex structure/K¨ahlerclass degeneration.This paper is to present a clean uniform proof of
Theorem 1.4.
In the setting 1.1, the uniform fibre diameter bound (1) holds.
This combined with Theorem 1.3 has implication on the pointed Gromov-Hausdorff limit around singular fibres.
Remark.
It should be emphasized that for the uniform fibre diameter bound tohold, the ‘at worst canonical singular fibre’ assumption is necessary , at least if [ ω X ] is a rational class. This is because on any smooth fibre X y , the rescaled fi-brewise metric t − ˜ ω t converges smoothly to the unique Calabi-Yau metric ω SRF,y on ( X y , [ ω X ]) as t →
0, with convergence rate depending on y ∈ Y ∖ S [25][26].Thus the uniformity in both t and y will imply a uniform diameter bound forall ω SRF,y in all y ∈ Y ∖ S , which is known to be equivalent to the ‘at worstcanonical singularity’ condition, assuming the rest of setting 1.1 and in additionthat [ ω X ] is an integral class up to a constant multiple [23]. For instance, thisuniform fibre diameter bound is not true around nodal elliptic curve fibres ona K3 surface. Remark.
In the motivating case [19] of Calabi-Yau 3-folds with Lefschetz K3fibrations, the uniform diameter bound and the Gromov-Hausdorff convergencestatements are consequences of the author’s gluing construction [20]. It is veryplausible that a similar construction can be made for higher dimensional Lef-schetz fibrations. But it seems unlikely that a gluing strategy can work in thefull generality of at worst canonical singularities.2he strategy for the uniform fibre diameter bound has two main new ingre-dients. The first is a uniform exponential integrability of the distance functionon the fibres, which amounts to proving the uniform fibre diameter bound mod-ulo a set of exponentially small measure. This method ( cf.
Theorem 3.1) is ofvery general nature and has its independent interest. The second is a judiciousapplication of Bishop-Gromov monotonicity to a critically chosen ball, whichprevents a subset of exponentially small measure staying far from the rest ofthe manifold.
Acknowledgement.
The author is a 2020 Clay Research Fellow, based at MIT.He thanks Valentino Tosatti for comments.
We now give an outline of Thm. 1.3 largely following [19] concerning how toidentify the pointed Gromov-Hausdorff limit of the neighbourhood of the (atworst canonical) singular fibre, in setting 1.1, assuming the uniform diameterbound (1). The key is that the uniform diameter bound implies a local non-collapsing condition around any given fibre, which enables the application ofmany standard geometric analysis arguments, in particular Cheeger-Coldingtheory.As useful background facts,
Proposition 2.1. [13][21][23] Assume the setting 1.1. Then1. The relative holomorphic volume form Ω y defined by Ω = Ω y ∧ dy satisfiesthe uniform bound A y = ∫ X y i ( n − ) Ω y ∧ Ω y ≤ C for all y around any givensingular fibre. In fact A y is continuous in y .2. The unique Calabi-Yau metrics ω SRF,y on X y in the class [ ω X ] have uni-formly bounded diameters independent of y , or equivalently, these metricsare uniformly volume non-collapsed.3. There exists p > such that ∫ X y ∣ Ω y ∧ Ω y ω n − X ∣ p ω n − X ≤ C. for all y around a given singular fibre.Morever, the Calabi-Yau metrics ω SRF,y are continuous in y in the Gromov-Hausdorff topology, including around singular fibres, where ω SRF,y is understoodas the metric completion of the regular locus for the singular Calabi-Yau metricconstructed in [6]. emark. The L p volume bound can be seen by passing to a log resolution.The uniform diameter bound is proved by the technique of [21], and under theprojective class condition it is known to be equivalent to the at worst canonicalsingular fibre assumption [23], as an application of Donaldson-Sun theory [4]. Write the Calabi-Yau metric in terms of the potential φ depending on t :˜ ω t = ω t + √ − ∂ ¯ ∂φ, ω t = tω X + π ∗ ω Y . The Calabi-Yau condition for ˜ ω t reads˜ ω nt = a t t n − i n Ω ∧ Ω , (2)where a t is a cohomological constant. Under the normalisation ∫ i n Ω ∧ Ω = ∫ X y [ ω X ] n − =
1, and since the base is 1-dimensional, a t = ∑ k = π ∗ [ ω Y ] k ⋅ [ ω X ] n − k ( nk ) t − k . (3)In the limit a t converges to a = n ∫ Y ω Y = n. From complex pluripotential theory,
Proposition 2.2. [7][3] There is a uniform constant such that ∥ φ ∥ L ∞ ≤ C. By a maximum principle argument based on the Chern-Lu formula,
Proposition 2.3.
There is a uniform bound Tr ˜ ω t π ∗ ω Y ≤ C . Consequently, the fibrewise restriction ˜ ω t ∣ X y has the pointwise volume den-sity upper bound˜ ω n − t ∣ X y ω n − SRF,y = ˜ ω n − t ∧ ω Y ω n − SRF,y ∧ ω Y ≤ C ˜ ω nt ( Tr ˜ ω t π ∗ ω Y ) Ω y ∧ Ω y ∧ ω Y ≤ Ct n − . (4)Define the oscillation to be osc = sup − inf. By applying Yau’s C -estimate fi-brewise, with ω SF R,y as the background metric (which has uniformly boundedSobolev and Poincar´e constants in the ‘at worst canonical singular fibre’ con-text),
Lemma 2.4.
The fibrewise oscillation satisfies the uniform bound osc X y φ ≤ Ct.
Next one introduces the fibrewise average function of φ : φ = ∫ X y φω X . ˜ ω t ( log Tr ˜ ω t ω X − Ct ( φ − φ )) ≥ Tr ˜ ω t ω X − Const t .
Now the fibrewise oscillation bound gives t ∣ φ − ¯ φ ∣ ≤ C , whence a maximumprinciple argument gives Theorem 2.5.
There is a uniform pointwise lower bound ˜ ω t ≥ Cω t . The severity of the singularity is measured by the function H = ω n − X ∧ ω Y ω nX ,whose zero locus is precise the π -critical points on X . By pointwise simultaneousdiagonalisation of ˜ ω t and ω t , Corollary 2.6.
There is a uniform upper bound ˜ ω t ≤ CH ω t . In particular, in the subset { H ≳ } ⊂ X , namely the region away from the π -critical points but not necessarily away from the singular fibres, there is auniform equivalence C − ω t ≤ ˜ ω t ≤ Cω t . (5)Around any given point in { H ≳ } , Evans-Krylov theory gives that t − ˜ ω t hasuniform C ∞ bound with respect to the background metric t − ω t . Corollary 2.7.
Inside { H ≳ } , ∥ ∇ ( k ) ω X t ˜ ω t ∣ X y ∥ L ∞ ≤ C ( k ) , ∥ ∇ ( k ) ω X ( Tr ˜ ω t ω Y )∣ X y ∥ L ∞ ≤ C ( k ) . Remark.
It should be emphasized that near the π -critical points, the metrics ω t and ˜ ω t are far from uniformly equivalent. Furthermore, the pointwise estimatefrom Cor. 2.6 cannot imply the uniform fibre diameter bound (1), nor do thefibres have any useful lower bound on the Ricci curvature to imply (1). Resolvingthis difficulty is the main concern of the present paper. From now on we assume (1) in the exposition.
Proposition 2.8.
Assuming (1), then t − ˜ ω t satisfies the local volume non-collapsing estimate: around any central point P , and for any ≲ R ≲ t − / , V ol ( B t − ˜ ω t ( P, R )) ≥ CR . (6) Morever
V ol ( B t − ˜ ω t ( P, R )) ≥ CR n for any R ≲ .Proof. Assume first that R ≳
1. Any fibre contains a subregion { H ≳ } where˜ ω t is uniformly equivalent to ω t . Thus if d ω Y ( y, y ′ ) ≲ Rt / / C , then the t − ˜ ω t -distance between the two fibres X y and X y ′ is O ( R ) . Using the fibre diameter5ound, we can reach any point on a nearby fibre within O ( R ) distance, sothe ball B t − ˜ ω t ( P, CR ) contains the preimage of B ω Y ( π ( P ) , Rt / ) . Since thevolume form of t − ˜ ω t is a t t − √ − ∧ Ω, we obtain the estimate (6). The R ≲ ω t .Thus non-collapsing Cheeger-Colding theory applies, and in particular aroundany point on X , including π -critical points, one can take non-collapsing pointedGromov-Hausdorff limits of ( X, t ˜ ω t , P ) , with all the standard consequences onits regularity. Let t ≪
1. We fix a central fibre X , which can be singular. The one-dimensionalbase condition will be crucially used. Consider a coordinate ball {∣ y ∣ ≤ R } ⊂ Y .Let ω Y, = A √ − dy ∧ d ¯ y be a Euclidean metric on {∣ y ∣ ≤ R } , where we recall A y = ∫ X y i ( n − ) Ω y ∧ Ω y .Chern-Lu inequality gives the subharmonicity∆ ˜ ω t ( log Tr ˜ ω t ω Y, ) ≥ . Using a slightly tricky argument based on the 3-circle inequality and the Harnackinquality (relying on the local non-collapsing), we deduce
Proposition 2.9.
Assuming (1), then we have a concentration estimate for Tr ˜ ω t ω Y, uniform for all choices of X : max ∣ y ∣≤ t / Tr ˜ ω t ω Y, ≤ + C ∣ log t ∣ , (7) t − n ∥ Tr ˜ ω t ω Y, − ∥ L ωt (∣ y ∣≲ t / ) ≤ C ∣ log t ∣ . (8)The concentration estimate easily entails that the two volume density on X , given by ( t − ˜ ω t ) n − and ω n − SRF,y , are close in the L -sense. By consideringthe fibrewise Monge-Amp`ere equation, one deduces that their relative K¨ahlerpotential is small in an integral sense. In the regular region { H ≳ } , thisimproves the smooth bounds in Cor. 2.7 to convergence bounds: Proposition 2.10.
For any small ǫ > , ∥ ∇ ( k ) ω X ( ω SRF,y − t ˜ ω t ∣ X y )∥ L ∞ ( X y ∩{ H ≳ }) ≤ C ( k, ǫ )∣ log t ∣ / − ǫ . (9)There is one extra bit of juice one can squeeze out of the Chern-Lu formulaand the concentration estimate, using an integration by part argument. We6ave a gradient bound, which shows that dπ is in some sense approximatelyparallel. ∫ ∣ y ∣≲ t / ( ∣ ∇ dπ ∣ Tr ˜ ω t ω Y, − ∣ ∂ log Tr ˜ ω t ω Y, ∣ ) ˜ ω nt ≤ Ct n − ∣ log t ∣ . (10)All these estimates are indepedent of the choice of X . Fix a point P on a (singular) fibre X , and look at the pointed sequence ofRicci flat spaces Z t = ( π − B ω Y ( , R ) ⊂ X, t − ˜ ω t ) . Local noncollapsing impliesthat after passing to subsequence, there is some complex n -dimensional Gromov-Hausdorff limit space ( Z, ω ∞ ) , with a Hausdorff codimension 4 regular locus Z reg which is connected, open, dense, where the limiting metric is smooth. Morever Z reg has a natural limiting complex structure, such that the limiting metric isK¨ahler. We shall suppress below mentions of subsequence to avoid overloadingnotation, and tacitly understand a Gromov-Hausdorff metric is fixed on thedisjoint union Z t ⊔ Z , which displays the GH convergence. Recall t ≪ X is pushed to infinity by scaling, the limit asa complex variety should be the normal neighbourhood of X , which is just thetrivial product X × C in the case of a smooth fibre, and the guess is that thesame is true for the singular fibre.More formally, we build comparison maps. Let u denote the standardcoordinate on C , and ω C refers to the standard Euclidean metric on C . Definethe holomorphic maps f t ∶ ( Z t , t − ˜ ω t ) → ( X × C , ω X + ω C ) , x ↦ ( x, u = t − / π ( x )) . Our scaling convention is that ω C agrees with t − ω Y, under the identification u = t − / y = t − / π ( x ) .By the uniform bound Tr ˜ ω t ω t ≤ C , there is a Lipschitz bound on f t inde-pendent of t , so the Gromov-Hausdorff limit inherits a Lipschitz map f ∞ into X × C . By the interior regularity of holomorphic functions, the limiting map f ∞ is holomorphic. As a rather formal consequence of the uniform fibre diameterbound, we can identify the image: Lemma 2.11.
The image of f ∞ is X × C . Recall the function H measures the severity of singular effect. Now H is acontinuous function on X , so defines a function on Z by pulling back via f ∞ .A qualitative consequence of the regularity in { H ≳ } is Proposition 2.12.
The map f ∞ is a biholomorphism { H > } ⊂ Z → X reg × C . t ˜ ω t ({∣ u ∣ ≤ D } ⊂ Z t ) ≥ Vol ω ∞ ({∣ u ∣ ≤ D } ⊂ Z ) . Using the explicit nature of the Calabi-Yau volume form, and the C ∞ loc conver-gence over { H > } , one finds Proposition 2.13. (Full measure property) The subset { H > } ≃ X reg × C inside Z must have full measure on each cylinder {∣ u ∣ ≤ D } ⊂ Z , so the set H = has measure zero in Z . In particular X reg × C is open and dense in Z . We now study the metric ω ∞ over the smooth region X reg0 × C . By passing(9) to the limit, and using the continuity of ω SRF,y at y = Proposition 2.14.
Over X reg × C , the limiting metric restricts fibrewise to theCalabi-Yau metric ω SRF, on X . We also need information about the horizontal component of the metric.By passing the concentration estimate in Prop. 2.9 to the limit,
Proposition 2.15.
The metric ω ∞ over X reg × C satisifies the Riemanniansubmersion property Tr ω ∞ ω C = . By passing the gradient estimate (10) to the limit,
Proposition 2.16.
Over X reg × C , the differential du is parallel with respect to ω ∞ . We can pin down the Riemannian metric on the regular locus:
Proposition 2.17.
The limiting metric ω ∞ = ω SRF, + ω C over X reg × C .Proof. The parallel diffential du induces a parallel ( , ) type vector field bythe complexified Hamiltonian construction: ι V ω ∞ = d ¯ u. In particular V is aholomorphic vector field. By assumption, there is no holomorphic vector fieldon X reg0 , so V must lie in the subbundle T C ⊂ T ( X reg0 × C ) . Morever, on eachfibre X reg × { u } , V must be a constant multiple of ∂∂u . We can then write V = λ ( u ) ∂∂u , where λ is a holomorphic function in u . Since du and V are bothparallel, the quantity λ = du ( V ) must be a constant.We know ω ∞ restricted to the fibres is just ω SRF, . By construction, thevector field V defines the Hermitian orthogonal complement of the holomorphictangent space of the fibres. Now V = λ ∂∂u where the constant is specified by theRiemmanian submersion property. The claim follows. There is still a small gap between Prop. 2.17 and the Gromov-Hausdorff conver-gence Theorem 1.3. By Prop. 2.17, we know the metric distance on X reg × C ⊂ Z
8s at most that of the product metric. We need to show that this is actuallyan equality, namely that one cannot shortcut the distance function by goingthrough the singular set in Z . (This is the only part of the argument not con-tained in the more restrictive setting of [19]). If so, then the density of X reg × C in Z ( cf. Prop 2.13) will imply that Z is isometric to ¯ X × C as required.Thus we concentrate on showing Proposition 2.18. (Geometric convexity) Given two points P , P in X reg × C ,which are GH limits of P t ∈ X and P t ∈ X respectively. Then for any given ǫ > , there is a small enough δ , such that for t → , there is a path containedin { H > δ } ⊂ X from P t to P t , whose t − ˜ ω t -length is at most d ( P t , P t ) + ǫ . This is precisely what allows one to reduce the distance function computa-tion to knowing the metric only in the regular region. Since P , P are fixed, wecan regard P t , P t ∈ { H ≳ } , and d ( P t , P t ) ≲
1. It is clear that the question onlyinvolves a local region of length scale O ( ) . The main techniques are developedby Song, Tian and Zhang [10][11].The following construction of a good cutoff function is taken from [10, Lem.3.7], and applied to the singular CY metric ( X , ω SRF, ) . Lemma 2.19.
Given λ > and any compact subset K contained in X reg . Thereis a cutoff function ρ λ ∈ C ∞ ( X reg ) compactly supported in X reg , with ≤ ρ λ ≤ ,which equals one on K , and satisfies the gradient bound ∫ X ∣ ∇ ρ λ ∣ ω n − SRF, < λ. By Cauchy-Schwarz, ∫ X ∣ ∇ ρ λ ∣ ω n − SRF, ≤ Cλ / . Applying the coarea formula to ∣ ∇ ρ λ ∣ as in [11, Lem. 2.5], we can find a levelset { ρ λ = a } compactly contained in X reg ∖ K , such thatArea ω SRF, ({ ρ λ = a }) ≤ Cλ / . Now since ρ λ is supported on the regular locus, we can regard it as a functionlocally on X , which is almost constant in the normal direction to X . Likewise { ρ λ = u } can be regarded as a hypersurface locally on X , separating { H ≳ } from the most curved region on X . For very small t depending on all previouschoices, the metric t − ˜ ω t is arbitrarily close to the product metric ω ∞ on thesupport of ρ λ , whenceArea t − ˜ ω t ({ ρ λ = a } ∩ d ( P, ⋅ ) ≲ ) ≤ Cλ / . (11) Proof. (Prop 2.18) By taking the compact set K large enough, we can ensure d ( P ti , { ρ λ = a }) ≳
1. The number λ can be taken very small depending on ǫ .9uppose there exists a point Q with d ( Q, P t ) ≲ ǫ , such that the minimal geodesicfrom P t to Q does not intersect { ρ λ = a } ∩ { d ( P, ⋅ ) ≤ d ( P t , P t ) + } . Then forlength reasons this minimal geodesic cannot intersect { ρ λ = a } , and since thesupport of ρ λ is compactly containted in the regular region, this geodesic muststay within { H ≳ δ } for δ depending only on ρ λ , and we can conclude Prop.2.18.Suppose the contrary, namely every minimal geodesic joining P t to anypoint in B t − ˜ ω t ( P t , ǫ ) intersects { ρ λ = a } ∩ { d ( P, ⋅ ) ≲ } . By a Bishop-Gromovcomparison argument, this would forceArea t − ˜ ω t ({ ρ λ = a } ∩ d ( P, ⋅ ) ≲ ) ≳ ǫ n − . This contradicts (11) by taking λ small enough in advance. For the moment, we step out of the setting 1.1, and consider a projective man-ifold M ⊂ P N of degree d and dimension n . Let ω F S = √− π log ∑ N ∣ Z i ∣ be thestandard Fubini-Study metric on ( M, c ( O ( )) , and ω = ω F S + √ − ∂ ¯ ∂φ be anysmooth K¨ahler metric in the same class. The following theorem of indepen-dent interest may be regarded as a Riemannian counterpart of uniform Skodaintegrability, discussed for instance in [5] recently. Theorem 3.1.
Assume the distance function d ω associated to ω satisfies ∫ M × M d ω ( y, y ′ ) ω nF S ( y ) ω nF S ( y ′ ) ≤ A, A ≥ . (12) Then there are constants C ( n ) depending only on n , and C ( n, N, d ) dependingonly on n, N and the degree d , such that ∫ M × M ω nF S ( y ) ω nF S ( y ′ ) exp ( d ω ( y, y ′ ) C ( n ) d ) ≤ e C ( n,N,d ) A . (13) Proof.
Our argument is inspired by Tian and Yau’s work on the α -invariant[24]. As a preliminary discussion, choose a ( N − n − ) -dimension projectivesubspace F ≃ CP N − n − ⊂ CP N , such that F ∩ M = ∅ . We project M onto an n -dimensional projective subspace F ⊥ , and call the projection π F . (The notationdoes not suggest perpendicularity for some fixed metric). If F and F ⊥ are chosengenerically, then π F ∶ M → F ⊥ is a finite map with covering degree d = deg ( M ) .Let φ F = d ∑ y ∈ π − F ( x ) φ ( y ) , x ∈ F ⊥ . ψ the relative potential between the Fubini-Study metric on CP N and F ⊥ , i.e. ψ F = π log ∥ Z ∥ ∥ π F ( Z )∥ , Z = ( Z , . . . , Z N ) , [ Z ∶ . . . ∶ Z N ] ∈ CP N ∖ F. Since F ∩ M = ∅ , we know ψ is smooth on M . We observe the pushforward of ω as a positive (1,1)-current is π F ∗ ( ω ) = d × ω F ⊥ + √ − ∂ ¯ ∂ ( dφ F + ∑ y ∈ F ⊥ ψ ( y )) . This defines a positive (1,1)-current with continuous potential in ( F ⊥ , c ( O ( d ))) ,and is smooth outside the branching locus. By the monotonicity formula in thetheory of Lelong numbers, applied to F ⊥ ≃ P n , we have ∫ π − F ( B ( r )) ω ∧ π ∗ F ω n − F ⊥ = ∫ B ( r ) π F ∗ ω ∧ ω n − F ⊥ ≤ C ( n ) dr n − . (14)Now for any x, x ′ ∈ F ⊥ , we consider the function ρ F ( x, x ′ ) = ∑ y ∈ π − F ( x ) ,y ′ ∈ π − F ( x ′ ) d ω ( y, y ′ ) . For fixed x ′ , this can be regarded as a function on x . Notice ∣ ∇ ω d ω ( ⋅ , y ′ )∣ ≤ ∣ ∇ ω F ⊥ ρ F ( x, x ′ )∣ ≤ d ∑ y ∈ π − F ( x ) ,y ′ ∈ π − F ( x ′ ) ∣ ∇ π ∗ F ω F ⊥ d ω ( y, y ′ )∣ ≤ d ∑ y ∈ π − F ( x ) ,y ′ ∈ π − F ( x ′ ) Tr ω F ⊥ ω = d ∑ y ∈ π − F ( x ) Tr ω F ⊥ ω Here the first inequality uses Cauchy-Schwarz, and the last inequality is becausethe traces are taken at y , with y ′ fixed. Since d ω comes from a smooth metricon M , it is easy to see ∇ ω F ⊥ ρ F has no distributional term supported on thebranching locus. Combining with (14), for any fixed x ′ , ∫ B ( r ) ∣ ∇ F ⊥ ρ F ∣ ω nF ⊥ ≤ C ( n ) d r n − . (15)By the John-Nirenberg inequality, ∫ F ⊥ exp ( ρ F ( ⋅ , x ′ ) − ¯ ρ F,x ′ d C ( n ) ) ω nF ⊥ ≤ C ′ ( n ) , (16)where ¯ ρ F,x ′ is the average number for fixed x ′ :¯ ρ F,x ′ = ∫ F ⊥ ρ F ( x, x ′ ) ω nF ⊥ ( x ) . ρ F = ∫ F ⊥ ∫ F ⊥ ρ F ( x, x ′ ) ω nF ⊥ ( x ) ω nF ⊥ ( x ′ ) . Clearly ¯ ρ F is the average of ¯ ρ F,x over all x ∈ F ⊥ . The above argument works also for the function ¯ ρ F,x to give ∫ F ⊥ ω nF ⊥ ( x ) exp ( ¯ ρ F,x − ¯ ρ F C ( n ) d ) ≤ C ′ ( n ) . Now by the change of variable formula,¯ ρ F = ∫ M × M d ω ( y, y ′ ) Jac ( π F )( y ) Jac ( π F )( y ′ ) ω nF S ( y ) ω nF S ( y ′ ) ≤ ∫ M × M d ω ( y, y ′ ) ω nF S ( y ) ω nF S ( y ′ ) ∥ Jac ( π F )∥ L ∞ ( M ) ≤ A ∥ Jac ( π F )∥ L ∞ ( M ) . We remark that the L ∞ -norm of the Jacobian factor is bounded on M because M ∩ F = ∅ , and as long as M is bounded away from F inside CP N then thisconstant stays uniform; this applies to small C -deformations of M , so by thecompactness of the Hilbert scheme, such constants can be made uniform forgiven n, N, d (possibly with changing choices of F, F ⊥ ).Thus ∫ F ⊥ ω nF ⊥ ( x ) exp ( ¯ ρ F,x C ( n ) d ) ≤ e C ( n,N,d ) A . (17)Combined with Cauchy-Schwarz and (16), ∫ F ⊥ × F ⊥ ω nF ⊥ ( x ) ω nF ⊥ ( x ′ ) exp ( ρ F ( x, x ′ ) C ( n ) d ) ≤ ( ∫ F ⊥ × F ⊥ ω nF ⊥ ( x ) ω nF ⊥ ( x ′ ) exp ( ρ F ( x, x ′ ) − ¯ ρ F,x C ( n ) d )) / × ( ∫ F ⊥ × F ⊥ ω nF ⊥ ( x ) ω nF ⊥ ( x ′ ) exp ( ¯ ρ F,x C ( n ) d )) / ≤ e C ( n,N,d ) A . Using the obvious inequality d ω ( y, y ′ ) ≤ ρ F ( π F ( y ) , π F ( y ′ )) , and changing thevalue of C ( n ) , ∫ M × M π ∗ F ω nF ⊥ ( y ) π ∗ F ω nF ⊥ ( y ′ ) exp ( d ω ( y, y ′ ) C ( n ) d ) ≤ e C ( n,N,d ) A . Notice this is already very close to our goal (13), in the sense that the exponentialintegrability ∫ ω nF S ( y ) ω nF S ( y ′ ) exp ( d ω ( y, y ′ ) C ( n ) d ) ≤ e C ( n,N,d ) A holds for pairs of points ( y, y ′ ) ∈ M × M where π ∗ F ω nF ⊥ ≳ ω nF S . (18)12ailure of this essentially means that the differential dπ F almost projects thetangent space of M at y or y ′ to a lower dimensional vector space. Now we recallthat the choice of ( F, F ⊥ ) is generic. By varying this choice, we can produce ( F , F ⊥ ) , . . . ( F l , F ⊥ l ) , with l suitably large depending on n, N , such that forany pair of ( y, y ′ ) ∈ M × M , the condition (18) holds for at least one choice of ( F i , F ⊥ i ) . (For instance, it is enough to take {( F i , F ⊥ i )} as a suitably dense ǫ -netin the product of Grassmannians.) Morever, the constants are robust for small C -deformation of M inside CP N , so by the compactness of the Hilbert schemeagain, the constant on the RHS of (13) is uniform in n, N, d . Remark.
Some a priori integral bound on d ω is necessary, for otherwise M maybe disconnected, or degenerating into a union of several components. The samereason shows it is not enough to have an L -bound on the distance function ona subset of M × M with say half of the Fubini-Study measure.However, we claim that it is enough to replace (12) with an L -bound on U × U for a large open subset U ⊂ M with ( − ǫ ) -percent of the Fubini-Studymeasure, for ǫ sufficiently small. To see this, first notice that in the John-Nirenberg inequality argument above, we can replace global average on F ⊥ bythe average on a subset V of F ⊥ with say half of the Fubini-Study measure. Itis enough to ensure that the L ( U × U ) -bound on d ω can bound the L ( V × V ) -norm on ρ F . This amounts to requiring that U contains π − F ( V ) for some V ⊂ F ⊥ with half measure, which would be true if U almost carries the full measure.This remark is quite convenient in situations where one can a priori boundthe metric in the generic region of M . Remark.
The above Theorem works for integral K¨ahler classes, but for irra-tional classes on projective manifolds it is often easy to reduce to the abovecase. For instance, consider M a complex submanifold of fixed degree inside aprojective manifold M ′ ⊂ CP N . Take an arbitrary fixed K¨ahler class χ on M ′ ,and consider K¨ahler metrics ω on M in the class χ ∣ M . We assume on a largeenough subset of M that ∫ d ω + ω FS ( y, y ′ ) ω nF S ( y ) ω nF S ( y ′ ) ≲ , and claim that there exists a uniform bound for all ( M, ω ) of the shape ∫ M × M ω nF S ( y ) ω nF S ( y ′ ) exp ( d ω ( y, y ′ ) C ) ≤ C ′ , To see this, we find a large integral multiple m , such that mc ( O ( )) − χ is aK¨ahler class on M , and we choose a K¨ahler representative ω ′ . Now ω ′ is boundedby some constant times ω F S . We can use the Theorem to get an exponentialintegrability bound for the distance function of ω + ω ′ . But it is obvious thatdistance functions increase with the metric, hence the claim.We can now return to the main setting 1.1.13 orollary 3.2. In the setting 1.1, there is a uniform exponential integrabilitybound for all fibres X y and for < t ≪ : ∫ X y × X y exp ( d t − ˜ ω t ( z, z ′ ) C ) ω n − X ( z ) ω n − X ( z ′ ) ≤ C ′ . Proof.
It suffices to prove this for all smooth fibres uniformly. By Cor. 2.6, thefibrewise metric has an upper bound t − ˜ ω t ≤ CH − ω X . Now on any fibre X y ,given a prescribed percentage 1 − ǫ , we can find a subset with at least 1 − ǫ ofthe ω n − X -measure, and demand H is bounded below on this subset. Since ω X is uniformly equivalent to the Fubini-Study metric, the claim follows from theRemarks above.The following Corollary asserts that modulo exponentially small probabil-ity, any point on X y is within O ( ) -distance to the regular region { H ≳ } ∩ X y . Corollary 3.3.
In the same setting, there are uniform constants such that ∫ X y exp ( d t − ˜ ω t ( z, { H ≳ } ∩ X y ) C ) ω n − X ( z ) ≤ C ′ . Proof.
By the Jensen inequality applied to the exp function, using also that ∫ X y ∩{ H ≳ } ω n − X ≥ , LHS ≤ ∫ X y exp ( C − ∫ ({ H ≳ }∩ X y ) d t − ˜ ω t ( z, z ′ ) ω n − X ( z ′ )) ω n − X ( z ) ≤ ∫ X y ×({ H ≳ }∩ X y ) exp ( d t − ˜ ω t ( z, z ′ ) C ) ω n − X ( z ) ω n − X ( z ′ ) ≤ C ′ . Here C changes from line to line as usual.However, what we need is the fibrewise Calabi-Yau volume measure, notsome Fubini-Study type measure. Proposition 3.4.
In the same setting, there are uniform constants such that ∫ X y exp ( d t − ˜ ω t ( z, { H ≳ } ∩ X y ) C ) i ( n − ) Ω y ∧ Ω y ≤ C ′ . Proof.
Combine item 3 of Prop. 2.1 with the above Corollary, and apply H¨olderinequality.
Remark.
Here we are working with the distance functions on X y induced bythe restriction of t ˜ ω t to X y . We can also study the distance function of t ˜ ω t on X and restrict it to X y . This function would be smaller, because the minimalgeodesics do not need to be contained in X y . Hence the distance bound canonly be better for the latter function, which is what we will use in the nextsection. 14 .2 Uniform fibre diameter bound We will now bridge the exponentially small gap between Prop. 3.4 and theuniform fibre diameter bound (1).
Proof. (Thm. 1.4) Take any point P on X y . All distances appearing below arecomputed on X , not on fibres. Let r be the smallest number such thatdist t − ˜ ω t ( B t − ˜ ω t ( P, r ) , { H ≳ } ⊂ X ) ≤ r. This exists because the diameter of X is finite (an a priori bound is known butnot necessary). If r ≤
1, then since t − ω t is uniformly equivalent to t − ˜ ω t in { H ≳ } ( cf. (5)), we can join P to { H ≳ } ∩ X y within O ( ) -distance, and weare done. So without loss of generality r ≥
1. The minimality of r shows thatin fact dist t − ˜ ω t ( B t − ˜ ω t ( P, r ) , { H ≳ } ⊂ X ) = r. (19)Our strategy is to derive two contrasting bounds on the volume of B t − ˜ ω t ( P, r ) .By Prop. 2.3, up to a constant factor the projection π ∶ X → Y decreasesdistance, so π ( B t − ˜ ω t ( P, r )) ⊂ B t − ω Y ( π ( P ) , Cr ) ⊂ Y. By Prop. 3.4 and the ensuing Remark, ∫ π − ( B t − ωY ( π ( P ) ,Cr )) exp ( d t − ˜ ω t ( z, { H ≳ } ∩ X y ) C ) i n Ω ∧ Ω = ∫ B t − ωY ( π ( P ) ,Cr ) √ − dy ∧ d ¯ y ∫ X y exp ( d t − ˜ ω t ( z, { H ≳ } ∩ X y ) C ) i ( n − ) Ω y ∧ Ω y ≲ ∫ B t − ωY ( π ( P ) ,Cr ) √ − dy ∧ d ¯ y ≲ r t. But from (19), the distance function in the exponent above is bounded belowby r on B t − ˜ ω t ( P, r ) . This forces ∫ B t − ωt ( P,r ) i n Ω ∧ Ω ≲ r te − C − r , or equivalently V ol t − ˜ ω t ( B t − ˜ ω t ( P, r )) ≲ r e − C − r (20)On the other hand, the ball B t − ˜ ω t ( P, r ) touches the regular region { H ≳ } where ˜ ω t is uniformly equivalent to ω t , whence by using the freedom to travelin the regular region, V ol t − ˜ ω t ( B t − ˜ ω t ( P, r )) ≳ r . Since X is Ricci-flat, Bishop-Gromov inequality gives V ol t − ˜ ω t ( B t − ˜ ω t ( P, r )) ≳ r . (21)Contrasting (20)(21) gives r ≲
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