Stability of the tangent bundle through conifold transitions
aa r X i v : . [ m a t h . DG ] F e b STABILITY OF THE TANGENT BUNDLE THROUGHCONIFOLD TRANSITIONS
TRISTAN C. COLLINS, SEBASTIEN PICARD, AND SHING-TUNG YAU
Abstract.
Let X be a compact, K¨ahler, Calabi-Yau threefold andsuppose X X X t , for t ∈ ∆, is a conifold transition obtainedby contracting finitely many disjoint ( − , −
1) curves in X and thensmoothing the resulting ordinary double point singularities. We showthat, for | t | ≪ T , X t admitsa Hermitian-Yang-Mills metric H t with respect to the conformally bal-anced metrics constructed by Fu-Li-Yau. Furthermore, we describe thebehavior of H t near the vanishing cycles of X t as t → Introduction
Let X be a compact, K¨ahler, Calabi-Yau threefold with trivial canoni-cal bundle. Around 1985 Clemens described a general procedure for con-structing new, possibly non-K¨ahler complex manifolds with trivial canoni-cal bundle by contracting a collection of disjoint ( − , −
1) curves and thensmoothing the resulting ordinary double point (ODP) singularities. Such ageometric transition is now called a conifold transition and we denote it by X X X t , where X is a singular variety with ODP singularities. Reid’sfantasy conjectures [74] that all complex threefolds with trivial canonicalbundle can be connected by a sequence of conifold transitions. The goal ofthis paper, motivated in part by the study of the Hull-Strominger system,is to show that the tangent bundle T , X t admits a Hermitian-Yang-Millsmetrics H t with respect to a class of balanced metrics constructed by Fu-Li-Yau [35], and to study the geometry of these metrics as | t | →
0. In order toput our work in context, it is useful to recall the origins and applications ofthe Hull-Strominger system.If X is a compact K¨ahler manifold with c ( X ) = 0 then the third author’ssolution of the Calabi conjecture [86] implies the existence of a unique Ricci-flat K¨ahler metric in any K¨ahler class on X . This result, which can beviewed as a higher dimensional analog of the Uniformization Theorem, yieldsa plethora of examples of compact Riemannian manifolds with zero Riccicurvature and holonomy contained in SU ( n ).Following the solution of the Calabi conjecture Candelas-Horowitz-Strominger-Witten [9] showed that compact K¨ahler manifolds with holonomy SU (3), in T.C.C is supported in part by NSF grant DMS-1810924, NSF CAREER grant DMS-1944952 and an Alfred P. Sloan Fellowship. particular, Calabi-Yau manifolds of complex dimension 3, are fundamentalbuilding blocks in torsion-free superstring compactifications. Precisely, [9]constructed superstring compactifications with the Standard Model gaugegroup from a Calabi-Yau threefold together with a holomorphic vector bun-dle E → X admitting a Hermitian-Yang-Mills connection and satisfyingthe topological constraints c ( E ) = 0, and c ( E ) = c ( X ). Of course, thenatural choice to make is E = T , X , but more generally the Donaldson-Uhlenbeck-Yau theorem [21, 83] implies that any slope stable vector bundlesatisfying the topological constraints will suffice. In total, these works leadto an abundance of a priori distinct superstring compactifications.Shortly thereafter the case of superstring compactifications with torsionwas analyzed by Hull [47] and Strominger [76]. In this case the compactifyingmanifold is a complex threefold X with a non-vanishing holomorphic (3 , E → X satisfying the topological constraints c ( E ) = 0 = c ( X )and c ( E ) = c ( X ). In order for this data to give rise to a supersymmet-ric compactification, X must admit a hermitian metric g , with associated(1 , ω , and E must admit a hermitian metric H solving the followingsystem of equations, called the Hull-Strominger system(1.1) d ( k Ω k ω ω ) = 0 , (1.2) ω ∧ F H = 0 , (1.3) √− ∂∂ω − α ′ Rm g ∧ Rm g − Tr F H ∧ F H ) = 0 . Here α ′ > F H denotes the curvature of the Chernconnection of ( E, H ), and k Ω k ω is the norm of Ω with respect to g . It isnatural to view Rm as the curvature of the Chern connection on T , X ,though other choices of unitary connection are admissible as discussed ine.g. [19, 28, 47]. Note that if g is K¨ahler, so that dω = 0, then (1.1) impliesthat ω solves the Monge-Amp`ere equation and is hence Ricci-flat, while (1.2)is the Hermitian-Yang-Mills equation for H , with respect to ω . Finally, inthe K¨ahler case, the anomaly cancellation equation (1.3) dictates thatTr Rm ∧ Rm − Tr F H ∧ F H = 0 . This coupling between the Hermitian-Yang-Mills equation and the Calabi-Yau equation is highly non-trivial, but it is automatically satisfied provided E = T , X . Thus, we can view the system (1.1), (1.2), (1.3) as a gener-alization of the Calabi-Yau equation to the setting of non-K¨ahler complexmanifolds. In particular, the Hull-Strominger system provides a set of equa-tions for uniformizing non-K¨ahler complex threefolds with trivial canonicalbundle which can be viewed as natural generalizations of the Calabi-Yauequation. The Hull-Strominger system has recently generated a great deal of interestin mathematics, both for its applications to the study of non-K¨ahler Calabi-Yau manifolds and its connections to string theory. Li-Yau [55] constructedsolutions on K¨ahler Calabi-Yau threefolds by deforming the complex struc-ture of T , X ⊕ C r . Deformations of more general bundles were consideredby Andreas-Garcia-Fernandez [1, 2] to more general bundles. Fu-Yau [34]constructed solutions to the Hull-Strominger system on Calabi-Eckmann-Goldstein-Prokushkin fibrations by using a certain ansatz to reduce the sys-tem to a non-linear PDE of Monge-Amp`ere type on a K Reid’s fantasy pro-poses that all complex 3-folds with trivial canonical bundle are connected bya sequence of conifold transitions. Recall that a conifold transition consistsof a contraction, followed by a smoothing:(1.4) X → X X t , where the contraction map X → X contracts a collection of disjoint rationalcurves C i ⊂ X , with normal bundle O P ( − ⊕ (called ( − , −
1) curves) toordinary double point (ODP) singularities, given locally by equations(1.5) ( X i =1 z i = 0 ) ⊂ C . By work of Friedman [32], under appropriate assumptions there is a smooth-ing(1.6) µ : X → ∆ , ∆ = { t ∈ C : | t | < } such that µ − (0) = X , and µ − ( t ) = X t is a smooth complex threefold withtrivial canonical bundle; see Section 2 for a more thorough discussion ofconifold transitions. Locally near the ODP singularities, this smoothing isgiven by(1.7) ( X i =1 z i = t ) ⊂ C × C . T. C. COLLINS, S. PICARD, AND S.-T. YAU
Note that in general, X t will no longer be K¨ahler (even topologically), evenif the initial manifold X was projective. These transitions were exploited byClemens to construct many new examples of compact, complex threefoldswith trivial canonical bundle; see [32].Green-H¨ubsch [41, 42] and Candelas-Green-H¨ubsch [8, 7] argued thatconifold transitions could be used to connect any two Calabi-Yau manifoldsrealized as complete intersections in products of projective spaces and thatconifold transitions were at finite distance in the moduli space of Calabi-Yaumanifolds for any of the common metrics on the moduli of Calabi-Yau vac-uua. In order to resolve the vaccum degeneracy problem for the heteroticstring it is essential to understand the solvability of the Hull-Stromingersystem through conifold transitions. In fact, the third author has advocatedthat the solvability of the Hull-Strominger system may provide a useful toolfor studying Reid’s fantasy as it provides an uniformization of non-K¨ahlerCalabi-Yau threefolds.The study of the Hull-Strominger system through conifold transitions wasinitiated by Fu-Li-Yau [35] who established the existence of metrics ω t on X t solving (1.1) assuming the input manifold X in (1.4) is a compact, K¨ahlerCalabi-Yau. Chuan [16] showed that if E → X is a holomorphic vector bun-dle which is Mumford-Takemoto stable with respect to some K¨ahler class,and holomorphically trivial in a neighborhood of the curves contracted bythe map X → X , then stability can be passed through the conifold transi-tion, in the following sense: there is a holomorphic vector bundle E t → X t with a hermitian metric H t solving the Hermitian-Yang-Mills equation (1.2)with respect to the Fu-Li-Yau metric. We remark that it is unclear whethersuch bundles E can be constructed so that, in addition c ( E t ) = c ( X t ) in H ( X t , R ). In any event, the metric H t constructed by Chuan is approxi-mately flat in a neighborhood of the vanishing cycles of the smooth X X t ,while the Fu-Li-Yau metric is modeled on a non-flat, K¨ahler Calabi-Yau conemetric.In this work we initiate the study of the Hull-Strominger system throughconifold transitions in the case when the gauge bundle E is taken to be T , X . Our main theorem is the following Theorem 1.1.
Let X → X X t be a conifold transition, with X t asin (1.6) . Equip X t with the Fu-Li-Yau balanced metric g t with associated (1 , form ω t . Then, for all | t | ≪ sufficiently small there exists a hermit-ian metric H t on T , X t solving ω t ∧ F H t = 0 . Furthermore, if p ∈ X is a ODP singularity, then after identifying a neigh-borhood of p ∈ X with the model smoothing (1.7) , there is a constant λ > such that on a neighborhood of the vanishing cycles given by R λ = {| t | k z k | t | λ } , for each k ∈ Z > there is a constant C k such that (1.8) (cid:12)(cid:12)(cid:12)(cid:12) ∇ kg t (cid:18) √− ∂∂ω t − α ′ Rm g t ∧ Rm g t − Tr F H t ∧ F H t ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) g t C k | t | λ k z k k )3 where k z k = P i =1 | z i | . Remark 1.2.
We have emphasized here the estimate (1.8) since it showsquantitative convergence of the pair ( g t , H t ) to a solution of the full Hull-Strominger system near the ordinary double points of X as t →
0. In fact,we have even more precise estimates which state that both g t and H t arelocally modeled on the Candelas-de la Ossa explicit K¨ahler Ricci-flat metric g co,t near the vanishing cycles; see Theorem 6.1. We show that, there is aconstant λ > p i ∈ X there are constants c i , d i > R λ we have |∇ kg co,t ( g t − c i g co,t ) | g co,t C k | t | λ k z k − k , |∇ kg co,t ( H t − d i g co,t ) | g co,t C k | t | λ k z k − k . where g co,t is the Candelas-de la Ossa Ricci-flat K¨ahler metric on the smooth-ing (1.7). This estimate shows that the local metric description of conifoldtransitions given by Candelas-de la Ossa [10] accurately describes globalnon-K¨ahler conifold transitions of heterotic strings near the vanishing cy-cles. For related work on Calabi-Yau metrics and K¨ahler conifold transitions,see [45, 69, 71, 80]. Remark 1.3.
A similar result holds if we replace the balanced Fu-Li-Yau metric g t with the conformally balanced metric ˇ g t obtained by confor-mally rescaling g t . In particular, the pair (ˇ g t , H t ) simultaneously solve (1.1)and (1.2), and satisfy an estimate similar to (1.8), but without the powerof | t | ; see Remark 6.13. Neverthless, this implies that near the ODP sin-gularities, at a suitable scale, the pair (ˇ g t , H t ) converges to the Calabi-Yausolution of the Hull-Strominger system on the conifold as | t | → Remark 1.4.
The existence of a metric H t on T , X t solving ω t ∧ F H t = 0implies that the tangent bundle of X t is stable with respect to the Fu-Li-Yaubalanced class [ ω t ] . In the case when X t is topologically k S × S [32, 43],the stability of the tangent bundle was noted in [5].The third author has conjectured [85] that if X is any complex threefoldwith trivial canonical bundle admitting a pair of metrics ( ω, H ) solving theconformally balanced equation (1.1) and the Hermitian-Yang-Mills equa-tion (1.2), then there is a solution of the full Hull-Strominger system. Wehope to return to this problem, in the setting of conifold transitions, infuture work.The outline of this paper is as follows. In Section 2 we discuss somebackground material, including the basic geometric properties of conifoldtransitions that will be important for our work. In Section 3 we construct a T. C. COLLINS, S. PICARD, AND S.-T. YAU metric H on the tangent bundle of T X → X reg . More precisely, the metric H is Hermitian-Yang-Mills with respect to a smooth, balanced metric g on X reg and, near the singular points of X , is uniformly equivalent to theCandelas-de la Ossa Ricci-flat K¨ahler metric on the conifold (1.5). Further-more, we show that H also satisfies scale invariant higher-order estimates.The metric H serves as the model metric for the Hermitian-Yang-Mills met-ric on X t , at least away from the vanishing cycles. In Section 4 we establishquantitative, polynomial decay of H towards a multiple of the Candelas-dela Ossa metric on the conifold. In Section 5 we use H to construct anapproximately Hermitian-Yang-Mills metric H t on X t with an explicit esti-mate for the decay rate, with respect to | t | , towards a Hermitian-Yang-Millsmetric. Finally, in Section 6 we show that, for | t | ≪ H t can be perturbed to a genuine Hermitian-Yang-Mills metric on T , X t . Forthe reader’s convenience we have provided an appendix detailing the aspectsof the Fu-Li-Yau construction which are important for our work. Acknowledgements:
We thank M. Garcia-Fernandez for helpful com-ments. T.C.C. is grateful to A. Jacob for helpful discussions concerning[49, 48] 2.
The geometry of conifold transitions
In this section we will discuss the basic geometry of conifold transitions.We begin with the following definition, which fixes the notion of Calabi-Yaumanifold to be considered in this paper.
Definition 2.1.
A smooth Calabi-Yau threefold is a smooth complex three-fold with finite fundamental group and trivial canonical bundle
The primary aim of this paper is to understand the solvability of theHermitian-Yang-Mills equation on the tangent bundle to a Calabi-Yau three-fold as it passes through a conifold transition.
Definition 2.2. A ( − , − curve C ⊂ X is a smooth rational curve C ≃ P such that the normal bundle N C/X ≃ O P ( − ⊕ . From [40, Satz 7] there is an open neighborhood U of C in X such that U is biholomorphic to a neighbourhood of the zero section in the total space of O P ( − ⊕O P ( − π C : X → X to a singular complex space X with an ordinary double point singularity at p such that π C : X \ C → X \ { p } and π C ( C ) = p . Concretely, if [ X : X ]denote homogeneous coordinates on P , then any non-zero point in the totalspace O P ( − ⊕ can be written uniquely as ( w X , w X , w X , w X ).This defines a biholomorphism from the complement of the zero section in O P ( − ⊕ to the complement of the origin in the conifold(2.1) ˆ V := { ˆ z ˆ z − ˆ z ˆ z = 0 } ⊂ C . This map can clearly be extended holomorphically over the zero section of O P ( − ⊕ by sending P to the origin in C . After a unitary change ofcoordinates we can rewrite (2.1) as the standard conifold(2.2) V := { z + z + z + z = 0 } ⊂ C . We now describe another realization of the affine variety (2.2). Considerthe Fano surface P × P . Denote by p i : P × P → P for i = 1 , i -th factor. The conifold can be realized as theblow-down of the zero section in the total space of p ∗ O P ( − ⊗ p ∗ O P ( − p ∗ O P (1) ⊗ p ∗ O P (1) define anembedding ι : P × P ֒ → P which is precisely the Segre embedding ι ( P × P ) = { X X − X X = 0 (cid:12)(cid:12) [ X : X : X : X ] ∈ P } . Taking the cone over this projective variety yields (2.1).The singular affine variety V given in (2.2) admits an explicit smoothinggiven by(2.3) V = { z + z + z + z = t } ⊂ C × C . We denote by µ : V → C the projection µ ( z, t ) = t , and let V t = µ − ( t ) bethe fiber over t .Now suppose that X is a Calabi-Yau threefold and let C , . . . , C k ⊂ X be a collection of disjoint ( − , −
1) curves. Let π : X → X be the mapcontracting the C i , so that X is a compact complex space with ordinarydouble point singularities at p i = π ( C i ). We have the following well-knownresult of Friedman [31, 32] Theorem 2.3 (Friedman, [32]) . There is a first order deformation of X smoothing p i if and only if there is a relation (2.4) k X i =1 λ i [ C i ] = 0 in H ( X, R ) where each λ i = 0 . When X is K¨ahler (or more generally, satisfies the √− ∂∂ -lemma) Kawa-mata [52], building on work of Ran [73], and independently Tian [79] showedthat the first order smoothings in Theorem 2.3 integrate to genuine smooth-ings. Furthermore, by [31, Lemma 8.2] if X is a K¨ahler, Calabi-Yau threefoldin the sense of Definition 2.1 then the fibers of smooth µ : X → ∆ are againCalabi-Yau. In particular, assuming (2.4) holds, there is a holomorphicfamily µ : X → ∆ := { t ∈ C : | t | < } T. C. COLLINS, S. PICARD, AND S.-T. YAU such that µ − ( t ) = X t is smooth for t = 0 and µ − (0) = X . By a re-sult of Kas-Schlessinger [51] the family X t is locally biholomorphic to themodel smoothing V near each ordinary double point. We make the followingdefinition. Definition 2.4.
Let X be a smooth, compact, complex -fold. A conifoldtransition of X , denoted X → X X t consists of a holomorphic map π : X → X and family µ : X → ∆ with µ − (0) = X such that (1) π : X → X contracts a collection of disjoint ( − , − curves C , . . . , C k to isolated, ordinary double point singularities p , . . . , p k ∈ X , and π is a biholomorphism X \ ∪ i C i → X \ { p , . . . , p k } . (2) µ : X → ∆ is a holomorphic smoothing of X = µ − (0) and X t = µ − ( t ) . Said informally, a conifold transition consists of contracting a collectionof disjoint ( − , −
1) curves followed by smoothing the resulting double pointsingularities. By the above discussion, Theorem 2.3 gives necessary andsufficient conditions for the existence of conifold transitions.2.1.
Metric geometry of the conifold.
Let us turn now to the discussionof some aspects of the metric geometry of conifold transitions. Recall thatthe conifold (2.2) can be viewed as the cone over P × P in the negativeline bundle p ∗ O P ( − ⊗ p ∗ O P ( − P × P is K¨ahler-Einstein withpositive Ricci curvature, it is well-known that the conifold admits a conicalCalabi-Yau metric. Explicitly, let h F S = X i =1 | X i | denote the Fubini-Study metric on O P ( − ι ∗ h F S = h KE where h KE = p ∗ h F S ⊗ p ∗ h F S and p ∗ i h F S is the pull-back of the Fubini-Studymetric on P for i = 1 ,
2. Define a function r : p ∗ O P ( − ⊗ p ∗ O P ( − → R > in the following way. If x ∈ P × P , and σ is a local section of p ∗ O P ( − ⊗ p ∗ O P ( − r ( x, σ ( x )) = (cid:0) | σ | h KE (cid:1) / . Clearly r − (0) is precisely the zero section of p ∗ O P ( − ⊗ p ∗ O P ( − r defines a function on the conifold (2.2). From the observation (2.5),we can write this function in terms of the coordinates on C as r = k z k / , k z k := X i =1 | z i | . An easy calculation shows that ω co, := 32 √− ∂∂r defines a conical Calabi-Yau metric g co, on V ; note that the factor of isa harmless scaling, but we have included it to be consistent with [35]. Moreprecisely, the Ricci-flat K¨ahler metric g co, is a cone metric over the link L := { r = 1 } ⊂ V and can be written as g co, = 32 (cid:0) dr + r g L (cid:1) where g L is (the pullback of) a Sasaki-Einstein metric on L := { r = 1 } ⊂ V .The cone ( V , g co, ) has a natural rescaling action generated by the vectorfield r ∂∂r . The vector field ξ = J ( r ∂∂r )is tangent the level sets or r and defines the Reeb vector field of the Sasakistructure on the link. We will use frequently the holomorphic vector field ξ C := r ∂∂r − √− ξ. Explicitly, the vector field ξ C generates the C ∗ action on V given by λ · ( z , z , z , z ) = ( λ / z , λ / z , λ / z , λ / z ) , and one can easily check that the cone metric g co, is homogeneous of degree2 under this action. In particular, we have Lemma 2.5.
The conical Calabi-Yau metric g co, on the conifold has thefollowing property: for every k ∈ Z > there is a constant C k so that |∇ k Rm( g co, ) | C k r − − k . Metric geometry of the local smoothings.
Candelas-de la Ossa[10] and independently Stenzel [75] constructed Calabi-Yau metrics on thesmoothings of the conifold (2.3) using ODE techniques. These metrics willplay an important role for us.
Proposition 2.6 (Candelas-de la Ossa, [10]) . Consider the smoothing ofthe conifold given by (2.3) , and set V t = { z + z + z + z = t } ⊂ C . Let k z k = P i =1 | z i | , and, for each t ∈ ∆ , set (2.6) f t ( s ) = 2 − | t | / Z cosh − ( s | t | )0 (sinh(2 τ ) − τ ) dτ Then ω co,t := √− ∂∂f t ( k z k ) defines a smooth Calabi-Yau metric g co,t on V t whose tangent cone at infinityis the conifold ( V , g co, ) . Let V denote the model smoothing (2.3) and let µ : V → C be theprojection to the t coordinate. There is a natural C ∗ action on the family V , given by(2.7) S λ ( z , z , z , z , t ) := ( λ / z , λ / z , λ / z , λ / z , λ t ) . so that S λ : V t → V λ t . Under this C ∗ action we have S ∗ λ f λ t ( k z k ) = | λ | f t ( k z k )and so, in particular, we have(2.8) g co,t = | t | / ( S t − / ) ∗ g co, . Note that, strictly speaking, we should fix a branch of log in the aboveexpression, but since the CO metric is manifestly S invariant such a dis-tinction is irrelevant. It follows that the CO metrics g co,t are generatedby the C ∗ action on V , up to rescaling. In particular, this shows (cf. [35,Lemma 5.1]) Lemma 2.7.
For each k ∈ Z > , and A > there is a constant C k,A > ,independent of t , so that the Calabi-Yau metrics g co,t satisfy sup k z k A | t | |∇ k Rm( g co,t ) | g co,t C k,A | t | − (2+ k ) . It will be important for us to understand the rate at which the CO metricconverges to its tangent cone at infinity. Consider the map(2.9) Φ t ( z ) : V \ (cid:26) k z k | t | (cid:27) −→ V t \ {k z k = | t |} ,z z + t ¯ z k z k . Tracing the definitions one can check that(2.10) Φ t = S t / ◦ Φ ◦ S t − / . We have
Lemma 2.8 (Conlon-Hein [18], Proposition 5.9) . Under the identification Φ we have that, for all k ∈ Z > , there is a constant C k such that |∇ kg co, (Φ ∗ g co, − g co, ) | g co, C k r − − k . Combining Lemma 2.8 with (2.8) and (2.10) we obtain estimates for thedecay rate of g co,t towards g co, . Corollary 2.9.
For all k ∈ Z > there is a uniform constant C k , independentof t , so that |∇ kg co, Φ ∗ t g co,t − g co, | g co, C k | t | r − − k . Proof.
From (2.8) and (2.10) we haveΦ ∗ t g co,t = | t | / ( S t − / ) ∗ Φ ∗ g co, . It follows that, if x ∈ V , then, from Lemma 2.8 we get |∇ kg co, (Φ ∗ t g co,t − g co, ) | g co, ( x ) = | t | / |∇ kg co, Φ ∗ t g co,t − g co, | | t | / g co, ( S t − / ( x )) C k | t | / | t | − · k r ( S − / t ( x )) − − k = C k | t | r ( x ) − − k (cid:3) One application of this result will be to transplanting estimates for tensorson V to estimates on the smooth varieties V t . For future reference, we record Lemma 2.10.
There is a constant
R > depending only on the constant C in Corollary 2.9, such that, if T is a tensor on some subset of V ∩ {k z k >R | t | } satisfying the estimate |∇ kg co, T | g co, M k r λ − k for some k ∈ Z > , constant M k > and some λ , then (Φ − t ) ∗ T defines atensor on V t ∩ {k z k > R +12 R | t |} satisfying the estimate |∇ kg co,t (Φ − t ) ∗ T | g co,t M ′ k r λ − k for constants M ′ k > depending only on M k and the the constants C k ap-pearing in Corollary 2.9. Note in particular that the estimate from Corollary 2.9 implies the fol-lowing estimate on V t (2.11) (cid:12)(cid:12) ∇ kg co,t [(Φ − t ) ∗ g co, − g co,t ] (cid:12)(cid:12) g co,t C | t | r − − k . It will be useful later in the paper to have a well adapted system ofcoordinates in which to carry out our analysis. The following, which werefer to as “holomorphic cylindrical coordinates” were used in [16]. Forcompleteness, we recall these coordinates and prove their existence in thefollowing
Lemma 2.11.
There are uniform constants ρ > and C k > , k ∈ Z > with the following effect: If ˆ z := (ˆ z , . . . , ˆ z ) ∈ V t and ˆ z = 0 , then there isan open neighborhood U ˆ z ∋ ˆ z and a holomorphic embedding ψ : B ρ (0) → U ˆ z such that, if ( w , . . . , w ) denote the local holomorphic coordinates on B ρ (0) ,then, setting ˆ r = r (ˆ z ) we have ( i ) We have ˆ r r ( w ) r on B ρ , and | ∂ k r | g euc ( w ) C k ˆ r where g euc denotes the Euclidean metric on B ρ . ( ii ) In these coordinates we have C − g euc ˆ r − g co,t C g euc and | ∂ k (ˆ r − g co,t ) | g euc C k Proof.
We begin by constructing some candidate coordinates. Fix ˆ z :=(ˆ z , . . . , ˆ z ) ∈ V t . Clearly | ˆ z i | > k ˆ z k for some 1 i
4, and hence,without loss of generality we may assume i = 4. We claim that w i = z i − ˆ z i ,for 1 i z . Indeed, consider the function F t ( z , z , z , z ) = X i =1 z i − t. By the implicit function theorem, the coordinates ( z , z , z ) form local co-ordinates on { F t = 0 } whenever ∂F∂z = 2 z = 0. Let us examine thesecoordinates. Since r = k z k / one can easily show that √− ∂∂r is aglobally defined, smooth metric on C \ { } , which we still denote by g co, .We claim that, up to scaling and translating, the estimates in ( i ) always holdin these coordinates. Indeed, from the scaling relation r ( λ · z ) = | λ | r ( z ) wemay as well assume that k ˆ z k = 1. It is easy to see that, for any multi-index α = ( k , k , k , k ) ∈ Z > we havesup { < k z k < } (cid:12)(cid:12)(cid:12)(cid:12) ∂ | α | ∂z α r (cid:12)(cid:12)(cid:12)(cid:12) g euc C ( α )Since 10 − < | z | <
10 on this region, the estimates in ( i ) will follow fromcomparing g euc to the Euclidean metric in the coordinates( w , w , w ) = ( z − ˆ z , z − ˆ z , z − ˆ z ) . Using the definition of the coordinates we have g euc = X i =1 | dw i | + | P i =1 z i ( w ) dw i | | z ( w ) | . For clarity, let us denote by g w = P i =1 | dw i | the Euclidean metric in the w coordinates. Then, from the Cauchy-Schwartz inequality, together with | z | > k ˆ z k = we have g w g euc g w . Noting that { < k z k < } implies that { ˆ r < r ( z ) < r } we see that ( i )holds in these coordinates.Next, we address ( ii ) in the special case t = 0. In this case the desiredestimates follow immediately from the above estimates for r , since rescalingpreserves V . It only remains to determine a bound for ρ such that { P i =1 | w | < ρ } implies { < k z ( w ) k < } . From the definition of the coordinates we have k z ( w ) k = X i =1 | w i + ˆ z i | + (cid:12)(cid:12) X i =1 ( w i + ˆ z i ) (cid:12)(cid:12) . On the one hand, we have X i =1 | w i + ˆ z i | > X i =1 | ˆ z i | − X i =1 | w i | , while on the other hand | X i =1 ( w i + ˆ z i ) | > (cid:12)(cid:12) X i =1 ˆ z i (cid:12)(cid:12) − X i =1 | w i | − X i =1 | ˆ z i | . Thanks to the fact that 1 = P i =1 | ˆ z i | + (cid:12)(cid:12) P i =1 ˆ z i (cid:12)(cid:12) we get k z ( w ) k > − X i =1 | w i | > P i =1 | w i | < ρ . The upper bound is similar.Next we consider the case when ˆ z ∈ V and {k ˆ z k > R } for some largeconstant R > z , z , z ), assuming as before that | ˆ z | > k ˆ z k . Sincewe have already established the estimates ( i ) in these coordinates, it sufficesto prove ( ii ). Let ˆ y ∈ V be defined byΦ (ˆ y ) = ˆ z ∈ V , | ˆ y | > . Note that from the definition of Φ we have k ˆ y k > R/
2. Define coordinates( x , x , x ) on V , near ˆ z by z i = | ˆ z | x i + ˆ z i and let ( w , w , w ) be the coordinates on V , centered at ˆ w constructedabove. Explicitly, z i = | ˆ y | w i + ˆ y i , i P i =1 | w i | < . The map Φ is given in these coordinates by(2.12) w i ˆ y kk ˆ z k w i + w i k ˆ y k + ˆ y i k ˆ y k · k z ( w ) k − ˆ y i k ˆ y k · k ˆ z k ! where k z ( w ) k = X i =1 (cid:12)(cid:12) k ˆ y k · w i + ˆ y i (cid:12)(cid:12) + (cid:12)(cid:12) X i =1 ( k ˆ y k · w i + ˆ y i | ) (cid:12)(cid:12) . When k y k > we have 1 k Φ ( y ) k k y k r (Φ ( y )) < r ( y ) < r (Φ ( y )) . From (2.12) it follows that there is a constant c ∈ [ ,
1] such that ∂∂w j x i ◦ Φ = cδ ij + O ( R − ) , ∂∂ ¯ w j x i ◦ Φ = O ( R − ) , and, for all multi-indices α with | α | > ∂ | α | ∂w α x i ◦ Φ = O ( R − ) , ∂ | α | ∂ ¯ w α x i ◦ Φ = O ( R − ) . Thus, by choosing R sufficiently large we can ensure that(2.15) 12 g w < Φ ∗ g x < g w . where g x denotes the Euclidean metric in x coordinates, and g w denotes theEuclidean metric in w coordinates. Furthermore, (2.14) implies that, for allmulti-indices α = 0 we have ∂ | α | ∂w α Φ ∗ g x = O ( R − ) . By Lemma 2.8 and (2.13) we can choose R sufficiently large so that(2.16) 12 g co, Φ ∗ g co, g co, . Combining (2.15), (2.16) and (2.13) we deduce the estimates in ( ii ) fromLemma 2.8 and the estimates for g co, in the w coordinates. For example,we have |∇ g x r (ˆ z ) − g co, | g x = |∇ Φ ∗ g x r (ˆ z ) − Φ ∗ g co, | Φ ∗ g x C (cid:0) |∇ g w r (ˆ z ) − Φ ∗ g co, | g w + | O ( R − ) r (ˆ y ) − g co, | g w (cid:1) C (cid:0) |∇ g co, (Φ ∗ g co, − g co, ) | g co, + | r (ˆ y ) − ∂g co, | g w + 1 (cid:1) C . Higher order derivatives follow similarly by induction. Note that if R issufficiently large then Φ ( { w : P i =1 | w i | < } ) ⊃ { x : P i =1 | x i | < } and so again, ρ can be chose uniformly.The portion of V given by { z ∈ V : k z k < R } is compact and hencethe desired coordinates can be constructed by a covering argument. It onlyremains to construct the coordinates on V t for 0 < | t | <
1. But for t =0 we can use the holomorphic rescaling map (2.7) to induce holomorphiccoordinates on V t from those on V . By (2.8) the metrics g co,t are generated,up to a scaling parameter, by the holomorphic rescaling map. On the otherhand, the estimates in ( i ) , ( ii ) are invariant under this rescaling, and hencethe lemma follows. (cid:3) Remark 2.12.
Note that the construction in Lemma 2.11 shows that thereis a constant
R > z ∈ {k z k > R | t |} and | ˆ z | > k ˆ z k , thenthe holomorphic cylindrical coordinates can be taken to be( w , w , w ) = 1 k ˆ z k ( z − ˆ z , z − ˆ z , z − ˆ z )Before moving on from this local discussion we state a lemma regardingextending some of the local objects introduced above. Lemma 2.13. If µ : X → ∆ is a global smoothing of a Calabi-Yau variety X = µ − (0) with ordinary double point singularities at points { p , . . . , p k } ,then there are disjoint open sets U i ⊂ X , with p i ∈ U i such that U i isbiholomorphic to a neighborhood of in the model smoothing (2.3) and, ( i ) there is a globally defined function r : X → R > such that, af-ter identifying U i with the model smoothing, r | U i = k z k / , and r − (0) = { p , . . . , p k } . ( ii ) There is a collection of closed sets U ′ i ⊂ U i , and closed sets C ti ⊂ X t smooth map Φ t : X \ ∪ i U ′ i → µ − ( t ) \ C ti such that, after identifying U i with the model smoothing we have U ′ i = (cid:26) k z k | t | (cid:27) , C ti = {k z k = | t |} and Φ t (cid:12)(cid:12) U i is the map z z + t ¯ z k z k .Proof. We only sketch the proof, since it is straightforward. Given a choiceof the open sets U i we extend the locally defined functions k z k / on X to globally defined positive functions which only vanish that the singularpoints p i . This establishes ( i ). To prove ( ii ) we just observe that the locallydefined maps are given by the flow of a vector field which lifts ∂∂t . We canextend this map globally by using a partition function to glue with any liftof ∂∂t to X \ U i (eg. by choosing a Riemannian metric on X ). (cid:3) Balanced metrics on conifold transitions.
We now review thework of Fu-Li-Yau [35] who constructed balanced metrics on non-K¨ahlerCalabi-Yau threefolds using conifold transitions and gluing.
Definition 2.14.
Let ( X, g ) be a complex manifold of complex dimension n with a hermitian metric. The metric g is said to be balanced if the associated (1 , form ω satisfies dω n − = 0 . We have the following theorem
Theorem 2.15 (Fu-Li-Yau [35], Theorem 1.2) . Let X be a smooth, K¨ahler,Calabi-Yau threefold, and suppose that X → X X t is a conifold transi-tion. Then, for | t | sufficiently small X t admits a balanced metric ω FLY ,t . We will need to recall some aspects of the proof of Theorem 2.15 as theywill play an important role in subsequent sections. The first step [35] is toconstruct a balanced metric on X = X by appropriately gluing a Calabi-Yau metric on X with the conical Calabi-Yau metric on the conifold. Tofix notation, for each ordinary double point p i ∈ X we fix an identificationof a neighborhood of p i with a neighborhood of the singular point in theconifold. Define U i ( ε ) = ( ( z , z , z , z ) ∈ C : X i =1 z i = 0 , and k z k ε ) . and let U ( ε ) = S i U i ( ε ). We state the following result of Fu-Li-Yau [35]; forthe reader’s convenience we have given a self-contained proof in Appendix A. Proposition 2.16 (Fu-Li-Yau [35], Proposition 2.6) . With the above nota-tion, for every < ε ≪ sufficiently small there is a constant M > suchthat there exists a hermitian metric g on X reg whose associated (1 , form ω has the following properties ( i ) On X \ U (1) we have ω = ω CY where ω CY is a Calabi-Yau metricon X . ( ii ) On U ( ε ) ∩ X reg we have ω = M / ε − / ω co, . ( iii ) On U (1) \ U ( ε ) , ω is √− ∂∂ -exact.In particular, g is balanced on X reg . For the remainder of the paper, ε > g . It will be useful to use to have a comparison of ω CY with ω on X , away from the contracted rational curves. In a neighborhoodof the curves, the metric ω CY is uniformly equivalent to the smooth refer-ence metric ω sm = √− ∂∂r + π ∗ ω F S (where we recall that r = k z k ).Rescaling by S λ ( z ) = λ / z gives S ∗ λ ω sm = λ √− ∂∂r + π ∗ ω F S , S ∗ λ ω co, = λ ω co, , and since the metrics ω co, and ω sm are uniformly equivalent on { r } ,we obtain(2.17) C − r ω CY ω Cr − ω CY on { < r < } .The next step in the proof of Theorem 2.15 is to construct approximatelybalanced metrics on the smooth fibers X t . Let U ( ) ⊂ X be a small openset containing the singular points p i . Note that X \ U ( ) is diffeomorphic to X t \ U ( √ ) by the map Φ t constructed in Lemma 2.13. If ω is the metricfrom Proposition 2.16, then (Φ ∗ t ω ) (2 , is positive definite for | t | sufficientlysmall and can be glued to the Candelas-de la Ossa metric ω co,t to obtain apositive (2 ,
2) form. The following result can be extracted from Fu-Li-Yau[35, Section 3], see for example [35, equation (3.4)]. Proposition 2.17 (Fu-Li-Yau [35]) . With notation as above, for ε, | t | suffi-ciently small and M sufficiently large there is a hermitian metric g t on X t such that the associated (1 , form ω t has ( i ) ω t = M / ε − / ω co,t is K¨ahler Ricci-flat in U ( ε ) ∩ X t . ( ii ) There is a constant C k , indepenedent of | t | so that | dω t | C k ( X t ,ω t ) C k | t | . ( iii ) As | t | → , Φ ∗ t ω t converges smoothly, in compact subsets of X \{ p , . . . , p k } to the balanced metric ω of Proposition 2.16. With this result, Theorem 2.15 is obtained by solving a 4-th order linearequation with estimates in order to perturb the approximately balancedmetric ω t of Proposition 2.17 to a genuine balanced metric ω FLY ,t for | t | sufficiently small.2.4. Balanced metrics on the small resolution.
The space X can beviewed as a small resolution of the singular space X , obtained by replacingthe ordinary double points with ( − , −
1) rational curves. It will be usefulto us to have a sequence of degenerating balanced metrics on X .Consider the total space π P : O P ( − ⊕ → P . Let h F S denote thestandard Fubini-Study metric on O P ( − z ∈ P , and ( u, v ) ∈ π − P ( z )denotes a point in the fiber over z , we define r ( u, v, z ) = (cid:0) | u | h F S + | v | h F S (cid:1) / . It is easy to check that r is the pull-back to O P ( − ⊕ of the potentialfor the conical Calabi-Yau metric on the conifold. Candelas-de la Ossa[10] constructed asymptotically conical Calabi-Yau metrics on the resolvedconifold via the ansatz ω co,a := √− ∂∂f a ( r ) + 4 a π ∗ P ω F S where ω F S = √− ∂∂ log h F S is the Fubini-Study metric on P . This metricis Calabi-Yau provided f a ( x ) satisfies the differential equation( xf ′ a ( x )) + 6 a ( xf ′ a ( x )) = x , f ′ a ( x ) > x >
0. From this expression it is straightforward to check that f a ( x ) = a f (cid:16) xa (cid:17) . In particular we see that(2.18) ω co,a = a S ∗ a − ω co,a =1 where S a ( u, v, z ) = ( a / u, a / v, z ) is the scaling generated by the holomor-phic Reeb vector field on the conifold. The argument of Fu-Li-Yau carriesover to give the following proposition; for the reader’s convenience, we haveprovided the details in Appendix A. Proposition 2.18 (Fu-Li-Yau [35], Proposition 2.6) . For every < ε ≪ sufficiently small there is a constant M , M > such that there exists asequence of balanced hermitian metric ω a on X with a → which has thefollowing properties: ( i ) On X \ U (1) we have ω a = ω CY where ω CY is a Calabi-Yau metricon X . ( ii ) On U ( ε ) ∩ X we have ω a = M / ε − / ω co,a . ( iii ) On U (1) \ U ( ε ) , ω a is √− ∂∂ -exact. ( iv ) As a → , ω a converges smoothly to ω on compact subsets of X \{ C , . . . , C k } . ( v ) There is an estimate M − Vol(
X, g a ) M . ( vi ) We have [ ω a ] = [ ω CY ] ∈ H ( X, R ) . Remark 2.19.
Regarding notation, following the conventions of [16], we willuse ω co,a to denote the Candelas-de la Ossa metric on the small resolutionof the conifold, while reserving ω co,t for the Candelas-de la Ossa metric onthe smoothing of the conifold.2.5. Notation.
Before beginning the construction, we establish notationand conventions that will be used throughout the paper. Throughout thepaper T , X will denote the holomorphic tangent bundle to X , and we willwrite the components of a hermitian metric H on T , X as H = H ¯ kj dz j ⊗ d ¯ z k , and we will denote the inner product by h V, W i H = H ¯ kj V j W k , V = V i ∂∂z i , W = W i ∂∂z i . The hermitian condition in this convention is H ¯ kj = H ¯ jk . The inverse of H will be denoted H p ¯ q , so that matrix multiplication is H j ¯ p H ¯ pk = δ j k . Ahermitian metric H induces metrics on all the associated bundles in theusual fashion.An endomorphism A : T , X → T , X has an adjoint A † defined by h AV, W i H = h V, A † W i H . When the dependence on the metric is emphasized, we will write this as A † H . An endomorphism is H -self-adjoint when A † = A . Note that, in thisnotation, the inner-product on endomorphisms is given by h A, B i H = Tr( AB † )The curvature F H of the Chern connection of the metric H will follow theconvention ( F H ) j ¯ kpq = − ∂ ¯ k ( H p ¯ r ∂ j H ¯ rq ) . From two hermitian metrics ˆ H and H we can form the relative endomor-phism denoted h = ˆ H − H , or in index notation denoted h pq = ˆ H p ¯ r H ¯ rq . A formula which is the starting point for many computations is for thedifference of the curvature tensors of ˆ H and H .(2.19) ( F H ) j ¯ k − ( F ˆ H ) j ¯ k = − ∂ ¯ k ( h − ∇ ˆ Hj h ) . Here the p, q endomorphism indices are omitted for ease of notation, and ∇ ˆ H denotes the Chern connection of ˆ H acting on h ∈ Γ(End T , X ) by ∇ ˆ Hj h = ∂ j h + [ ˆ H − ∂ j ˆ H, h ] . In this paper, g will typically denote a balanced hermitian metric with as-sociated form ω = √− g ¯ kj dz j ∧ d ¯ z k . The contraction operator Λ ω acts on F H by ( √− ω F H ) pq = g j ¯ k ( F H ) j ¯ kpq . The curvature of g will be denoted ( R g ) j ¯ kpq = − ∂ ¯ k ( g p ¯ r ∂ j g ¯ rq ). Since g is notK¨ahler, it will have non-zero torsion, which we denote by( T g ) rij = ( A g ) irj − ( A g ) j ri where A g = g − ∂g is the Chern connection of g .We will denote the ( n, n ) form corresponding to g by d vol g = ω n n ! . We will denote the complex Laplacian acting on functions by ∆ g f = g j ¯ k ∂ j ∂ ¯ k f .The balanced condition dω n − = 0 allows us to integrate by parts so that Z X ψ ∆ g ϕ d vol g = Z X ϕ ∆ g ψ d vol g for any ψ, ϕ ∈ C ∞ ( X ).3. Hermitian-Yang-Mills metrics on the central fiber
In this section, we construct a Hermitian-Yang-Mills metric on the tangentbundle of the singular space X with respect to the Fu-Li-Yau balanced metric ω . We will prove: Theorem 3.1.
There exists a hermitian metric H on T X reg satisfying F H ∧ ω = 0 where ω is the Fu-Li-Yau metric of Proposition 2.16 and F H is the curva-ture of the Chern connection of H . The metric H satisfies the estimates (3.1) | H | g + | H − | g C , |∇ kg H | g C k r − k , where r : X → R > is as in Lemma 2.13. We will produce H by extracting a limit from a sequence { ( H a , ω a ) } ofHermitian-Yang-Mills metrics with respect to the degenerating sequence ofbackground metrics { ω a } from Proposition 2.18. A similar approach is takenin [16]. Let X be a simply-connected, compact K¨ahler manifold of dimension n = 3 with trivial canonical bundle. By Yau’s theorem [86], the bundle T , X has a Ricci-flat metric ω CY , and therefore T , X is polystable withrespect to [ ω CY ]. In fact, T , X is stable because it cannot holomorphicallysplit. As noted in [87] , the de Rham decomposition theorem implies that ifthe tangent bundle to Calabi-Yau manifold splits holomorphically, then themanifold itself splits holomorphically and metrically as a product. In dimen-sion n = 3, at least one factor in this decomposition must be 1-dimensional,and hence a torus. When X is simply connected, this is impossible. Thus,( X, [ ω CY ]) satisfies the stability condition1rk( F ) Z X c ( F ) ∧ ω < F ⊆ T , X of rank 1 or 2. It followsfrom Proposition 2.18 that the same inequality holds if we replace ω CY by ω a for any a > T , X is stable with respect to the balanced classes [ ω a ] ∈ H , ( X, R ). By the Li-Yau [54] generalization to Gauduchon metrics of theDonaldson-Uhlenbeck-Yau theorem [21, 83], there exists a family of metrics H a on T , X such that F H a ∧ ω a = 0 . Our goal is to obtain a limiting metric H as a → Reference metrics.
To study the sequence ( g a , H a ), we will use asequence of reference metrics ˆ H a , given byˆ H a = e ψ a g a where ψ a satisfies(3.2) ∆ g a ψ a = 13 Tr √− ω a F g a , Z X ψ a d vol g a = 0 . The solvability of (3.2) follows from the balanced condition since Tr F g a isexact. The advantage of ˆ H a is that these metrics now have the propertythat Tr Λ ω a F ˆ H a = 0 . This follows from(3.3) √− ω a F ˆ H a = − (∆ g a ψ a ) I + √− ω a F g a . If we form the relative endomorphism ˆ h a = ˆ H − a H a , then (2.19) implies∆ g a log det ˆ h a = Tr √− ω a F H a − Tr √− ω a F ˆ H a = 0 . Therefore, det ˆ h a is constant, and we will choose a normalization for H a suchthat det ˆ h a ≡ . We prove a uniform C estimate for the conformal factor. Lemma 3.2.
The sequence { ψ a } satisfies a uniform bound k ψ a k L ∞ ( X ) C. Proof:
Let U ( δ ) = { r < δ } . Since g a = Rg co,a on U ( ε ), we have Λ ω a F g a =0 on U ( ε ) and ∆ g a ψ a = 0 on U ( ε ) . By the maximum principle,sup U ( ε ) | ψ a | sup ∂U ( ε ) | ψ a | . Define ϕ a ( z ) = sup X ψ a − ψ a ( z ) . Let p ∈ X be a point where 0 = ϕ a ( p ) = inf X ϕ a . By the above estimate, wemay assume p ∈ X \ U ( ε ). Fix a finite open cover { V i } of coordinate chartsof K = X \ U ( ε ) such that the eigenvalues of ( g a ) i ¯ j are uniformly boundedabove and below on each chart. Denote by B i coordinate balls each compactcontained in V i which still cover K . Note that since the metrics g a convergesmoothly and uniformly on X \ U ( ε ) to the metric g the sets B i , V i can bechosen independent of a .Suppose p ∈ B so that inf B ϕ = 0. By the Harnack inequality for ellipticPDE (e.g. [44, Theorem 5.10]),sup B ϕ a C (cid:18) inf B ϕ a + k ∆ g a ϕ a k L ∞ (cid:19) = C k ∆ g a ϕ a k L ∞ . From Proposition 2.18, the function Tr Λ ω a F g a is supported on U (1) \ U ( ε )and is bounded uniformly, independent of a . Thereforesup X | ∆ g a ϕ a | C. It follows that sup B ϕ a C . Let B be another coordinate ball with B ∩ B = ∅ . By the Harnack inequality,sup B ϕ a C (cid:18) inf B ∩ B ϕ a + k ∆ g a ϕ a k L ∞ (cid:19) C (cid:18) sup B ϕ a + 1 (cid:19) C. Continuing this process for each B i , we concludesup K ϕ a C, and hence sup X ϕ a C . This gives a bound on the oscillationsup X ψ a − inf X ψ a C. Since R X ψ a d vol g a = 0, there is a point q ∈ X where ψ a ( q ) = 0. Thereforesup X | ψ a | C where C is independent of a > (cid:3) We record some corollaries of this estimate.
Corollary 3.3.
Let H a , ˆ H a , g a , h a , ψ a be as above, for < a ≪ . Then wehave ( i ) Let Z = ∪ i C i be the union of all ( − , − curves contracted duringthe conifold transition. On compact sets K ⊂ X \ Z , the metrics ˆ H a converge smoothly to a limiting metric ˆ H = e ψ g . ( ii ) For all a ≪ , the metrics ˆ H a are uniformly equivalent to thebackground metrics g a . That is, there is a uniform constant C , in-dependent of a such that C − g a ˆ H a Cg a . In particular, since det ˆ H − a H a = 1 , the endomorphism h a = g − a H a = e ψa ˆ H − a H a satisfies C − det h a C for a uniform constant C , independent of a ≪ . ( iii ) The Hermitian-Yang-Mills tensor is bounded along the sequence sup X |√− ω a F ˆ H a | ˆ H a C. The full curvature of the limiting ˆ H satisfies | F ˆ H | ˆ H Cr − on X reg .Proof. To prove convergence of ˆ H a on a compact set K , we cover K byfinitely many coordinate charts and apply interior estimates for the Laplaceequation (3.2) to ψ a . On K , the metrics g a converge uniformly to g ,and hence after a subsequence ˆ H a = e ψ a g a converges to a limiting met-ric ˆ H . The uniform bounds for ˆ H a and det h a follow from the C estimate k ψ a k L ∞ C .On a neighborhood U ( ε ) containing the holomorphic curves, we haveΛ ω a F g a = 0 and Λ ω a F ˆ H a = 0. Outside of U ( ε ), the metrics ( ˆ H a , ω a ) areuniformly bounded, hence | Λ ω a F ˆ H a | ˆ H a C .The full curvature F ˆ H does not vanish on U ( ε ), however in this neigh-borhood | Rm g | g Cr − . In U ( ε ) we have ∆ g ψ = 0, and estimates forthe Laplacian in cylindrical coordinates imply | ∂ ψ | g euc C and hence |∇ g ψ | g Cr − by Lemma 2.11. It follows that | F ˆ H | ˆ H Cr − since g and ˆ H are uniformly equivalent. (cid:3) Uhlenbeck-Yau C estimate. In this section, we derive the followingestimate: Proposition 3.4.
Along the sequence of endomorphisms h a = g − a H a , wehave the uniform C estimate (3.4) C − I h a CI where I denotes the identity endomorphism. We will prove this by following the argument of Uhlenbeck-Yau [83].Thanks to the estimate C − det h a C , it suffices to show(3.5) Tr h a C. Rather than h a = g − a H a , we will work with the reference metric ˆ H a = e ψ a g a from the previous section and relative endomorphism ˆ h a = ˆ H − a H a . Theestimate k ψ a k L ∞ ( X ) C in Lemma 3.2 shows that a bound Tr ˆ h a C implies (3.5).To prove (3.5), suppose on the contrary that Tr ˆ h a → ∞ as a →
0. Let˜ h a = ˆ h a sup X Tr ˆ h a . The starting point in the proof of the C estimate of Uhlenbeck-Yau is thefollowing inequality (see [83, equation (4.6)]); Lemma 3.5.
Fix < σ , and any two metrics ˆ H, H on T , X → X .Let h = ˆ H − H , and let g be a Hermitian metric on X . Then we have (3.6) | h − σ/ ˆ ∇ h σ | H,g g j ¯ k h h − ˆ ∇ j h, ˆ ∇ k h σ i ˆ H where ˆ ∇ is the Chern connection of ˆ H . We rewrite (3.6) using the identity(3.7) ∂ j Tr h σ = σ h h − ˆ ∇ i h, h σ i ˆ H , which implies 1 σ ∆ g Tr h σ = g j ¯ k ∂ ¯ k h h − ˆ ∇ j h, h σ i ˆ H . Therefore, (3.6) is equivalent to | h − σ/ ˆ ∇ h σ | H,g − σ ∆ g Tr h σ g j ¯ k h h − ˆ ∇ j h, ˆ ∇ k h σ i ˆ H − g j ¯ k ∂ ¯ k h h − ˆ ∇ j h, h σ i ˆ H = − g j ¯ k h ∂ ¯ k ( h − ˆ ∇ j h ) , h σ i ˆ H . (3.8)We will make use of inequality (3.8) by relating the right-hand side to thecurvature tensor. With the same notation as Lemma 3.5, the differencebetween curvatures of the Chern connections (2.19) defined by H, ˆ H showsthat the key inequality (3.8) can be written as | h − σ/ ˆ ∇ h σ | H,g − σ ∆ g Tr h σ h ( √− ω F − √− ω ˆ F ) , h σ i ˆ H . In our case, applying this inequality to the Hermitian-Yang-Mills metric H a and the reference metric ˆ H a , we obtain(3.9) | ˜ h − σ/ a ˆ ∇ ˜ h σa | H a ,g a − σ ∆ ω a Tr ˜ h σa −h√− ω a F ˆ H a , ˜ h σa i ˆ H a . Corollary 3.3 gives the bound | Λ ω a F ˆ H a | ˆ H a C which together with and0 < ˜ h a I yields the estimate(3.10) | ˜ h − σ/ a ∇ ˜ h σa | H a ,g a − σ ∆ g a Tr ˜ h σa C, where C is independent of a, σ . Integrating both sides using the balancedcondition gives(3.11) Z X | ˜ h − σ/ a ˆ ∇ ˜ h σa | H a ,g a d vol g a C by Proposition 2.18 ( v ). Since 0 ˜ h a I , this implies(3.12) Z X | ˆ ∇ ˜ h σa | H a ,g a d vol g a C. Let K be a compact set which is the closure of an open set K o satisfying K o ∩ Z = ∅ , where Z = ∪ ki =1 C i is the union of all ( − , −
1) curves beingcontracted. The metrics ( g a , ˆ H a ) are uniformly equivalent to ( g , ˆ H ) on K .Then(3.13) Z K | ˆ ∇ ˜ h σa | H ,g d vol g C K , where ∇ is with respect to g . For each 0 < σ
1, we have weak conver-gence ˜ h σa k ⇀ ˜ h σ ∞ in W , ( K ) along a subsequence; here W , ( K ) denotesthe Sobolev space defined by ( g , ˆ H ). By a diagonal argument, there is asubsequence a i → h σa i ⇀ ˜ h σ ∞ in W , ( K ) for all σ ∈ { /n : n ∈ N } . By semicontinuity of weak conver-gence, we have the estimate Z K |∇ h σ ∞ | g d vol g lim sup i Z K |∇ ˜ h σa i | g d vol g C K . Let σ i = 1 /i , and define π ∈ W , ( K ) by( I − h σ i ∞ ) ⇀ π in the weak limit i → ∞ in W , ( K ). Exhausting X \ Z with compact sets K , we obtain an endomorphism π ∈ Γ( X \ Z, End T , X ) with regularity π ∈ W , loc ( X \ Z ). The definition of π is such that it is the projection ontoKer h ∞ , and it satisfies π = π † ˆ H = π almost everywhere.We need to verify Lemma 3.6.
The projection π is not trivial, in the sense that it is neitherthe identity nor the zero projection. Proof:
We show that h ∞ is not identically zero. We will use repeatedlythe following inequality, which is a consequence of (3.9) with σ = 1;(3.14) ∆ g a Tr ˜ h a > h√− ω a F ˆ H a , ˜ h a i ˆ H a . By its normalization, we have Tr ˜ h a x a ∈ X suchthat (Tr ˜ h a )( x a ) = 1. In U ( ε ), we have Λ ω a F ˆ H a = 0 and hence∆ g a Tr ˜ h a > , in U ( ε ) . In particular, by the maximum principle, sup U ( ε ) Tr ˜ h a sup ∂U ( ε ) Tr ˜ h a .Thus, we may assume x a ∈ { r > ε } .The metrics g a are uniformly equivalent on { r > ε/ } . In particular,we can fix a uniform number 0 < δ ≪ B δ ( x a ) ⊆ { r > ε/ } and, in local coordinates on B δ ( x a ) there is auniform constant M , independent of a , such that the eigenvalues of ( g a ) ¯ kj are bounded above by M and below by M − .From (3.14) we obtain the estimate∆ g a Tr ˜ h a − C Tr ˜ h a > . Applying the Moser iteration (e.g. [44, Theorem 4.1]) gives1 = sup B δ ( x a ) Tr ˜ h a C k Tr ˜ h a k L ( { r > ε/ } ,d vol g ) for a uniform constant C . Let K = { r > ε/ } . By (3.13) and Rellich’stheorem, we have ˜ h a → h ∞ in L ( K, d vol g ) and k Tr h ∞ k L ( K,d vol g ) > C − , therefore h ∞ is not identically zero.Finally, note that since ˜ h a converges to h ∞ pointwise almost everywhereon K , and det ˜ h a = (sup X Tr ˆ h a ) − → h ∞ has a non-trivialkernel almost everywhere on K . Hence π = 0. (cid:3) We have thus constructed a nontrivial projection π which projects ontothe kernel of h ∞ . To obtain a holomorphic subbundle, we need a furtherholomorphic condition on π . Following Uhlenbeck-Yau [83] we have, Lemma 3.7.
The projection π satisfies ( I − π ) ¯ ∂π = 0 on X \ Z .Proof. Following [83], rather than work with ( I − π ) ¯ ∂π , we differentiate( I − π ) π = 0 to obtain | ( I − π ) ¯ ∂π | H ,g = | ¯ ∂ ( I − π ) π | H ,g . Taking the adjoint with respect to ˆ H and using π † = π and ( ˆ ∇ i s ) † = ∂ ¯ i s for self-adjoint endomorphisms s , we obtain | ( I − π ) ¯ ∂π | H ,g = | π ˆ ∇ ( I − π ) | H ,g where ˆ ∇ is the covariant derivative with respect to ˆ H . We approximate theintegral of the quantity on the right-hand side by Z X | ( I − ˜ h sa ) ∇ ˆ H a ˜ h σa | H a ,g a d vol g a . The elementary inequality˜ h − σ/ > s + σs ( I − ˜ h sa )and inequality (3.11) implies Z X | ( I − ˜ h sa ) ∇ ˆ H a ˜ h σa | H a ,g a d vol g a (cid:18) s s + σ (cid:19) Z X | h − σ/ a ∇ ˆ H a h σa | H a ,g a d vol g a C (cid:18) s s + σ (cid:19) . (3.15)Let U δ = { r > δ } . Then since ˜ h sa → ˜ h s ∞ in L ( U δ ) by (3.13) and Rellich’stheorem, and ˜ h sa ⇀ ˜ h s ∞ weakly in W , ( U δ ), we have that( I − ˜ h sa ) ∇ ˆ H a ˜ h σa ⇀ ( I − ˜ h s ∞ ) ˆ ∇ ˜ h σ ∞ weakly in L ( U δ ) . We let a → Z U δ | ( I − ˜ h s ∞ ) ˆ ∇ ˜ h σ ∞ | H ,g d vol g C (cid:18) s s + σ (cid:19) . We now let s →
0, which implies Z U δ | π ˆ ∇ ˜ h σ ∞ | H ,g d vol g , and then taking σ →
0, we conclude Z U δ | π ˆ ∇ ( I − π ) | H ,g d vol g = 0 , using semi-continuity of weak convergence. (cid:3) Altogether, we have produced an endomorphism π ∈ Γ(End T , X (cid:12)(cid:12) X \ Z )such that • π ∈ W , loc ( X \ Z ) with respect to the norms ( ˆ H , g ). • π = π † = π , where † is with respect to ˆ H . • ( I − π ) ¯ ∂π = 0We will need a more precise L bound on | ˆ ∇ π | . Lemma 3.8.
For any δ > , we can estimate Z { r>δ } | ˆ ∇ π | H ,g d vol g Z { r>δ } (Tr Λ ω F ˆ H π ) d vol g . Proof:
We work on the set U δ = { r > δ } where I − ˜ h σ i a i converges weaklyas i → ∞ to π in W , ( U δ ) and ˆ H a converges to ˆ H in C ∞ ( U δ ). Z U δ (Tr √− ω F ˆ H π ) d vol g = Z U δ Tr [ √− ω F ˆ H ( π − I )] d vol g = − lim i →∞ Z U δ Tr ( √− ω i F ˆ H i ˜ h σ i i ) d vol g i (3.16)In the first equality we used Tr Λ ω F ˆ H = 0. Using the formula (2.19) forthe difference between the curvature tensors F ˆ H and F H , we obtain Z U δ (Tr Λ ω F ˆ H π ) d vol g = − lim i →∞ Z U δ Tr g j ¯ k ∂ ¯ k (˜ h − i ˆ ∇ j ˜ h i )˜ h σ i i ) d vol g i . The inequality (3.6) can be written as | h − σ/ ˆ ∇ h σ | H,g − σ ∆ g Tr h σ − g j ¯ k h ∂ ¯ k ( h − ˆ ∇ j h ) , h σ i ˆ H . Since h † = h (with respect to ˆ H ) and h u, v i ˆ H = Tr( uv † ), we obtain Z U δ (Tr Λ ω F ˆ H π ) d vol g > lim i →∞ Z U δ | ˜ h − σ/ i ˆ ∇ ˜ h σ i i | H i ,g i d vol g i − lim i →∞ Z U δ σ ∆ ω i Tr ˜ h σi d vol g i . By the balanced condition, this is Z U δ (Tr Λ ω F ˆ H π ) d vol g > lim i →∞ Z U δ | ˜ h − σ/ i ˆ ∇ ˜ h σ i i | H i ,g i d vol g i + lim i →∞ Z { r<δ } σ ∆ ω i Tr ˜ h σi d vol g i . Let 0 < δ < ε , where ε is the transition radius in the construction of ω a , sothat Λ ω a F ˆ H a = 0 on { r < δ } . It follows from (3.9) that1 σ ∆ ω i Tr ˜ h σi > , on { r < δ } . Combining this with | ˆ ∇ ˜ h σ | | ˜ h − σ/ ˆ ∇ ˜ h σ | , we obtain Z U δ (Tr Λ ω F ˆ H π ) d vol g > lim i →∞ Z U δ | ˆ ∇ ˜ h σ i i | H i ,g i d vol g i . We conclude by semi-continuity of weak convergence. (cid:3)
We now apply the work of Uhlenbeck-Yau [83] (see also [68]) to concludethat, at least over X \ Z , the projection π defines a coherent subsheaf E ⊂ T , X | X \ Z , which is locally free outside of a complex codimension 2 set. Let k > E . We can view E as defining a meromorphicmap µ E : X \ Z → Gr( k, T , X )to the Grassmann bundle of k -planes in T , X . Locally near a point in Z wecan trivialize T , X and, by taking Pl¨ucker coordinates on Gr( k, T , X ), weview µ E as a collection of meromorphic functions defined on the complementof Z . On the other hand, since Z has complex codimension 2, a classicalresult of Levi [53] (see also [20, Chapter 2]) implies that µ extends over Z . It follows that E extends over Z to a coherent sheaf (also denoted by E ) by taking the direct image of the tautological bundle over Gr( k, T , X ).We have thus produced a coherent sheaf E ⊂ T , X , locally free outside acodimension 2 set Z ′ . We will show that this sheaf contradicts the stabilityof T , X .To contradict stability, we need to show that c ( E ) · [ ω CY ] > . The only reason this does not follow immediately from the standard argu-ment is that the metrics ˆ H and ω are not smooth on X . Thus we needto show that the singularities do not contribute. Denote by ˆ H ′ the met-ric induced by ˆ H on the subbundle E| X \ Z ′ ⊂ T , X | X \ Z ′ . We begin bycomputing the slope defined by ˆ H ′ and ω . Let us introduce the notation c ( E , ˆ H ′ ) · ω = Z X \ Z ′ Tr √− F ˆ H ′ ∧ ω . The identity for the curvature of the induced connection on a subbundledefined by a projection π is ( see, e.g. [83, equation (4.16)])Tr √− ω F ˆ H ′ = Tr √− ω F ˆ H π − | ˆ ∇ π | H ,g . Therefore c ( E , ˆ H ′ ) · ω = Z X \ Z ′ (cid:20) (Tr √− ω F ˆ H π ) − | ˆ ∇ π | H ,g (cid:21) d vol g . By letting δ → c ( E , ˆ H ′ ) · ω > . We note that this quantity is finite. We can estimate the endomorphismΛ ω ( F ˆ H π ) by using that | F ˆ H | ˆ H ,g Cr − , thanks to Corollary 3.3, and | π | ˆ H C since π = π † ˆ H = π . Therefore (cid:12)(cid:12)(cid:12)(cid:12) Z X \ Z (Tr √− ω F ˆ H π ) d vol g (cid:12)(cid:12)(cid:12)(cid:12) C Z X \ Z r − d vol g C, using that, near { r = 0 } , the metric g is a cone over a five-dimensional link.The next step is to show that c ( E , ˆ H ′ ) · ω is equal to c ( E ) · [ ω CY ] .Recall that E defines a line bundle L = det E , and if e ϕ is a smooth metricon L then β = −√− ∂∂ϕ is a representative of c ( E ). For concreteness, welet β be the curvature form associated to the metric g CY (cid:12)(cid:12) E . We write thedifference as c ( E ) ∧ [ ω CY ] − c ( E , ˆ H ′ ) .ω = Z X \ Z ′ ( β − √− F ˆ H ′ ) ∧ ω + Z X \ Z β ∧ ( ω − ω ):= (I) + (II) . We will treat each term individually. Recall Z ′ is a codimension 2 analyticset containing the singularities of E (which contains Z = ∪ C i ). Let η δ bea cutoff function such that η δ ≡ { dist g CY ( Z ′ , · ) > δ } and η δ ≡ { dist g CY ( Z ′ , · ) < δ } with |√− ∂∂η δ | g CY Cδ − . • Term (I). Working near a point where E is locally free we have β − √− F ˆ H ′ = −√− ∂∂ log det g CY (cid:12)(cid:12) E det ˆ H (cid:12)(cid:12) E ! . Note that by the AM-GM inequality we have det g CY (cid:12)(cid:12) E det ˆ H (cid:12)(cid:12) E ! E ) E ) Tr (cid:16) ˆ H (cid:12)(cid:12) − E g CY (cid:12)(cid:12) E (cid:17) E ) Tr (cid:16) ˆ H − g CY (cid:17) where in the second inequality we used that ˆ H − g CY is positive definite.Similarly we have det ˆ H (cid:12)(cid:12) E det g CY (cid:12)(cid:12) E ! E ) E ) Tr (cid:16) g − CY ˆ H (cid:17) . On the other hand from (2.17) we havedist g CY ( Z, · ) / g CY g dist g CY ( Z, · ) − / g CY since r = k z k ∼ dist g CY ( Z, · ) near the singular points. Hence the sameestimates hold for ˆ H and so we have (cid:12)(cid:12)(cid:12)(cid:12) log det g CY (cid:12)(cid:12) E det ˆ H (cid:12)(cid:12) E ! (cid:12)(cid:12)(cid:12)(cid:12) − C log dist g CY ( Z, · ) + C for a uniform constant C . Integrating by parts gives Z X \ Z ′ η δ ( β − √− F ˆ H ′ ) ∧ ω = Z X \ Z ′ log det g CY (cid:12)(cid:12) E det ˆ H (cid:12)(cid:12) E ! √− ∂∂η δ ∧ ω . From the definition of η δ and the bound ω dist g CY ( Z, · ) − / ω CY we get (cid:12)(cid:12)(cid:12)(cid:12) Z X \ Z ′ η δ ( β − √− F ˆ H ′ ) ∧ ω (cid:12)(cid:12)(cid:12)(cid:12) Cδ − − / ( − log( δ ))since Vol g CY ( { x ∈ X : δ < dist g CY ( Z ′ , x ) < δ } ) ∼ δ . It follows that term(I) vanishes. • Term (II). By Proposition 2.18,(II) = Z U \ Z β ∧ ( ω CY − ω )where U is a tubular neighborhood of Z , which is a disjoint union of tubularneighborhoods of the ( − , −
1) rational curves. Since U retracts onto Z ,which has complex dimension 1 (and hence H ( U ) = 0) we can write ω CY = d Φ for a smooth 3-form Φ. On the other hand, by Proposition 2.18 ω co = √− ∂∂ Φ ′ where, Φ ′ is smooth in U \ Z and near each ( − , −
1) curve C i there is a constant λ i > ′ = λ i r √− ∂∂r = λ i r ω where we recall that r = k z k / . Thus, we have Z X \ Z η δ β ∧ ( ω CY − ω ) = Z U dη δ ∧ β ∧ Φ − Z U \ Z √− ∂∂η δ ∧ β ∧ Φ ′ . The first integral is easily seen to be of order δ . For the second integral,we use the bound g dist g CY ( Z, · ) − / g CY together with the definition of η δ and r to conclude (cid:12)(cid:12) Z U \ Z √− ∂∂η δ ∧ β ∧ Φ ′ (cid:12)(cid:12) Cδ − − . It follows that term (II) vanishes, and hence c ( E , ˆ H ′ ) · ω = c ( E ) · [ ω CY ] and hence c ( E ) · [ ω CY ] >
0, which contradicts the stability of T , X . Weconclude that sup X Tr ˆ h a C as a →
0, which proves the C estimate (3.4).3.3. Gradient estimate.
In this section, we will show that, along the se-quence ( g a , H a ), there holds an estimate of the form(3.17) |∇ g a H a | g a Cr − . We will use the ideas from Calabi’s C estimate [6], as applied in complexgeometry by Yau [86] and further developed by Phong-Sesum-Sturm [67](for other applications of this technique, see e.g. [26, 70, 81]). We will workwith the quantity S = |∇ H a h a h − a | g a ,H a , where h a = g − a H a and by the mixed norms we mean |∇ h a h − a | g a ,H a = ( g a ) j ¯ k ( H a ) ¯ βα ( H a ) µ ¯ ν ( ∇ j h a h − a ) αµ ( ∇ k h a h − a ) βν . The quantity S can also be understood as the difference of connections bythe formula(3.18) ∇ H hh − = A H − A g , where A H = H − ∂H, A g = g − ∂g. For ease of notation, in this section we omit the sequence subscript a . Wewill obtain the following estimate. Proposition 3.9.
Let ( X, g ) be a compact Hermitian complex manifold withsmooth function r : X → [0 , ∞ ) satisfying |∇ r | g Λ for Λ > . Let H be a second Hermitian metric on T , X satisfying g j ¯ k ( F H ) j ¯ kαβ = 0 and C − g H C g . Let ε > . Suppose that on { r ε } , the metric g isK¨ahler Ricci-flat, and satisfies | Rm g | g C r − . Suppose on the set { r > ε } , we have the estimate | T g | + | R g | + |∇ g R g | Λ , where T g is the torsion of g and R g is the curvature of g . Then (3.19) r |∇ hh − | H C ( C , C , Λ , ε ) where h = g − H and ∇ is the Chern connection of H . The sequence ( g a , H a ) satisfies the hypothesis of the proposition on X with a function r which is an extension of k z k / from U (1 /
2) to all of X with r − (0) = ∪ C i , and the constants are uniform in a . Indeed, the uniformbounds | Rm g a | g a Cr − , |∇ r | g a C can be seen by a scaling argument. First, the bounds hold on { r > ε } since the geometry of g a is uniform there. Second, on { r < ε } these boundshold when a = 1, and to obtain uniform bounds for all a we work in co-ordinates ( u, v, z ) used previously on O P ( − ⊕ and use the scaling map S a − ( u, v, z ) = ( a − / u, a − / v, z ). Since r = (1 + | z | )( | u | + | v | ), wehave S ∗ a − r = a − r , and we also have S ∗ a − ω co, = a − ω co,a , see, e.g. (2.18).Pulling back | Rm g | g Cr − gives the uniform estimate in a , and similarlyfor |∇ r | .Therefore, by proving Proposition 3.9, we can conclude the gradient esti-mate (3.17) since |∇ g a H a | g a = | ( ∇ g a − ∇ H a ) H a | g a = | ( ∇ H a h a h − a ) H a | g a Cr − . Laplacian of S . The proof of Proposition 3.9 will occupy the remain-der of this section. To estimate S = |∇ hh − | H , we start by differentiatingit once ∇ ¯ k S = h∇ ¯ k ( ∇ hh − ) , ∇ hh − i g,H + h∇ hh − , ∇ k ( ∇ hh − ) i g,H . The covariant derivative ∇ k here is the Chern connection of H on indicesmeasured with H , and the Chern connection of g on indices measured with g . Concretely, we mean ∇ k ( ∇ j hh − ) αβ = ∂ k ( ∇ j hh − ) αβ − ( ∇ r hh − ) αβ ( A g ) krj +( A H ) kαγ ( ∇ j hh − ) γ β − ( ∇ j hh − ) αγ ( A H ) kγβ . (3.20)Differentiating S twice gives g j ¯ k ∇ j ∇ ¯ k S = |∇ ( ∇ hh − ) | g,H + | ¯ ∇ ( ∇ hh − ) | g,H + g j ¯ k h∇ j ∇ ¯ k ( ∇ hh − ) , ∇ hh − i g,H + h∇ hh − , g k ¯ j ∇ ¯ j ∇ k ( ∇ hh − ) i g,H where |∇ ( ∇ hh − ) | g,H = g j ¯ k H p ¯ q H µ ¯ ν H ¯ βα ∇ j ( ∇ p hh − ) αµ ∇ k ( ∇ q hh − ) βν . Our curvature conventions imply the commutator relations[ ∇ j , ∇ ¯ k ] V α = F j ¯ kαγ V γ , [ ∇ j , ∇ ¯ k ] V α = − V γ F j ¯ kγα , which gives ∇ ¯ j ∇ k ( ∇ r hh − ) αβ = ∇ k ∇ ¯ j ( ∇ r hh − ) αβ + ( ∇ s hh − ) αβ ( R g ) k ¯ jsr − F k ¯ jαγ ( ∇ r hh − ) γ β + ( ∇ r hh − ) αγ F k ¯ j γβ . (3.21)Since Λ ω F H = 0 and we write g jk ( R g ) ¯ kj sr = ( R g ) sr , we have g k ¯ j ∇ ¯ j ∇ k ( ∇ r hh − ) pq = g j ¯ k ∇ j ∇ ¯ k ( ∇ r hh − ) pq + ( ∇ s hh − ) αβ ( R g ) sr . Therefore∆ g S = 2Re h g j ¯ k ∇ j ∇ ¯ k ( ∇ hh − ) , ∇ hh − i + |∇ ( ∇ hh − ) | g,H + | ¯ ∇ ( ∇ hh − ) | g,H + g s ¯ r H ¯ βα H µ ¯ ν ( ∇ s hh − ) αµ ( ∇ p hh − ) βν ( R g ) pr . We relate the highest order terms of order ∇ h to the curvatures R g , F H and their derivatives. Lemma 3.10.
We have the following equation g j ¯ k ∇ j ∇ ¯ k ( ∇ i hh − ) αβ = g j ¯ k ( ∇ g ) j ( R g ) i ¯ kαβ + g j ¯ k ( ∇ j hh − ) αγ ( R g ) i ¯ kγβ − g j ¯ k ( R g ) i ¯ kαγ ( ∇ j hh − ) γ β + g j ¯ k ∂ ¯ k ( ∇ r hh − ) αβ ( T g ) rij − g j ¯ k ( F g ) r ¯ kαβ ( T g ) rij . (3.22) Proof.
By (3.18),(3.23) ∂ ¯ k ( ∇ i hh − ) = ( R g ) i ¯ k − ( F H ) i ¯ k . Therefore(3.24) g j ¯ k ∇ j ∇ ¯ k ( ∇ i hh − ) αβ = g j ¯ k ∇ j ( R g ) i ¯ kαβ − g j ¯ k ∇ j ( F H ) i ¯ kαβ . Recall that our notation (3.20) is such that ∇ acts by the Chern connectionof H on α, β, γ indices and acts by the Chern connection of g on i, j, k indices.We will need the Bianchi identity(3.25) ∇ j ( F H ) i ¯ kαβ = ∇ i ( F H ) j ¯ kαβ + ( F H ) r ¯ kαβ ( T g ) rij where ( T g ) rij = ( A g ) irj − ( A g ) j ri is the torsion of the balanced metric g . This can be seen as follows. Ournotation for ∇ means that ∇ j ( F H ) i ¯ kαβ = ∂ j ( F H ) i ¯ kαβ − ( F H ) r ¯ kαβ ( A g ) j ri + ( A H ) j αγ ( F H ) i ¯ kγβ − ( F H ) i ¯ kαγ ( A H ) j γ β . Since F j ¯ k = − ∂ ¯ k A j , this is ∇ j ( F H ) i ¯ kαβ = − ∂ ¯ k ∂ j ( A H ) iαβ − ( F H ) r ¯ kαβ ( A g ) j ri − ( A H ) j αγ ∂ ¯ k ( A H ) iγβ + ∂ ¯ k ( A H ) iαγ ( A H ) jγ β . Therefore ∇ j ( F H ) i ¯ kαβ − ∇ i ( F H ) j ¯ kαβ = ( F H ) r ¯ kpq [( A g ) irj − ( A g ) j ri ] − ∂ ¯ k (cid:20) ∂ j ( A H ) iαβ − ∂ i ( A H ) j αβ + ( A H ) j αγ ( A H ) iγβ − ( A H ) iαγ ( A H ) jγ β (cid:21) The term on the last line is − ∂ ¯ k [ F jiαβ ], which is zero for the Chern connec-tion since both i, j are unbarred. This proves the Bianchi identity (3.25).Contracting (3.25) and using Λ ω F H = 0, we obtain g j ¯ k ∇ j ( F H ) i ¯ kαβ = g j ¯ k ( F H ) r ¯ kαβ ( T g ) rij . Substituting this into (3.24) gives,(3.26) g j ¯ k ∇ j ∇ ¯ k ( ∇ i hh − ) αβ = g j ¯ k ∇ j ( R g ) i ¯ kαβ − g j ¯ k ( F H ) r ¯ kαβ ( T g ) rij . We will convert the Chern connection of H into the Chern connection of g in the first term via ∇ j ( R g ) i ¯ kαβ = ( ∇ g ) j ( R g ) i ¯ kαβ + [( A H ) j αγ − ( A g ) j αγ ]( R g ) i ¯ kγ β − ( R g ) i ¯ kαγ [( A H ) j γβ − ( A g ) j γ β ] . By (3.18), (3.26) becomes g j ¯ k ∇ j ∇ ¯ k ( ∇ i hh − ) αβ = g j ¯ k ( ∇ gj R g ) i ¯ kαβ + g j ¯ k ( ∇ j hh − ) αγ ( R g ) i ¯ kγ β − g j ¯ k ( R g ) i ¯ kαγ ( ∇ j hh − ) γ β − g j ¯ k ( F H ) r ¯ kαβ ( T g ) rij . (3.27)Using (3.23), we obtain the statement in the lemma. (cid:3) Altogether, the Laplacian of S is(3.28)∆ g S = |∇ ( ∇ hh − ) | g,H + | ¯ ∇ ( ∇ hh − ) | g,H + 2Re (cid:20) (I) + (II) + (IIIa) + (IIIb) (cid:21) where(I) = g i ¯ ℓ H ¯ µα H β ¯ ν g j ¯ k ∂ ¯ k ( ∇ r hh − ) αβ ( T g ) rij ( ∇ ℓ hh − ) µν , (II) = g i ¯ ℓ H ¯ µα H β ¯ ν [ g j ¯ k ( ∇ gj R g ) i ¯ kαβ − g j ¯ k ( R g ) r ¯ kαβ ( T g ) rij ]( ∇ ℓ hh − ) µν (IIIa) = g i ¯ ℓ H ¯ µα H β ¯ ν [ g j ¯ k ( ∇ j hh − ) αγ ( R g ) i ¯ kγ β ]( ∇ ℓ hh − ) µν − g j ¯ k ( R g ) i ¯ kαγ ( ∇ j hh − ) γ β ( ∇ ℓ hh − ) µν (IIIb) = g i ¯ ℓ H ¯ µα H β ¯ ν ( ∇ i hh − ) αβ ( R g ) ¯ ℓ ¯ p ( ∇ p hh − ) µν Recall ε > { r ε } and { r > ε } . • On { r > ε } , the geometry ( X, g ) is uniformly bounded by a constant Λ.The C estimate C − g H C g allows us to use norms with respect to g or H interchangeably up to the cost of constant. Hence on { r > ε } we have (cid:12)(cid:12)(cid:12)(cid:12) (I) + (II) + (IIIa) + (IIIb) (cid:12)(cid:12)(cid:12)(cid:12) C Λ ,C (cid:20) | ¯ ∇ ( ∇ hh − ) | g,H |∇ hh − | g,H + |∇ hh − | g,H + |∇ hh − | g,H (cid:21) and (cid:12)(cid:12)(cid:12)(cid:12) (I) + (II) + (IIIa) + (IIIb) (cid:12)(cid:12)(cid:12)(cid:12)
14 ( |∇ ( ∇ hh − ) | g,H + | ¯ ∇ ( ∇ hh − ) | g,H ) + C ( S + 1)(3.29)where C depends on C and Λ. • On { r < ε } , the metric g is K¨ahler Ricci-flat. Term (I) vanishes sincethe torsion T g = 0. Term (II) vanishes by the Bianchi identity g j ¯ k ∇ j R i ¯ kαβ = g j ¯ k ∇ i R j ¯ kαβ = 0combined with the Ricci-flat condition. Term (IIIb) also vanishes and weare left with∆ g S = |∇ ( ∇ hh − ) | g,H + | ¯ ∇ ( ∇ hh − ) | g,H + 2Re (IIIa) . By uniform equivalence of the metrics g and H , we may estimate this as | (IIIa) | C ( C ) | Rm g | g S. By the estimate | Rm g | C r − , we see that on the entirety of X we canestimate (cid:12)(cid:12)(cid:12)(cid:12) (I) + (II) + (IIIa) + (IIIb) (cid:12)(cid:12)(cid:12)(cid:12)
14 ( |∇ ( ∇ hh − ) | g,H + | ¯ ∇ ( ∇ hh − ) | g,H ) + Cr − ( S + 1)(3.30)where C depends on C , C , Λ, ε .3.3.2. Test function.
To construct a test function to control S , we will useTr h . Contracting the formula (2.19) for the difference of curvature tensorsand using Λ ω F H = 0, we see that √− ω Rm g = − g j ¯ k ∂ ¯ k ( h ∇ Hj h − ) = g j ¯ k ∂ ¯ k ( ∇ Hj hh − ) . Therefore Tr ( √− ω Rm g ) h = ∆ g Tr h − g j ¯ k Tr ∇ j hh − ∇ ¯ k h. We note that since Λ ω Rm g = 0 in { r ε } , we have the bound | Λ ω Rm g | C (Λ) . Let δ >
0. Let ζ ( s ) : [0 , ∞ ) → [0 ,
1] be a cutoff function satisfying ζ ( s ) ≡ s > ζ ( s ) ≡ s
1, and | ζ ′ | ζ . We will use the testfunction P ( z ) = ζ δ ( z ) S ( z ) + Aδ Tr h ( z ) , ζ δ ( z ) = ζ (cid:18) r ( z ) δ (cid:19) , for A (Λ , C , C ) ≫ g P = ζ δ ∆ g S + S ∆ g ζ δ + 2Re g j ¯ k ∂ j ζ δ ∂ ¯ k S + Aδ − ∆ g Tr h. By (3.28),∆ g P = ζ δ ( |∇ ( ∇ hh − ) | g,H + | ¯ ∇ ( ∇ hh − ) | g,H ) + Aδ − g j ¯ k Tr ∇ j hh − ∇ ¯ k h +2 ζ δ Re (cid:20) (I) + (II) + (III) (cid:21) + (IV) + (V) + (VI)(3.31)where(IV) = S ∆ g ζ δ (V) = 2Re g j ¯ k ∂ j ζ δ [ h∇ ¯ k ( ∇ hh − ) , ∇ hh − i + h∇ hh − , ∇ k ( ∇ hh − ) i ](VI) = Aδ − Tr ( √− Rm g ) h We want to show that P is bounded by C ( C , C , Λ) δ − . If P attains amaximum on { r δ } , then ζ δ = 0 and P is bounded by Aδ − sup X Tr h .Suppose P attains a maximum at a point x ∈ { r > δ } . Our good term willbe(3.32) Aδ − g j ¯ k Tr ∇ j hh − ∇ ¯ k h > Aδ C ( C ) |∇ hh − | g,H = ACδ S using C − g H C g and h † = h . For the term (V), | (V) | > − CS / |∇ ζ δ | g ( |∇ ( ∇ hh − ) | g,H + |∇ ( ∇ hh − ) | g,H ) . We have |∇ ζ δ | g = δ − | ζ ′ ||∇ r | g C (Λ) δ − | ζ ′ | . Since | ζ ′ | | ζ | / , we have | (V) | > − Cδ − S / | ζ δ | / ( |∇ ( ∇ hh − ) | + |∇ ( ∇ hh − ) | )Using 2 ab a + b , we can estimate | (V) | > − | ζ δ | ( |∇ ( ∇ hh − ) | + |∇ ( ∇ hh − ) | ) − Cδ − S. In (3.30), we showed the estimate2 ζ δ Re (cid:20) (I)+(II)+(III) (cid:21) > − | ζ δ | ( |∇ ( ∇ hh − ) | + |∇ ( ∇ hh − ) | ) − C | ζ δ | r − S. We are working in the region where r − < δ − , hence we obtain at x theinequality0 > ∆ P ( x ) > ζ δ |∇ ( ∇ hh − ) | g,H + | ¯ ∇ ( ∇ hh − ) | g,H ) + AC δ − S − Cδ − S − CAδ − . For A ≫ C , C , Λ, we obtain S ( x ) C. It follows that P ( x ) is bounded by Cδ − . From the bound of P , we obtaina bound for S on { r > δ } since ζ ≡ { r > δ } S Cδ − . It follows that for any point x ∈ X , we have the estimate S ( x ) Cr ( x ) − . This completes the proof of Proposition 3.9.3.4.
Higher estimates.
By combining the C and C estimates along thedegenerating sequence ( g a , H a ), we can apply regularity theory of ellipticequations to obtain higher order estimates and obtain a limit as a → Proof of Theorem 3.1:
To extract a limit from ( g a , H a ), we fix δ > U δ = { r > δ } . Since g a → g smoothly uniformly on U δ ,we can cover U δ by finitely many charts where the matrices representing themetrics g a satisfyΛ − δ δ kj ( g a ) ¯ kj Λ δ δ kj , k ( g a ) ¯ kj k C k Λ δ,k . The C estimate (3.4) for H a implies that on this cover of U δ , the localmatrices satisfy C − δ kj ( H a ) ¯ kj Cδ kj . The C estimate (3.17) gives a uniform bound k ∂ ( H a ) ¯ µν k L ∞ ( U δ ) C uniform in a on the local matrices ( H a ) ¯ µν . The equation Λ ω a F H a = 0 isgiven in local charts as( g a ) j ¯ k ∂ j ∂ ¯ k ( H a ) ¯ µν = ( g a ) j ¯ k ∂ ¯ k ( H a ) ¯ µγ ( H a ) γ ¯ α ∂ j ( H a ) ¯ αν . The right-hand side is bounded in L ∞ . By the local C ,α estimate for ellipticPDE, the local matrices ( H a ) ¯ µν satisfy k ( H a ) ¯ µν k C ,α ( U δ ) C. By the local Schauder estimates, we can take a smooth limit of the sequence( g a , H a ) on U δ as a →
0. The limiting metric H satisfiesΛ ω F H = 0 on U δ . We can now let δ → H on X reg . The C and C estimates imply(3.33) | H | g + | H − | g + r |∇ g H | g C. To obtain higher estimates, we work near the singularities of X , which canbe identified with a neighborhood of V with g = g co, . In holomorphiccylindrical coordinates (see Lemma 2.11), we have r g − co, = O ( I ) (notation O ( I ) is used for a matrix uniformly equivalent to the identity) and theequation r ( g co, ) j ¯ k ∂ j ∂ ¯ k ( H ) ¯ µν = r ( g co, ) j ¯ k ∂ ¯ k ( H ) ¯ µγ ( H ) γ ¯ α ∂ j ( H ) ¯ αν . Estimate (3.33) in these coordinates is H = r O ( I ) and ∂H = r O (1).Therefore this local equation is of the form a i ¯ j ∂ i ∂ ¯ j H = f, f = r O (1) , a i ¯ j = O ( I ) . Local C ,α estimates for elliptic PDE imply k ( H ) ¯ µν k C ,α ( B / ,g euc ) C ( k H ¯ µν k L ∞ ( B ,g euc ) + k f k L ∞ ( B ) ) Cr . Local Schauder estimates then imply k ( H ) ¯ µν k C k,α ( B / ,g euc ) C k r . Con-verting these local estimates to norms using g co, gives |∇ kg co, H | g co, C k r − k which completes the estimate. (cid:3) Quantitative convergence to the tangent cone
In this section we show that the estimates for the Hermitian-Yang-Millsmetric H constructed in Theorem 3.1 can be improved to obtain the decayof H towards the Candelas-de la Ossa metric. This will be an essentialingredient in the perturbation argument later in the paper. The main goalof this section is to prove Theorem 4.1.
Let ( V , g co, ) denote the conifold equipped with Candelas-de la Ossa Ricci-flat K¨ahler cone metric, and let ∈ V denote the tipof the cone. Suppose H is a Hermitian-Yang-Mills metric on T , V over B (0) \ { } . Assume that there is a constant C > so that H satisfies C − g co, < H < Cg co, . Then there are constants c > , λ ∈ (0 , , and for each k ∈ Z > a constant C k > , such that the following estimate holds |∇ kg co, ( H − c g co, ) | g co, C k r λ − k . where, as usual, r ( x ) = dg co, ( x, is the distance to ∈ V with respect to g co, . The proof of Theorem 4.1 follows closely the work of Jacob-S´a Earp-Walpuski [48] who studied related quantitative convergence results in thecase of punctured balls in C n . Related results for stationary Yang-Mills con-nections were obtained by Yang using a Lojasiewicz inequality [84]. Chen-Sun [12, 13, 14, 15] obtained a general characterization tangent cones ofHermitian-Yang-Mills connections on reflexive sheaves on the ball in C n without estimates for the convergence rate. For our applications, the poly-nomial decay rate, as well as the convergence at the level of metrics (ratherthan connections) obtained in Theorem 4.1 is crucial.The first step towards establishing Theorem 4.1 is to prove the followingPoincar´e inequality. Lemma 4.2.
Let ( V , g co, ) be the conifold equipped with the Candelas-de laOssa metric. There is a uniform constant C > with the following property:for any ρ ∈ (0 , and any s ∈ C ∞ ( { r = ρ } , √− su ( T , V , g co, )) we have Z { r = ρ } | s | g co, dS ( ρ ) g co, Cρ Z { r = ρ } |∇ g co, s | g co, dS ( ρ ) g co, where dS ( ρ ) g co, denotes the surface measure on { r = ρ } induced by g co, .Proof. The result follows from standard elliptic theory and scaling providedwe can show that there are no parallel sections of √− su ( T , V , g co, ) onthe link of the cone { r = 1 } .To begin, recall from Section 2 that V can be identified with the comple-ment of the zero section in ι ∗ O P ( −
1) where ι : P × P → P is the Segre embedding. In particular, there is a projection π : V \ { } → P × P whose fibers are orbits of the holomorphic Reeb vector field. Let E → P × P be the holomorphic vector bundle generated by the invariantsections of T , V , so that T , V = π ∗ E . We can describe E explicitly; if L ⊂ T , V denotes the trivial line bundle generated by the non-vanishingholomorphic Reeb field, then we have an exact sequence0 → L → T , V → π ∗ T , ( P × P ) → . Note that the L -valued (1 ,
0) form on V given by ξ ⊗ ∂ log r is preciselythe orthogonal projection T , V → L given by the Calabi-Yau metric g co, .Therefore the second fundamental form of L ⊂ T , V is represented by ξ ⊗ √− ∂∂ log r = ξ ⊗ π ∗ ω KE where ω KE is the K¨ahler-Einstein metric on P × P satisfyingRic( ω KE ) = 3 ω KE . Since L is trivial, we can view E as the bundle corresponding to c ( P × P )under the isomorphismsExt ( T , ( P × P ) , O P × P ) ∼ = H ( T , ( P × P ) ∨ ) ∼ = H , ( P × P , C ) Thus E sits in an exact sequence(4.1) 0 → O P × P → E → T , ( P × P ) → . Furthermore, since g co, is Calabi-Yau on the cone, one can easily show that E admits a natural Hermitian-Yang-Mills metric with respect to the K¨ahlerclass c ( p ∗ O P (1) ⊗ p ∗ O P (1)), see e.g. [78] for related discussion.It is easy to show, by direct calculation, that any parallel section of s ∈ C ∞ ( { r = ρ } , √− su ( T , V , g co, )) descends to a trace-free, global holo-morphic section s ∈ H ( P × P , End( E )). Thus, it suffices to show thatthe only global holomorphic endomorphisms of E are multiples of the iden-tity map. This will follow from the usual result for stable vector bundlesprovided we can show that E is indecomposable [77]. Let H E denote theHermitian-Yang-Mills connection on E and let ω = p ∗ ω F S + p ∗ ω F S where ω F S denotes the Fubini-Study metric on P . By the exact sequence (4.1),the slope of E is given by µ ( E ) = c ( E ) ∪ [ ω ]rk( E ) = 2 R P × P ω E can be holomorphically decomposed as E = E ⊕ E where rk( E ) = 1 ,
2. A standard computation shows that the decomposition E = E ⊕ E is orthogonal with respect to the Hermitian-Yang-Mills metricand the restriction H E (cid:12)(cid:12) E is Hermitian-Yang-Mills with slope µ ( E ) = µ ( E )[77]. Thus we have c ( E ) ∪ [ ω ] = 43 rk( E ) . However, since c ( E ) , [ ω ] ∈ H ( P × P , Z ) this implies that rk( E ) ∈ Z ,which is impossible since rk( E ) = 1 ,
2. Therefore E is indecomposable andhence stable. The result follows. (cid:3) In the remainder of this section we will show that Lemma 4.2 impliesTheorem 4.1. Much of the argument is based on the following well-knownformula: if H, ˆ H are Hermitian metrics on a holomorphic vector bundle E ,then the positive definite, hermitian (with respect to either H, ˆ H ) endomor-phism h = H − ˆ H satisfies(4.2) F j ¯ k − ˆ F j ¯ k = ∂ ¯ k ( h − ∇ j h )where F (resp. ˆ F ) denotes the curvature of the Chern connection, ∇ (resp.ˆ ∇ ) defined with respect to H (resp. ˆ H ). In our case we will take H = g theCalabi-Yau cone metric on V . We begin with the following lemma, whichshows that, at least at the level of the determinant, the metric H decaystowards g co, . Lemma 4.3.
Let ( V , g ) be a Calabi-Yau cone of real dimension n > .Suppose H is a Hermitian-Yang-Mills metric on T , V → B (0) \ { } with slope . Suppose there is a constant C > such that the h = g − H sat-isfies C − Id < h < CId . Then, there are constants C k , C ∗ , γ > , with γ depending only on ( V , g co, ) so that, for each k ∈ N we have (cid:12)(cid:12)(cid:12)(cid:12) ∇ kg (log (det h ) − C ∗ ) (cid:12)(cid:12)(cid:12)(cid:12) g C k r γ − k Proof.
Since g is Ricci flat, and H is Hermitian-Yang-Mills with slope 0, itfollows from (4.2) that g j ¯ k ∂ j ∂ ¯ k log det h = − g j ¯ k F j ¯ k = 0 . On the other hand, since log det h is bounded the result follows from sep-aration of variables. To see this recall that if ϕ λ is a function on the link L := ∂B (0) ⊂ V satisfying ∆ g L ϕ λ + λϕ λ = 0with λ >
0, then we can produce harmonic functions u ± λ on V given by u ± λ = r a ( λ ) ± ϕ λ where 2 a ( λ ) ± = − ( n − ± p ( n − + 4 λ (since n > { ϕ λ } λ ∈ Spec(∆ gL ) , standard elliptic theorysays we can write log det h (cid:12)(cid:12) L = X λ ∈ Spec(∆ gL ) c λ ϕ λ for constant c λ . Now, we claim that since log det h is bounded we havelog det h = X λ ∈ Spec(∆ gL ) c λ u + λ := u. This follows from the Caccioppoli inequality; let η be a standard, smoothcut-off function on R with η ( x ) = 0 for x
1, and η ( x ) = 1 for x >
2. Consider η ε ( y ) = η ( ε − d g ( y, u := log det h − u is harmonic,bounded and vanishes on the link L we have0 = − Z B η ε ˆ u ∆ g ˆ ud vol g = Z B h∇ ( η ε ˆ u ) , ∇ ˆ u i d vol g Applying Cauchy-Schwarz we obtain Z B η ε |∇ ˆ u | C Z B ε \ B ε |∇ η ε | ˆ u . Now we have |∇ η ε | Cε − , while | B ε \ B ε | Cε n and so, since n > ε → |∇ ˆ u | ≡ k = 0 follows from the fact that the only harmonicfunction on the link { r = 1 } is a constant. Combining this with standardestimates for harmonic functions and scaling, we obtain the result for k > (cid:3) Next we prove that the relative endomorphism h = g − co, H is W , on thecone. Lemma 4.4.
With h = g − co, H as above, we have |∇ g co, h | g co, ∈ L ( B (0) , g co, ) .Proof. To ease notation let us denote g = g co, and ∇ = ∇ g co, . To provethe L bound, observe that (4.2) implies g j ¯ k ∂ j ∂ ¯ k Tr h = g j ¯ k Tr ( ∂ ¯ k hh − ∇ j h ) . On the other hand, since h is hermitian and bounded we have C − |∇ h | g g j ¯ k Tr ( ∂ ¯ k hh − ∇ j h ) C |∇ h | g . This implies an L estimate for |∇ h | g . To see this, let η ( x ) be a standardcut-off function such that η ( x ) ≡ x ∈ [0 , η ( x ) ≡ x ∈ [2 , ∞ )and | η ′ | + | η ′′ | <
10. For y > η y = η (cid:0) y − r (cid:1) . Since Tr h is uniformlybounded we have Z B g j ¯ k Tr ( ∂ ¯ k hh − ∇ j h ) d vol g lim ε → Z B (1 − η ε ) ( η ) g j ¯ k Tr ( ∂ ¯ k hh − ∇ j h ) d vol g = lim ε → Z B (1 − η ε ) η (∆ g Tr h ) d vol g = lim ε → Z B (cid:0) ∆ g (1 − η ε ) η (cid:1) Tr hd vol g C lim ε → ε − Vol g ( B ε \ B ε ) + C and the result follows since ε − Vol g ( B ε \ B ε ) Cε . (cid:3) The next step is to establish some decay for the endomorphism h = g − co, H . Lemma 4.5.
Define a hermitian endomorphism s by e s = (cid:18) det g co, det H (cid:19) · g − co, H. Then are constants
C > , α ∈ (0 , depending on ( V , g co, ) and | s | L ∞ ( B (0)) such that µ ( τ ) := Z B τ r − n |∇ g co, s | g co, d vol g co, Cτ α for all τ .Proof. Again, we denote g = g co, and ∇ = ∇ g co, to ease notation. It is nothard to check that h e − s ∇ j e s , s i g = h∇ s, s i . Therefore, from (4.2) we have g j ¯ k ∂ j ∂ ¯ k | s | g = g j ¯ k ∂ ¯ k h∇ j s, s i g + g j ¯ k h s, ∂ ¯ j s i g = g j ¯ k ∂ ¯ k h e − s ∇ j e s , s i g + g j ¯ k h s, e − s ∂ ¯ j e s i g = g j ¯ k h e − s ∇ j e s , ∇ k s i g + g j ¯ k h ∂ ¯ k s, e − s ∂ ¯ j e s i g Note that the bound C − g < H < Cg co, implies that | s | < C . Thanks to[83, Lemma 2.1] there is a uniform constant A > C sothat g j ¯ k h e − s ∇ j e s , ∇ k s i g + g j ¯ k h ∂ ¯ k s, e − s ∂ ¯ j e s i g > A − |∇ s | . In summation, we have show that(4.3) ∆ | s | g > A − |∇ s | g We now show that µ ( τ ) < C for some constant C independent of τ . Let η y ( r ) be the cut-off function from Lemma 4.3. First note that, for any ε > g r − n = 0 and | ∆ g η τ | Cτ − , togetherwith the bound for | s | yields(4.4) Z B τ \ B ε r − n |∇ s | g d vol g A Z B τ \ B ε (1 − η ε ) η τ r − n ∆ | s | g d vol g A ′ ε − n Z B ε \ B ε | s | dV g + Z B τ \ B τ r − n | s | d vol g ! which is bounded thanks to the L ∞ bound for s . To improve the estimatewe decompose the second integral appearing on the right of (4.4) as Z B τ \ B τ | s | d vol g Z ττ dr · Z ∂B r (0) | s | dS g ( r ) ! where dS g ( r ) denotes the surface measure on ∂B r . Since s is trace free wecan apply the Poincar´e inequality in Lemma 4.2 to get Z ∂B r | s | dS g ( r ) Cr Z ∂B r (0) |∇ T s | dS g ( r ) Cr Z ∂B r |∇ s | dS g ( r )where we wrote ∇ T for the covariant derivative tangent to ∂B r . Thus, wehave Z B τ \ B τ r − n | s | d vol g C Z B τ \ B τ r − n |∇ s | d vol g = C ( µ (2 τ ) − µ ( τ ))Arguing similarly for the first term yields Z B ε \ B ε ε − n | s | d vol g Cµ (2 ε )All together this implies µ ( τ ) CC + 1 ( µ (2 τ ) + µ (2 ε )) . Since this estimate holds for all ε > µ ( ε ) → µ ( τ ) CC + 1 µ (2 τ ) . The lemma follows by a standard iteration argument. (cid:3) Theorem 4.1 will now follow from Lemma 4.5 together with the regularitytheory for the Hermitian-Yang-Mills equation, which we recall below. Theregularity theory is originally due to Bando-Siu [3, Proposition 1], but werefer the reader to the paper of Jacob-Walpuski [49, Theorem C.1] for theprecise statement which implies the one below.
Proposition 4.6.
Let ( Y, g, J ) be a K¨ahler manifold of dimension n withbounded geometry, and let E → Y be a holomorphic vector bundle. If H , H are hermitian metrics on E , H is Hermitian-Yang-Mills and s :=log( H − H ) ∈ C ∞ ( Y, √− su ( E, H )) , then, for all k ∈ N and p ∈ (1 , ∞ ) there is a function f k,p ( y ) > depending only on k, p and the geometry of ( Y, g ) such that f k,p (0) = 0 and r k +2 − np k∇ k +2 H s k L p ( B r ( x )) f k,p k s k L ∞ ( B r ( x )) + k X i =0 r i k∇ iH F H k L ∞ ( B r ( x )) ! . We can now prove Theorem 4.1.
Proof of Theorem 4.1.
Let H be as in the statement of the theorem, and set g = g co, . Throughout the proof C will denote a constant which can changefrom line to line, but depends only on ( V , g ) and the positive upper andlower bounds for g − H . Define s by e s = (cid:18) det g co, det H (cid:19) · g − co, H. so that s ∈ C ∞ ( V , √− su ( T , V , g )).Fix 0 < R ≪ B R \ B R/ . Let m R : V → V be the map m R ( p ) = R − · p where · denotes the natural scaling action onthe cone. Let ˆ s = m ∗ R s . From the scale invariance of µ ( τ ) we have Z B \ B |∇ ˆ s | g d vol g CR α while the Poincar´e inequality proved in Lemma 4.2 implies(4.5) Z B \ B | ˆ s | C Z B \ B |∇ ˆ s | CR α . Note that the L ∞ bound for ˆ s together with the interior estimates, Propo-sition 4.6, yield k∇ k +2 ˆ s k L p ( B \ B / ) C k,p and hence, by the Sobolev imbedding theorem we get(4.6) k ˆ s k C k ( B \ B / ) C k . To improve this bound to a decay estimate we appeal to the Hermitian-Yang-Mills equation. From (4.2), we have(4.7) g j ¯ k ∇ ¯ k ∇ j ˆ s + B ( ∇ ˆ s ⊗ ∇ ˆ s ) = 0 where B ( · ) is linear with coefficients depending on ˆ s , but not on any of itsderivatives. The result now follows from standard elliptic regularity andbootstrapping. By elliptic regularity we have k ˆ s k W , ( B . \ B / ) C (cid:16) k ∆ˆ s k L ( B \ B / ) + k ˆ s k L ( B \ B / ) (cid:17) . On the other hand, from (4.7) we have k ∆ˆ s k L ( B \ B / ) C Z B \ B / |∇ ˆ s | ! C Z B \ B / |∇ ˆ s | ! where, in the second inequality, we used (4.6) with k = 2. From (4.5) weconclude that k ˆ s k W , ( B . \ B / ) CR α , for a uniform constant C >
0. By differentiating (4.7) a straightforwardboot-strapping argument yields k ˆ s k W k, ( B \ B ) C k R α for uniform constants C k >
0. All together we obtain | ˆ s | C k ( B \ B ) C k R α from the Sobolev imbedding theorem. Rescaling yields(4.8) | s | C k ( B R \ B R ) C k R α − k . Finally, Theorem 4.1 follows from this estimate together with Lemma 4.3. (cid:3)
Remark 4.7.
In the setting where we apply Theorem 4.1, where the metric H obtained as a limit of ( H a , g a ), we can apply Theorem 3.1 to obtain (4.6)bypassing the Bando-Siu regularity theorem.5. Approximate Hermitian-Yang-Mills metrics
In the previous section, we started from a Calabi-Yau metric ω CY on asimply connected K¨ahler Calabi-Yau threefold ( X, Ω) and constructed a pairof metrics ( g , H ) on a singular space X obtained by contracting ( − , − C i . These metrics satisfy dω = 0 and Λ ω F H = 0. Let { p i } denotethe nodal points of X . There are constants c i , R i , λ > • The Fu-Li-Yau construction gives g = R i g co, , near node p i . • By (4.8) and Lemma 4.3, for k ∈ Z > there exists M k > |∇ kg co, ( H − c i g co, ) | g co, M k r λ − k , near node p i . For ease of notation, in this section we will work at a single node point withscale constants c = R = 1. The metric H on X has conical singularitieswhich we will desingularize by gluing in the asymptotically conical metric g co,t on X t (for other work in geometry using this technique, see e.g. [11, 50]).This will provide an approximate Hermitian-Yang-Mills metric H t on thesmoothings X t for t sufficiently small.Recall that under the assumption of Theorem 2.3 there is a smoothing µ : X → ∆ with µ − ( t ) = X t and the family X is locally described by { ( z, t ) : z ∈ V t } near the nodes, with V t = { P i =1 z i = t } ⊂ C .We will show there exists γ, ε ∈ (0 ,
1) and
C > H t satisfies k Λ ω t F H t k C ,aβ − C | t | γ for all 0 < | t | ε , and for suitably defined weighted H¨older spaces with β ∈ [ − , < a <
1; see Section 5.2 below for a precise definition.We recall that we denote by ω t the Hermitian metric from Proposition 2.17constructed by Fu-Li-Yau, which satisfies ω t = ω co,t near the nodes andconverges back to the balanced metric ω on compact sets as t → Definition of the approximate solution.
To construct a Hermit-ian metric H t on X t which approximately solves the Hermitian-Yang-Millsequation, we will glue g co,t to a deformation of the singular metric H onthe annulus region {| t | α k z k | t | α } ⊂ V t . Here 0 < α <
1, and specifically we will take α = (1 + λ/ − where λ > χ ( z ) = ζ ( | t | − α k z k )be a cutoff function on this annulus region, i.e. the function ζ : [0 , ∞ ) → [0 ,
1] satisfies ζ ≡ ,
1] and ζ ≡ , ∞ ). Our glued Hermitianmetric on X t is(5.2) H t = χg co,t + (1 − χ ) K t , where K t = [(Φ − t ) ∗ H ] , is the J t -invariant part of the pullback (Φ − t ) ∗ H . Explicitly, we define( A , ) αβ = ( A αβ + J µα A µν J νβ ) for a symmetric 2-tensor A . Recall Φ t isdefined in Lemma 2.13, and note that K t is defined on X t \{k z k = | t |} andso H t is defined on all of X t .We will need estimates on the glued metric H t which are uniform in t . Lemma 5.1.
There exists constants ε, C k > such that for all < | t | < ε we have (5.3) C − g t H t C g t , |∇ kg t H t | g t C k r − k . Proof:
We work region-by-region. • Region {k z k | t | α } . Here H t = g t = g co,t so the estimates are trivial. • Region {| t | α k z k } . Here g t = g co,t and k z k ≫ | t | for all t smallenough. The estimate (3.1) reads | H | g co, C and | H − | g co, C , and sopulling back by by Φ − t and using Lemma 2.10 gives C − g co,t (Φ − t ) ∗ H Cg co,t . Since | [(Φ − t ) ∗ H ] , | g co,t | (Φ − t ) ∗ H | g co,t and similarly for H − , this provesthat C − g co,t H t Cg co,t . Next, pulling back |∇ kg co, H | C k r − k byLemma 2.10 gives |∇ kg co,t K t | g co,t Cr − k . This proves (5.3) in the region { | t | α k z k } where H t = K t =[(Φ − t ) ∗ H ] , . In the transition region {| t | α k z k k t k α } we have, ∇ H t = ( ∇ χ )( g co,t − K t ) + (1 − χ ) ∇ K t and |∇ g t H t | g t C |∇ χ | g t + C |∇ g t K t | g t . We estimate(5.4) |∇ χ | g t C | t | − α |∇ r | g t C | t | − α r Cr − . Here we used |∇ r | g t C and r = k z k | t | α in the transition region.Thus(5.5) |∇ g co,t H t | g co,t Cr − and the higher order estimates are similar. • Region { r } . Here H t = [(Φ − t ) ∗ H ] , and the metrics Φ ∗ t g t convergesmoothly uniformly to g as t → H t follow from pulling back the estimates for H obtained in Theorem 3.1. (cid:3) Weighted H¨older spaces.
In the upcoming analysis we will workin weighted H¨older spaces on X t using the weight function r , the metric g t which is equal to the model metric g co,t near the nodes, and the gluedmetric H t . We will use the connection ∇ H t when differentiating. For endo-morphisms h ∈ Γ(End T , X t ), we use the norm k h k C kβ ( g t ,H t ) = k X i =0 sup X t | r − β + i ∇ iH t h | g t . For Φ ∈ Γ((
T X t ) p ⊗ ( T ∗ X t ) q ), we define the semi-norm[Φ] C ,aβ = sup x = y (cid:20) min( r ( x ) , r ( y )) − β | Φ( x ) − Φ( y ) | g t d ( x, y ) a (cid:21) where the sup is taken over points x, y with distance less than the injectivityradius and Φ( x ) − Φ( y ) is understood by ∇ g t -parallel transport along the minimal g t geodesic connecting x and y . The weighted H¨older norms arethen k h k C k,aβ ( g t ,H t ) = k h k C kβ ( g t ,H t ) + [ ∇ kH t h ] C ,aβ − k − a . This definition is well adapted to work on annuli U ˆ r = { (1 / r r ( z ) r } at a given scale ˆ r >
0. The norm over U ˆ r is equivalent to(5.6) k h k C ,aβ ( U ˆ r ) = ˆ r − β (cid:20) sup U ˆ r k h k C (ˆ r − g t ) + sup x,y ∈ U ˆ r | h ( x ) − h ( y ) | ˆ r − g t d ˆ r − g t ( x, y ) a (cid:21) , where norms on the endomorphism h ∈ Γ(End T , X t ) are now all withrespect to the rescaled metric ˆ r − g t . We will often estimate global H¨oldernorms by estimating them on local annuli U ˆ r . Lemma 5.2.
Let β and h ∈ Γ(End T , X t ) . Suppose there is a uniformbound on the local estimates k h k C ,aβ ( U ˆ r ) K for all ˆ r > , where U ˆ r = { (1 / r r r } ⊂ X t . Then k h k C ,aβ ( X t ) CK for a constant C > which is independent of t .Proof: The local bounds imply k h k C β ( X t ) K , so we need to estimatethe global H¨older semi-norm. Let x, y ∈ X t and suppose r ( x ) r ( y ). If y lies in the set U = { (1 / r ( x ) < r < r ( x ) } , the estimate is assumed. If r ( x ) > ε >
0, the geometry is uniform in t and the estimate holds for C ( ε ).In the remaining case, we assume x, y ∈ V t with 2 r ( x ) < r ( y ) and g t = g co,t and we claim that d g t ( x, y ) > C − r ( x )for C > t . Indeed, this inequality holds at t = 1 for aconstant C >
1, and the uniform bound in t follows from the bound when t = 1 by scaling S ( z ) = t − / z with S ∗ g co, = | t | − / g co,t and S ∗ r = | t | − / r .Therefore r ( x ) − β + a | h ( x ) − h ( y ) | d g t ( x, y ) a is bounded by C k h k C β ( X t ) since β (cid:3) We end this discussion with the following remark: if r − β | h | g t + r − β +1 |∇ g t h | g t K, then for 0 < a < k h k C ,aβ ( X t ) CK where C is indepen-dent of t . This can be seen for example from expression (5.6), since ˆ r − g t isuniformly (in t ) equivalent to the Euclidean metric in holomorphic cylindri-cal coordinates (Lemma 2.11). Also, the difference of connections satisfiesthe bound r | A g t − A H t | g t C by Lemma 5.1, so to estimate k h k C ,aβ ( X t ) wecould equivalently estimate r − β +1 |∇ H t h | H t instead of r − β +1 |∇ g t h | g t . Smallness of the approximate solution.
The main objective of thissection is to show that the glued metric H t has small Hermitian-Yang-Millstensor. Proposition 5.3.
Let H t be the glued metric as in (5.2). There exists C > and ε > such that for any < a < , and any t ∈ C ∗ with | t | ε we have (5.7) k Λ ω t F H t k C ,a − C | t | αλ . where λ > is the rate in (5.1) (see Theorem 4.1), and α = (1 + λ ) − . The approximate solution will be estimated in four regions. • Region {k z k | t | α } . Here H t = g co,t and Λ ω t F H t = 0. • Region { | t | α k z k | t | α } . This contains the transition region, andwe will show that here k Λ ω t F H t k C ,a − C | t | ( αλ ) / in Lemma 5.4 below. • Region { | t | α k z k } . In this region, H t = (Φ − t ) ∗ H and we needto control the Hermitian-Yang-Mills tensor of (Φ − t ) ∗ H . We will estimate k Λ ω t F H t k C ,a − C | t | − α in this region in Lemma 5.5 below. • Region { r > } . In this region the geometry is smoothly varying, andso since Λ ω F H = 0 then k Λ ω t F H t k C ,a − C | t | . By Lemma 5.2, it suffices to check the H¨older estimate on these local piecesto obtain the global estimate.We start by estimating the Hermitian-Yang-Mills tensor in the transitionregion.
Lemma 5.4.
With notation as in Proposition 5.3, the estimate k Λ ω t F H t k C ,a − ( U ) C | t | αλ holds in the region U = { | t | α k z k | t | α } .Proof: If we decompose H = g co, + E , then the glued metric is H t = g co,t + (1 − χ ) (cid:20) (Φ − t ) ∗ E + (Φ − t ) ∗ g co, − g co,t (cid:21) . Since Λ ω t F g co,t = 0, the formula (2.19) for the difference of curvature tensorsgives(5.8) √− ω t F H t = − ( g co,t ) j ¯ k ∂ ¯ k ( h − t ( ∇ g co,t ) j h t ) where(5.9) h t = I + (1 − χ ) (cid:20) g − co,t (Φ − t ) ∗ E + g − co,t [(Φ − t ) ∗ g co, − g co,t ] (cid:21) . During this proof, we simply write ∇ = ∇ g co,t . We claim that(5.10) C − I h t CI, |∇ k h t | g co,t Cr − k | t | λα/ . Assuming this, (5.8) and | t | λα/ < | Λ ω t F H t | g co,t | h − ∇ h | g co,t + C | h − ∇ h t | g co,t Cr − | t | λα/ . Similarly |∇ Λ ω t F H t | g co,t Cr − | t | λα/ and this proves the estimate | Λ ω t F H t | C a − ( U ) C | t | λα/ .We now prove the claim (5.10). Estimate (5.1) implies |∇ k E | g co, C k r λ − k , which by Lemma 2.10 and (2.11) yields |∇ k (Φ − t ) ∗ E | g co,t + |∇ k [(Φ − t ) ∗ g co, − g co,t ] | g co,t Cr − k ( r λ + | t | r − ) . Since r ∼ | t | α , from (5.9) we have h t = I + (1 − χ ) E , |∇ k E| g co,t Cr − k ( | t | λα/ + | t | − α ) . We choose α such that λα/ − α , so that |E| C | t | λα/ and | h t − I | g co,t C | t | λα/ ≪ C − I h t CI . Taking a derivative gives ∇ h t = −E∇ χ + (1 − χ ) ∇E . Since |E| C | t | λα/ , |∇E| Cr − | t | λα/ and |∇ χ | Cr − (e.g. (5.4)), weobtain |∇ h t | Cr − | t | λα/ . The higher order estimates in the claim (5.10) are similar. (cid:3)
We now consider the next region past the transition zone.
Lemma 5.5.
Let F K t be the curvature of K t = [(Φ − t ) ∗ H ] , . Then on D = {| t | α k z k } , we can estimate (5.11) k Λ ω t F K t k C ,a − ( D ) C | t | − α . Proof:
Let (ˆ z , ˆ z , ˆ z , ˆ z , t ) be a point in X = { ( z, t ) : P i =1 z i = t } with k ˆ z k > | t | α , and suppose without loss of generality that ˆ z = 0. Let λ = k ˆ z k and ˆ r = λ / . We take local coordinates on U ˆ r = { λ k z k λ } ⊂ X t given by w i = λ z i . These coordinates land in (cid:26) | w | (cid:27) ⊂ C where | w | = | ( w , w , w ) | is the Euclidean norm on C . The formula for thecurvature on V t in coordinates is( F K t ) j ¯ k = − K − t ∂ j ∂ ¯ k K t + K − t ∂ j K t K − t ∂ ¯ k K t . We showed in Lemma 2.11 that in { w i } coordinates, we have g t = ˆ r O ( I ),and by Lemma 5.1 the metric K t = ˆ r O ( I ). Here we write O ( I ) for a matrixwhich is positive-define with positivity and derivative bounds independentof t, λ . Therefore(5.12) | F K t | g t = ˆ r − O (1) , where O (1) denotes a function with smooth bounds independent of t, λ . • We claim:(5.13) ∂∂t (cid:20) [(Φ − t ) ∗ g co, ] j ¯ k ( F K t ) j ¯ k (cid:21) = ˆ r − O (1) . Here [(Φ − t ) ∗ g co, ] j ¯ k are the local matrix entries of the inverse of (Φ − t ) ∗ g co, (not the raised indices with respect to g t ). Assuming this for now, we com-plete the proof of the lemma. Since at t = 0 we have [(Φ − t ) ∗ g co, ] j ¯ k ( F K t ) j ¯ k =0, this implies [(Φ − t ) ∗ g co, ] j ¯ k ( F K t ) j ¯ k = t ˆ r − O (1) . We can then write √− ω t F K t = [(Φ − t ) ∗ g co, ] j ¯ k ( F K t ) j ¯ k + (cid:20) g j ¯ kco,t − [(Φ − t ) ∗ g co, ] j ¯ k (cid:21) ( F K t ) j ¯ k , and by (2.11) and (5.12) we have √− ω t F K t = t ˆ r − O (1)in { w i } coordinates. Therefore |∇ ( √− ω t F K t ) | g t = | t | ˆ r − O (1) . Then for any 0 < a < k Λ ω t F K t k C ,a − ( U ˆ r ) C | t | ˆ r − C | t | − α using that | t | α λ λ = ˆ r . By Lemma 5.2, this gives the H¨olderestimate on all of D = {| t | α k z k } . • We now prove the claimed (5.13). We start with the variation of K t .The metric H is defined on V = { P i x i = 0 } and here we use coordinates( x , x , x ) given by x i = k ˆ x k x i where ˆ x ∈ V is the point such that Φ t (ˆ x ) =ˆ z . The map Φ t appears in coordinates { x i } and { w i } as(5.14) Φ it ( x ) = (cid:18) x i + t k ˆ x k ¯ x i P i =1 | x i | + | P i =1 ( x i ) | (cid:19) k ˆ x k λ . Recall that k x k k Φ t ( x ) k k x k , and so k ˆ x k ∼ λ and coordinates z i are in the range { | z | } ⊂ C . We may assume the coordinates x i on V are in the range { | x | } ⊂ C . Abusing notation, we simply write w i = w i ◦ Φ t ( x ). The change of coordinates is of the form ∂w i ∂x j = k ˆ x k λ δ ij + tλ O (1) and hence(5.15) ∂∂t ∂w i ∂x j = λ − O (1) , δ ij ∂w i ∂x j δ ij . Differentiating the inverse Jacobian then also gives ∂∂t ∂x i ∂w j = λ − O (1) . We now compute the variation of K t in these coordinates. In components K t = ( K t ) ¯ kj dw j ⊗ d ¯ w k , we have ∂∂t ( K t ) ¯ kj ( w ) = ∂∂t (cid:20) ∂x p ∂w k ( H ) ¯ pq ( x ( w )) ∂x q ∂w j (cid:21) . We note ∂∂t (cid:20) ( H )( x ( w )) (cid:21) = ∂H ∂x i ( x ( w )) ∂x i ∂w p ∂w p ∂t = O ( λ − ) ∂H ∂x i . Recall that in these coordinates, we have that H = r (ˆ x ) O ( I ), and wenoted earlier that r (ˆ x ) ∼ r (ˆ z ) = ˆ r . Putting everything together, we have(5.16) ∂∂t ( K t ) ¯ kj ( w ) = ˆ r λ − O (1) . Since K t = ˆ r O ( I ) in these coordinates, it follows that ∂∂t ( F K t ) j ¯ k = λ − O (1) . Thus ∂∂t (cid:20) [(Φ − t ) ∗ g co, ] j ¯ k ( F K t ) j ¯ k (cid:21) = (cid:20) ∂∂t [(Φ − t ) ∗ g co, ] j ¯ k ( F K t ) j ¯ k (cid:21) + (cid:20) [(Φ − t ) ∗ g co, ] j ¯ k ∂∂t ( F K t ) j ¯ k (cid:21) = (cid:20) ∂∂t [(Φ − t ) ∗ g co, ] j ¯ k ( F K t ) j ¯ k (cid:21) + λ − ˆ r − O (1) . (5.17)Here we used (2.11) and g − co,t = ˆ r − O ( I ) in these coordinates. The samecomputation as (5.16) gives ∂∂t ((Φ − t ) ∗ g co, ) ¯ kj ( w ) = ˆ r λ − O (1)and therefore ∂∂t (cid:20) [(Φ − t ) ∗ g co, ] j ¯ k ( F K t ) j ¯ k (cid:21) = λ − ˆ r − O (1) . Since λ = ˆ r , this completes the proof of (5.13). (cid:3) Perturbation
At this stage in the construction, we have a pair of metrics ( g t , H t ) onthe smoothing X t such that both of these metrics agree with a scaling of g co,t near the vanishing cycles {k z k = t } . The metric ω t is not balancedon all of X t and the metric H t is not Hermitian-Yang-Mills with respect to ω t away from the vanishing cycles, but by the construction they are closeto solving these equations. In this section we will perturb ( g t , H t ) to a pair( g FLY ,t , ˇ H t ) solving the Hermitian-Yang-Mills equation. We will prove: Theorem 6.1.
There exists ε > such that for all < | t | < ε , there existson X t a pair of hermitian metrics ( g FLY ,t , ˇ H t ) solving dω ,t = 0 , F ˇ H t ∧ ω ,t = 0 . Near the vanishing cycles, these metrics have the following local description.There exists λ, c i , d i > such that for any k ∈ Z > , there exists C k > suchthat for all | t | < ε (6.1) |∇ kg co,t ( g FLY ,t − c i g co,t ) | g co,t C k | t | / r − k and (6.2) |∇ kg co,t ( ˇ H t − d i g co,t ) | g co,t C k | t | λ r − k in the region R λ = {| t | k z k | t | λ } . Remark 6.2.
The decay estimates (6.1) and (6.2) imply that, at an appro-priate scale, the Hermitian-Yang-Mills metrics H t converge smoothly to amultiple of the CO metric g co, as | t | → The θ -perturbed Fu-Li-Yau metric. We recall the construction ofFu-Li-Yau [35] which perturbs ω t to a balanced metric ω FLY ,t . The Fu-Li-Yau balanced metric is obtained via the ansatz(6.3) ω ,t = ω t + θ t + ¯ θ t . The (2 ,
2) form θ t is constructed to satisfy ∂θ t = 0 , ¯ ∂θ t = − ¯ ∂ω t . More specifically, the correction θ t is of the form θ t = ∂ ¯ ∂ † ∂ † γ t where adjoints are with respect to g t and γ t ∈ Λ , ( X t ) satisfies ∂γ t = 0.Estimates for γ t and θ t were obtained by Fu-Li-Yau [35]. We will use theversions stated in [16]. [16, Proposition 3.6] states that | θ t | g t C k z k − / | t | , which using k z k > | t | implies(6.4) | θ t | g t C | t | (2 / . The proof of [16, Proposition 3.8] uses(6.5) Z X t | γ t | g t C | t | . We will need the following higher estimate on ∇ θ . Lemma 6.3. |∇ k θ t | g t C k | t | / r − k . Proof:
This is similar to [16, Proposition 3.7]. We first show the estimateon a compact set K which does not intersect the vanishing cycles. Theoperator E t given by E t = ∂ ¯ ∂ ¯ ∂ † ∂ † + ∂ † ¯ ∂ ¯ ∂ † ∂ + ∂ † ∂, where † is with respect to g t , is a 4th order elliptic operator. The form γ t satisfies E t ( γ ) = ¯ ∂ω t . In fact, it is obtained in [35] by solving this equation. On K , the geometryis uniform in t , hence by elliptic estimates we have k γ k C ( K ) C ( k γ k L ( K ) + k ¯ ∂ω k W k,p ( K ) )for some k, p >
1. As noted in Lemma 2.17, the construction of ω t is suchthat(6.6) | ¯ ∂ω t | C k ( X t ,g t ) C k | t | . By (6.6) and (6.5), we have k γ k C ( K ) C | t | and hence k∇ θ t k L ∞ ( K ) C | t | .Similarly, |∇ k θ t k L ∞ ( K ) C k | t | .We now prove the estimate stated in the lemma on a set U ( ε ) ∩ X t con-taining the vanishing cycles and assume g t = R g co,t . Here ¯ ∂θ = ¯ ∂ω co,t = 0,and we also have ¯ ∂ † θ = ¯ ∂ † ∂ ¯ ∂ † ∂ † γ t = 0since ∂ and ¯ ∂ † commute because g t is K¨ahler on this set. Therefore∆ ¯ ∂ θ t = 0 . Working in holomorphic cylindrical coordinates (see Lemma 2.11), we canverify that the coefficients of the equation r ∆ ¯ ∂ θ t = 0 are uniformly boundedin C α . Indeed, by the Bochner-Kodaira formula,∆ ¯ ∂ θ = − g i ¯ jt ∂ i ∂ ¯ j θ + Γ ∗ ∂θ + ∂ Γ ∗ θ + Γ ∗ Γ ∗ θ + Rm g t ∗ θ, and the uniform boundedness of the coefficients of r ∆ ¯ ∂ follows from Lemma2.11. By the Schauder estimates in this coordinate chart, we obtainsup B / | ∂θ t | g euc C sup B | θ t | g euc . Using g co,t = r O ( I ) in these coordinates, we obtain |∇ θ t | g t Cr − sup X t | θ t | g t . Since | θ t | g t C | t | / , we obtain the lemma for k = 1 and higher k > (cid:3) .We now note some general facts on (2 ,
2) forms constructed via the ansatz˜ ω = ω + θ + ¯ θ . Lemma 6.4.
On a complex manifold of dimension n , the equation ω n − = Ψ > has solution g ¯ kj = (det Ψ) / ( n − (Ψ − ) ¯ kj , where ω = √− g ¯ kj dz j ∧ d ¯ z k and Ψ is written as X k,j c kj Ψ k ¯ j dz ∧ d ¯ z ∧ · · · ∧ d dz k ∧ d ¯ z k ∧ · · · ∧ dz j ∧ c d ¯ z j ∧ · · · ∧ dz n ∧ d ¯ z n with c kj = ( √− n − ( n − sgn ( k, j ) .Proof: See e.g. [56] or [62]. Direct computation of ω n − gives (det g ) g j ¯ k =Ψ j ¯ k and the result follows from taking the determinant of both sides andsolving for g . (cid:3) Lemma 6.5.
Let ∇ be the Chern connection with respect to a Hermitianmetric ω . Let η = √− η ¯ kj dz j ∧ d ¯ z k be a positive (1 , form solving η = ω + θ + ¯ θ , where θ = 14 θ s ¯ rj ¯ k dz s ∧ d ¯ z r ∧ dz j ∧ d ¯ z k . Then ∇ i η ¯ kj = − η s ¯ r ( ∇ i θ s ¯ rj ¯ k + ∇ i ¯ θ s ¯ rj ¯ k ) + 18 (cid:20) η p ¯ q η s ¯ r ( ∇ i θ s ¯ rj ¯ k + ∇ i ¯ θ s ¯ rp ¯ q ) (cid:21) η ¯ kj . Proof:
A similar computation can be found in [63]. In components, theequation η = ω + θ + ¯ θ is − η ¯ rs η ¯ kj + 2 η ¯ rj η ¯ ks = ( ω + θ + ¯ θ ) s ¯ rj ¯ k . Differentiating this equation leads to −∇ i η ¯ rs η ¯ kj − η ¯ rs ∇ i η ¯ kj + ∇ i η ¯ rj η ¯ ks + η ¯ rj ∇ i η ¯ ks = 12 ( ∇ i θ s ¯ rj ¯ k + ∇ i ¯ θ s ¯ rj ¯ k )Contracting by η s ¯ r gives − ( η s ¯ r ∇ i η ¯ rs ) η ¯ kj − ∇ i η ¯ kj = 12 η s ¯ r ( ∇ i θ s ¯ rj ¯ k + ∇ i ¯ θ s ¯ rj ¯ k )Contracting again by η j ¯ k gives − η s ¯ r ∇ i η ¯ rs = 12 η j ¯ k η s ¯ r ( ∇ i θ s ¯ rj ¯ k + ∇ i ¯ θ s ¯ rj ¯ k ) . Combining the previous two identities proves the lemma. (cid:3) Using what we have obtained so far in this subsection, we can derive themain estimate of this subsection which shows that the difference between g − t and ( g FLY ,t ) − is small. Lemma 6.6.
There exists C k > and ε > with the following property.For all < | t | < ε , the θ -perturbed Fu-Li-Yau metric g FLY ,t satisfies theestimates: − g t g FLY ,t g t , and |∇ kg t ( g FLY ,t − g t ) | g t C k | t | / r − k , for k ∈ Z > . Furthermore, we have k ω − ,t − ω − t k C ,α C | t | / . Proof: If | θ t + ¯ θ t | g t , then by Lemma 6.4 we have | g FLY ,t | g t , | g − ,t | g t . Next, we write the difference of metrics as ω FLY ,t − ω t = Z dds η s ds where η s solves η s = ω t + s ( θ t + ¯ θ t ). By the variation formula in Lemma6.5, we have dds ( η s ) ¯ kj = − η s ¯ rs ( θ + ¯ θ ) s ¯ rj ¯ k + 18 [ η p ¯ qs η s ¯ rs ( θ + ¯ θ ) s ¯ rp ¯ q ] ( η s ) ¯ kj . The same argument as above shows that | η s | g t C and | η − s | g t C for s ∈ [0 , | g FLY ,t − g t | g t C | θ t | g t C | t | / by (6.4). Next, by Lemma 6.5 and Lemma 6.3, we have |∇ g t ( g FLY ,t ) | g t C |∇ g t θ | g t C | t | / r − . Higher order estimates for |∇ kg t ( g FLY ,t ) | g t are similar.It remains to estimate the difference, which can be done by: k ω − ,t − ω − t k C ,α = k ω − ,t ( ω FLY ,t − ω t ) ω − t k C ,α k ω − ,t k C ,α k ω FLY ,t − ω t k C ,α k ω − t k C ,α . (6.7)Since |∇ ω − ,t | g t C | t | / r − , we have k ω − ,t k C ,α C (1 + r |∇ ω − ,t | g t ) C. The estimate | ω FLY ,t − ω t | + r |∇ g t ( ω FLY ,t − ω t ) | C | t | / implies k ω FLY ,t − ω t k C ,α C | t | /
36 T. C. COLLINS, S. PICARD, AND S.-T. YAU which proves the lemma. (cid:3)
Uniform weighted H¨older estimates.
Let g t be the metric con-structed by Fu-Li-Yau which satisfies g t = g co,t near the vanishing cycles(for ease of notation, in this section we assume that the constant M / ε − / in Proposition 2.17 is equal to 1). Let H t be the metric on X t constructed inthe previous section, i.e. the glued approximate solution to the Hermitian-Yang-Mills equation. We will use a linear operator L t which acts on endo-morphisms h ∈ Γ(End T , X t ) by L t h = ( g t ) j ¯ k ∂ ¯ k ∇ H t j h + 12 [ √− ω t F H t , h ] . The motivation for this operator is that it is close to the linearization ofthe Hermitian-Yang-Mills equation √− ω FLY ,t F H = 0 at the approximatesolution H t , the difference being the use of g t instead of g FLY ,t . We start byproving uniform Schauder estimates independent of t on the deformations X t . Proposition 6.7.
Let β . There exists C > and a ∈ (0 , such thatfor all t ∈ C ∗ and h ∈ Γ(End T , X t ) , we can estimate k h k C ,aβ ( X t ) C ( k h k C β ( X t ) + k L t h k C ,aβ − ( X t ) ) where the weighted H¨older norms are defined in § ( g t , H t ) .Proof: On X t ∩ { r > } , the geometry is uniform in t and the estimateholds by the usual Schauder estimates. Let ˆ x ∈ X t ∩ { r ( x ) } . Wedenote the scale of this point by the constant ˆ r := r (ˆ x ). We will work inholomorphic cylindrical coordinates { w i } given in Lemma 2.11 on the set U ˆ r = { (1 / r < r < r } . By Lemma 2.11 and Lemma 5.1, the operator ˆ r L t is uniformly elliptic withuniform derivative estimates in coordinates { w i } . The standard Schauderestimates applied to each matrix entry ˆ r − β h ij imply k ˆ r − β h k C ,αg euc ( B / ) C ( k ˆ r − β h k C g euc ( B ) + k ˆ r L t (ˆ r − β h ) k C αg euc ( B ) )with usual (non-scaled) norms k u k C kg euc ( B ) = k X i =1 sup B | D i u | , [ u ] C αg euc ( B ) = sup x = y | u ( x ) − u ( y ) || x − y | α . As observed in (5.6), since ˆ r − g t is uniformly and smoothly equivalent to g euc in coordinates { w i } , the weighted H¨older norms are equivalent to theselocal Euclidean norms, and we have k h k C ,aβ ( U ˆ r ) C ( k h k C β ( U ˆ r ) + k L t h k C aβ − ( U ˆ r ) ) . The norm k · k C ,aβ ( U ˆ r ) involves connection terms from ∇ H t , but these arebounded in coordinates { w i } by Lemma 5.3. By Lemma 5.2, these localestimates on sets U ˆ r imply the global bound. (cid:3) The next step is to improve this estimate for endomorphisms orthogonalto the identity. For a related argument used in a gluing construction ofK¨ahler-Einstein metrics on nodal surfaces, see [72].
Proposition 6.8.
Let β ∈ ( − , . There exists C > and α ∈ (0 , withthe following property. Let t ∈ C with < | t | < be arbitrary. We canestimate k h k C ,αβ ( X t ) C k L t h k C αβ − ( X t ) , for all h ∈ Γ(End T , X t ) satisfying h † Ht = h and R X t (Tr h ) d vol g FLY ,t = 0 .Proof: Suppose there exists a sequence of t i → k h i k C ,αβ > M i k L t i h i k C αβ − with M i → ∞ and h i defined on X t i . Replacing h i with h i / k h i k C ,αβ , wehave a sequence with k h i k C ,αβ ( X ti ) = 1 , k L t i h i k C αβ − ( X ti ) → . Let K ⊂ X be a compact set on the central fiber disjoint from the singularpoints. For all t small enough, we have a sequence Φ ∗ t i h i of endomorphismsdefined on K . Since C − r ( x ) r (Φ t ( x )) Cr ( x ), we have a uniform boundon k Φ ∗ t i h i k C ,αβ ( K, Φ ∗ ti g i , Φ ∗ ti H i ) C. By Lemma 2.16, Φ ∗ t i g t i → g smoothly uniformly on compact sets, and thedefinition of H t implies that Φ ∗ t i H t i → H smoothly uniformly on compactsets. We can thus extract a limiting endomorphism h ∈ C ,α/ loc ( X ) whichsatisfies the growth estimates | h | g Cr β , |∇ H h | g Cr β − and the identities(6.8) ( g ) j ¯ k ∂ ¯ k ∇ H j h = 0 , h † = h where † is with respect to H . We now show that(6.9) Z X (Tr h ) d vol g = 0 . For this, we let δ >
0, so that h i , g i converge uniformly on { r > δ } . ByLemma 6.6, we also have that (Φ t i ) ∗ g FLY ,t i → g uniformly in the C normon { r > δ } . Therefore Z X ∩{ r>δ } (Tr h ) d vol g = lim i Z X ti ∩{ r>δ } (Tr h i ) d vol g FLY ,ti . Since R X ti Tr h i = 0, then(6.10) Z X (Tr h ) d vol g = − lim δ → lim i Z X ti ∩{ r<δ } (Tr h i ) d vol g FLY ,ti . From d vol g FLY ,t Cd vol g t (Lemma 6.6) and | h i | r β , we have (cid:12)(cid:12)(cid:12)(cid:12) Z X ti ∩{ r<δ } (Tr h i ) d vol g FLY ,ti (cid:12)(cid:12)(cid:12)(cid:12) Cδ β Z X ti ∩{ r<δ } d vol g ti = Cδ β Z X ∩{ r<δ | t i | − / } S ∗ t / d vol g ti = Cδ β Z X ∩{ r<δ | t i | − / } | t i | d vol g co, where S t / : V → V t , S t / ( z ) = t / z is the scaling action (2.7) whichsatisfies S ∗ t / g co,t = | t | / g co, . We have Vol g co, ( { r < R } ) = O ( R ) since g co, is asymptotically conical, and so (cid:12)(cid:12)(cid:12)(cid:12) Z X ti ∩{ r<δ } (Tr h i ) d vol g FLY ,ti (cid:12)(cid:12)(cid:12)(cid:12) Cδ β which together with (6.10) proves (6.9).Next, (6.8) implies that the identity∆ g | h | H = 2 |∇ h | H ,g holds pointwise away from the nodes. Let η δ be a cutoff function such that η δ ≡ { r < δ } , η δ ≡ { r > δ } and | ∆ g η δ | Cδ − . Then2 Z U δ |∇ h | H ,g d vol g Z X η δ ∆ g | h | H d vol g . Recall that g is balanced on X and so we can integrate by parts.2 Z U δ |∇ h | H ,g d vol g Cδ − Z { δ 0. The uniform Schauder estimates in Proposition6.7 imply 1 C ( k h i k C β + M − i ) , and hence | r − β h i | ( z i ) > C − for a sequence z i ∈ X t i . • Case 1: Suppose lim inf r ( z i ) > 0. It then follows that after taking asubsequence, we have z i → z ∈ X with r ( z ) > 0. Then | h ( z ) | > C − r ( z ) β > h ≡ • Case 2: Suppose r ( z i ) → 0. In this case, we can assume that all points z i are in the region of X t which can be identified with a subset of V t = { P z i = t } and where g t = g co,t . Define the function u i : V t i ∩ {k z k } → R givenby u i = | h i | H ti . The sequence { u i } satisfies the uniform growth estimate(6.11) k u i k C ,α β ( g t ) C, which written in full is | u | + r − |∇ u | g co,t + r − |∇ g co,t u | g co,t + [ ∇ g co,t u ] C ,aβ − − a Cr β . This definition of k u k C ,α β ( g t ) is slightly different than the weighted H¨oldernorms used previously for h t , since we use ∇ with respect to g t (rather than H t ). These estimates for u i follow from k h i k C ,αβ ( g t ,H t ) H t to norms in g t .Direct computation gives the identity∆ g i u i = 2Re h L i h i , h i i H i + 2 |∇ h i | H i ,g i . We are assuming k L t i h i k C αβ − ε i with ε i → 0, hence(6.12) ∆ g i u i > − Cε i r β − . We will rescale the functions u i to take a limit. For ease of notation, wewrite λ i = r ( z i ). We will use the scaling map S λ i : V t i λ − i → V t i from (2.7)given by S ( x ) = λ / i x . We rescale and pullback u i via S λ i to obtain afunction ˜ u i : V t i λ − i ∩ {k x k λ − i } → R defined by ˜ u i ( x ) = λ − βi u ( λ / i x ) . The rescaling is setup so that the estimates for u i imply estimates for ˜ u i .For example, we have(6.13) ∆ g tiλ − i ˜ u i > − Cε i r β − . Indeed, pulling back the Laplacian gives S ∗ λ i (∆ g ti u i ) = λ βi ∆ S ∗ g ti ˜ u i = λ βi λ − i ∆ g tiλ − ˜ u i by using the rescaling relation S ∗ λ g t = λ g tλ − . Using r ( λ / x ) = λr ( x ), weobtain (6.13) from (6.12). Similar computations show that(6.14) | ˜ u i | + r − |∇ ˜ u i | g co,tiλ − i + r − |∇ ˜ u i | g co,tiλ − i + [ ∇ ˜ u i ] C ,aβ − − a Cr β . Recall that the points z i satisfy | h i | ( z i ) > C − λ βi . Then the points x i = λ − / i z i satisfy ˜ u i ( x i ) > C − . The sequence { t i λ − i } lies in [0 , k z k > | t | implies r ( z i ) > | t i | .After taking a subsequence, we have convergence t i λ − i → κ for κ ∈ [0 , • Case 2a: t i λ − i → κ > 0. The points x i = λ − / i z i satisfy k x i k = 1. Wemay assume x i → x ∞ ∈ V κ , k x ∞ k = 1 . We can take a limit of { ˜ u i } on compact sets and obtain a C ,αloc limitingfunction u ∞ > V κ satisfying u ∞ Cr β and ∆ g κ u ∞ > 0. By themaximum principle, sup r R u ∞ sup r = R u ∞ R β . Letting R → ∞ , we obtain that u ∞ ≡ β < 0. This contradicts u ∞ ( x ∞ ) > • Case 2b: t i λ − i → 0. In this case, the points x i = λ − / i z i converge aftera subsequence to x i → x ∞ ∈ V , k x ∞ k = 1 . Let v i : {k x k > | t i | λ − i } ∩ V → R with v i = Φ ∗ t i λ − i ˜ u i be the correspondingsequence of functions on the cone V . Since k x k k Φ( x ) k k x k , wehave the growth estimate v i Cr β . By pulling-back (6.14), on compact sets K we have the estimate k v i k C ,α ( K, Φ ∗ g tiλ − i ) C ( K ) , ∆ Φ ∗ g tiλ − i v i > − ε i C ( K ) . Corollary 2.9 implies Φ ∗ g t i λ − i → g co, uniformly on K . Taking a limit of { v i } on compact sets produces a C ,αloc limiting function v ∞ > V satisfying ∆ g co, v ∞ > , v ∞ Cr β for β ∈ ( − , v ∞ ≡ v ∞ ( x ∞ ) > (cid:3) Lemma 6.9. Let V be a Riemannian cone of dimension n > with metric g = dr + r g L . Let u be a C function satisfying ∆ g u > and u > .Suppose there exists M > such that u M r − δ where δ ∈ (0 , n − . Then u ≡ .Proof: This is a standard PDE result (e.g. [44]), but we give the prooffor completeness. Recall that the real Laplacian is∆ g u = ∂ r ∂ r u + ( n − r − ∂ r u + r − ∆ g L u. Let B R (0) = { x ∈ V : r ( x ) < R } . We start by noting that for any ϕ ∈ C ( ∂B R (0) , R ), there exists h ∈ C ( B R (0) , R ) such that h | ∂B R (0) = ϕ and∆ g h = 0 , sup B R (0) | h | sup ∂B R (0) | ϕ | . To obtain such a harmonic function h , we start by expanding ϕ | ∂B R (0) = P λ ∈ Spec(∆ gL ) c λ ψ λ , where ψ λ are an L orthogonal basis of eigenfunctionsof ∆ g L on the link L = { r = 1 } , with eigenvalue convention ∆ L ψ λ = − λψ λ .We then let h = X λ ∈ Spec(∆ gL ) c λ (cid:18) rR (cid:19) a ( λ ) ψ λ where a ( λ ) = ( − ( n − 2) + p ( n − + 4 λ ) > 0. Direct computation gives∆ g h = 0, and by the maximum principlesup B R \ B ε | h | max { sup ∂B R | ϕ | , sup ∂B ε | h |} . for any ε > 0. As ε → 0, we see that sup ∂B ε | h | selects the λ = 0 mode c ψ .This is a constant equal to B R ) R ∂B R ϕ , which is bounded by sup ∂B R | ϕ | .We now prove that the subharmonic function u given in the lemma isconstant. Let R > 1. Let h R be the harmonic function mentioned abovewith h R | ∂B R = u . Then(6.15) | h R | sup ∂B R | u | M R − δ . Let 0 < ε < 1, and consider v = u − h R − M ε n − − δ r − ( n − defined on B R \ B ε . Since ∆ g r − ( n − = 0 we have that ∆ g v > 0, and v | ∂B ε | u | + | h R | − M ε − δ M ε − δ + M R − δ − M ε − δ . We also have v | ∂B R = − M ε n − − δ R n − < . By the maximum principle, v B R \ B ε . We now fix x ∈ B R \ B ε . Then v ( x ) u ( x ) h R ( x ) + 2 M ε n − − δ r n − ( x ) . This holds true for all 0 < ε < 1, hence taking ε → u ( x ) h R ( x ) . By (6.15), we conclude u ( x ) M R − δ . We now take R → ∞ to conclude u 0. Since u > u ≡ (cid:3) Inverting the linearized operator. Let(6.16) W t = (cid:26) u ∈ Γ(End T , X t ) : u † = u, Z X t (Tr u ) d vol g FLY ,t = 0 (cid:27) where † is the adjoint with respect to H t . When linearizing the equationΛ ω FLY ,t F H = 0 at the approximate solution H t we obtain an operator L thatacts on endomorphisms by L u = − ( g FLY ,t ) j ¯ k ∂ ¯ k ∇ H t j u − 12 [ √− ω FLY ,t F H t , u ] . This operator L involves g FLY ,t rather than g t , so it is a perturbation of theoperator L in Proposition 6.8.We note that L : W t → W t . Indeed, since u † = u , ( √− F ) † = √− F and ( g j ¯ k ∇ j ∇ ¯ k u ) † = g j ¯ k ∇ ¯ k ∇ j u , we have( L u ) † = − ( g FLY ,t ) j ¯ k ∇ H t j ∂ ¯ k u + 12 [ √− ω FLY ,t F H t , u ] . The commutator identity for [ ∇ j , ∇ ¯ k ] now shows that ( L u ) † = L u . Next,since ω FLY ,t is balanced, we do have Z X t (∆ g FLY ,t Tr u ) ( ω FLY ,t ) = 0 . This verifies that L : W t → W t preserves the subspace W t .Having obtained the estimate k h k C k Lh k from Proposition 6.8, we caninvert L with a bound on the inverse. Lemma 6.10. Let L : C ,αβ ( W t ) → C ,αβ − ( W t ) for α ∈ (0 , and β ∈ ( − , .There exists | t | > such that for all < | t | | t | , then L is invertible andthe operator norm (6.17) kL − k C is bounded independent of t . Proof: We start by discussing the uniform estimate. Let L be the operatorthat was estimated in Proposition 6.8. For u ∈ C ,αβ ( W t ), we have k u k C ,αβ C k Lu k C ,αβ − C k (( g FLY ) j ¯ k − g j ¯ kt ) ∂ ¯ k ∇ H t j u k C ,αβ − + C k g − − g − t k C ,α k F H t k C ,α − k u k C ,αβ + C kL u k C ,αβ − C | t | / k u k C ,αβ + C | t | / k F H t k C ,α − k u k C ,αβ + C kL u k C ,αβ − by Lemma 6.6. By Lemma 5.3, we have r | F H t | g t + r |∇ g t F H t | g t C andhence(6.18) k F H t k C ,α − C uniformly in t . We conclude that for t small enough, then(6.19) k u k C ,αβ C kL u k C ,αβ − , u ∈ W t . The proof of the lemma now follows from standard elliptic PDE theory onthe smooth compact manifold X t . We will use the space of sections denotedby H = (cid:26) u ∈ Γ(End T , X t ) : u † = u (cid:27) with L inner product h s, h i L = Z X h s, h i H t d vol g FLY ,t , s, h ∈ H . Then we can orthogonally decompose H = W ⊕ C I . We consider P : C ,α ( H ) → C ,α ( H ) with P u = L u . From (6.19), we see that C I = ker P .Furthermore, by the balanced condition of ω FLY ,t we see thatim P ⊆ W. Therefore (ker P † ) ⊥ ⊆ W and so C I ⊆ ker P † . We will show C I = ker P † .An integration by parts argument using the balanced property shows thatthe operator ∆ = ( g FLY ,t ) j ¯ k ∂ ¯ k ∇ H t j is L self-adjoint and so has degree zero.Thus P also has degree zero. Therefore dim ker P † = 1 and C I = ker P † . Itfollows that im P = W . Hence L : W → W is invertible, and the bound onthe inverse in weighted norms is (6.19). (cid:3) Fixed point theorem. In this section, we use the glued metric H t tosolve the Hermitian-Yang-Mills equation √− ω FLY ,t F H = 0 on the smooth-ing X t via a fixed point theorem.Our space of deformations will be the space W t defined in (6.16). Weintroduce the operator F : W t → W t given(6.20) F ( u ) = e u/ ( √− ω FLY ,t F u ) e − u/ , where F u is the curvature of the metric H t,u = H t e u .We note that F : W t → W t . Indeed, for u ∈ W t , we have R Tr F ( u ) = 0since Z X t c ( T , X t ) ∧ ω ,t = 0as X t has trivial canonical bundle. The adjoint action of e u/ in (6.20) isadded to ensure that F ( u ) † = F ( u ). Indeed, √− F H t,u is self-adjoint withrespect to the metric H t,u . In coordinates where H t = I and e u is diagonal,it is straight-forward to check that( e u/ ( √− ω FLY ,t F u ) e − u/ ) † = e u/ ( √− ω FLY ,t F u ) e − u/ where † is the adjoint with respect to H t .A well-known computation gives the linearization of F u . δ ( F u ) = ¯ ∂∂ H t,u ( e − u δe u ) . The linearization of F is then( δ F ) | u ( δu ) = − e u/ [( g FLY ,t ) j ¯ k ∂ ¯ k ∇ H t,u j ( e − u δe u )] e − u/ +( δe u/ )( √− ω FLY ,t F u ) e − u/ +( e u/ )( √− ω FLY ,t F u ) δe − u/ . Let L be the linearization ( δF ) | at u = 0. That is, L : W t → W t with L w = − ( g F LY,t ) j ¯ k ∂ ¯ k ∇ H t j w − 12 [ √− ω F LY,t F H t , w ] . We previously inverted L and gave a bound on the inverse L − uniform in t . We can write F ( u ) = F (0) + L ( u ) + Q ( u )where by definition(6.21) Q ( u ) = F ( u ) − F (0) − L ( u ) . We define N : C ,αβ ( W ) → C ,αβ ( W )given by N ( u ) = L − ( −F (0) − Q ( u )) . To solve F ( u ) = 0, it is equivalent to find a fixed point N ( u ) = u. Proposition 6.11. Let a ∈ (0 , . There exists ε > and β ∈ ( − , withthe following property. Let U t = { u ∈ C ,aβ ( W ( X t )) : k u k C ,aβ < | t | (2 / | β | } . Then for all < | t | < ε , the mapping N preserves U t and satisfies (6.22) kN ( u ) − N ( v ) k C ,aβ k u − v k C ,aβ for all u, v ∈ U t .Proof: We start by assuming (6.22) and prove that N preserves U t . Wehave F (0) = √− ω FLY ,t F H t and we can estimate kF (0) k C ,a − k Λ ω t F H t k C ,a − + k ω − t − ω − ,t k C ,a k F H t k C ,a − . Proposition 5.3, Lemma 6.6 and (6.18) imply kF (0) k C ,a − C ( | t | / + | t | | αλ | / ) . The contribution | t | | αλ | / is the slowest rate. For any β ∈ ( − , kF (0) k C ,aβ − C | t | | αλ | / . Since kL − k C by (6.17), it follows that kN (0) k C ,aβ C | t | | αλ | / , and hence (6.22) implies that for u ∈ U t then kN ( u ) k kN ( u ) − N (0) k + kN (0) k | t | (2 / | β | + C | t | | αλ | / < | t | (2 / | β | for β = − | αλ | and t small enough.Thus it remains to prove (6.22). By definition N ( u ) − N ( v ) = L − ( Q ( v ) − Q ( u )) . Since kL − k C by (6.17), we have(6.23) kN ( u ) − N ( v ) k C ,aβ C kQ ( u ) − Q ( v ) k C ,aβ − . So we need to estimate Q ( u ) − Q ( v ). One way to write this is Q ( u ) − Q ( v ) = Z dds Q ( w s ) ds where w s = su + (1 − s ) v . By the definition (6.21) of Q , its variation is dds Q ( w s ) = ( δ F ) | w s ( u − v ) − L ( u − v ) . We claim that the approximate linearized operator L is close enough to theactual linearization δ F , so that for all w, s ∈ U then(6.24) k ( δ F ) | w ( s ) − L ( s ) k C ,aβ − C k w k C ,a k s k C ,aβ . Assuming this, we conclude kQ ( u ) − Q ( v ) k C ,aβ − C ( k u k C ,a + k v k C ,a ) k u − v k C ,aβ . By (6.23), we see that kN ( u ) − N ( v ) k C ,aβ C ( k u k C ,a + k v k C ,a ) k u − v k C ,aβ . If u ∈ U , then k u k C ,aβ < | t | | β | , and since r > | t | , we have k u k C ,a X t r | β | k u k C ,aβ | t | | β | / | t | (2 / | β | ε | β | / . Thus if ε is small enough, then N is a contraction map. It remains to prove(6.24). Lemma 6.12. There exists C > such that for all t ∈ C ∗ and u, s ∈ W t with k u k C ,a ( X t ) , we can estimate k ( δ F ) | u ( s ) − L ( s ) k C ,aβ − ( X t ) C k u k C ,a ( X t ) k s k C ,aβ ( X t ) . Proof: To simplify notation, for this estimate we will write g = g FLY ,t , H = H t and H u = H t,u . The linearization was computed in (6.21), and thedifference is explicitly given by( δ F ) | u ( s ) − L ( s )= (cid:20) [ δ exp] | u ( s/ (cid:21) ( √− ω F H u ) e − u/ − s ( √− ω F H )+( e u/ )( √− ω F H u ) (cid:20) [ δ exp] | u ( − s/ (cid:21) + 12 ( √− ω F H ) s − e u/ [ g j ¯ k ∂ ¯ k ∇ H u j e − u [ δ exp] | u ( s )] e − u/ + g j ¯ k ∂ ¯ k ∇ H j s = (I) + (II) + (III) , where (I) , (II) , (III) denotes the terms on each line and the derivative of thematrix exponential is given by the formula[ δ exp] | u ( δu ) = Z e λu ( δu ) e (1 − λ ) u dλ. • Terms (I) and (II). These two terms are estimated in the same way, sowe only give the estimate for (I). We start by writing(I) = [ δ exp] | u ( s/ √− ω F H u ) e − u/ − s ( √− ω F H )= Z ddγ (cid:20) [ δ exp] | γu ( s/ √− ω F H γu ) e − ( γ/ u (cid:21) dγ. (6.25) There are 3 terms depending on where ddγ lands. The first is(Ia) γ = (cid:20) ddγ [ δ exp] | γu ( s/ (cid:21) ( √− ω F H γu ) e − ( γ/ u . This can be estimated as k (Ia) γ k C ,aβ − C k F H γu k C ,a − k u k C ,a k s k C ,aβ Next, we consider(Ib) γ = [ δ exp] | γu ( s/ (cid:20) ddγ ( √− ω F H γu ) (cid:21) e − ( γ/ u . Since ddγ F H γu = ¯ ∂∂ H γu u , we have k (Ib) γ k C ,aβ − C k ¯ ∂ ∇ H γu u k C ,a − k s k C ,aβ . The other term is(Ic) γ = [ δ exp] | γu ( s/ √− ω F H γu ) (cid:20) ddγ e − ( γ/ u (cid:21) and can be estimated by k (Ic) γ k C ,aβ − C k F H γu k C ,a − k u k C ,a k s k C ,aβ . Altogether, using k u k C ,a 1, the formula for the difference of connections A H γu − A H = e − γu ∇ H e γu which gives a formula for F H γu , and k F H k C ,a − C , we have k (I) k C ,aβ − C k u k C ,a k s k C ,aβ • Term (III). We write(III) = − Z ddγ (cid:20) e ( γ/ u [ g j ¯ k ∂ ¯ k ∇ H γu j e − γu [ δ exp] γu ( s )] e − ( γ/ u (cid:21) dγ. Using k u k C ,a 1, from here we can derive the estimate k (III) k C ,aβ − C k u k C ,a k s k C ,aβ . To do this, the covariant derivative ∇ H γu can be converted to ∇ H via ∇ H γu −∇ H = e − γu ∇ H e γu and we can use the variational formula ddγ ∇ H γu = ∇ H γu u . (cid:3) By Proposition 6.11 and the Banach fixed point theorem, there existsˇ u ∈ C ,aβ ( W ( X t )) with k ˇ u k C ,aβ < | t | (2 / β such that N (ˇ u ) = ˇ u , meaning thatΛ ω FLY ,t F ˇ u = 0 where F u is the curvature of H t e u . This proves the existenceof a pair solving dω ,t = 0 , F ˇ H t ∧ ω ,t = 0on X t . To complete the proof of the main theorem, we describe the behaviorof ( g FLY ,t , ˇ H t ) near the vanishing cycles. Proof of Theorem 6.1: The local estimate (6.1) follows from Lemma 6.6since g t = c i g co,t in a fixed neighborhood of the vanishing cycles. TheHermitian-Yang-Mills metric is given by ˇ H t = H t e ˇ u where ˇ u is the fixed-point solving N (ˇ u ) = ˇ u and satisfying k ˇ u k C ,aβ | t | (2 / | β | . Near the nodes,by the gluing construction (see (5.2)) we have ˇ H t = d i g co,t e ˇ u in the region R λ = {| t | k z k | t | λ } , which implies | ˇ H t − d i g co,t | g co,t Cr −| β | | t | (2 / | β | C | t | | β | / since r > | t | . We obtain similar estimates for ∇ k ˇ H t for k = 1 and k = 2.For the higher order estimates, we write the equation g j ¯ k ( F ˇ H ) j ¯ k = 0 inholomorphic cylindrical coordinates as r ( g FLY ,t ) j ¯ k ∂ j ∂ ¯ k ˇ H t = r ( g FLY ,t ) j ¯ k ∂ j ˇ H t ˇ H − t ∂ ¯ k ˇ H t . Note that since g t = r O ( I ) (see Lemma 2.11), by (6.1) we also have g FLY ,t = r O (1) for small t . Schauder estimates and a bootstrap argument imply | ∂ ℓ ˇ H ¯ kj | g euc C ℓ | t | | β | / . Converting g euc in holomorphic cylindrical coordinates to g co,t gives thestated estimate with λ = | β | / (cid:3) Remark 6.13. To simultaneously solve the Hermitian-Yang-Mills and theconformally balanced equations d ( k Ω t k ˇ ω t ˇ ω t ) = 0 , F ˇ H t ∧ ˇ ω t = 0 , we can set ˇ ω t = k Ω t k − ω FLY ,t ω FLY ,t so that k Ω t k ˇ ω t ˇ ω t = ω ,t . Here Ω t is a holomorphic volume form on X t ,whose existence through conifold transitions is proved in [31], normalizedby R X t √− t ∧ Ω t = 1. It is straightforward to show that this conformallybalanced metric ˇ g t associated to ˇ ω t satisfies a decay estimate near the van-ishing cycles. Namely, near any ODP p i ∈ X there is a constant c i such thatwe have the estimates | ˇ g t − c i g FLY ,t | g FLY ,t C ( r / + | t | )and for k > |∇ kg F LY,t (ˇ g t − c i g FLY ,t ) | g FLY ,t C k r − k . Note that, unlike Lemma 6.6, one can no longer expect decay in | t | for fixed r .This is due to the the fact that, near the ODP singularities, the holomorphicvolume form Ω t will not necessarily converge to a constant multiple of thenatural equivariant holomorphic volume form on the conifold. Nevertheless,these estimates still imply that there is a constant d i > g t , ˇ H t ) converges to ( g co, , d i g co, ) as | t | → Appendix A. The Fu-Li-Yau Gluing Construction In this appendix we will explain the gluing result of Fu-Li-Yau, whichestablishes Propositions 2.16 and 2.18. Before beginning, we recall the no-tation. We are primarily interested in the small resolution of the conifold,given by p : O P ( − ⊕ → P . Let h F S denote the Fubini-Study metric on O P ( − π : O P ( − ⊕ → V be the map contracting P . The pull-back radial function of the conicalCalabi-Yau metric on V is given by r = (cid:16) | u | h fs + | v | h fs (cid:17) / where ( u, v ) are fiber coordinates on O P ( − ⊕ . The holomorphic Reebfield on V induces a holomorphic C ∗ action given by S λ ( x, u, v ) = ( x, λ / u, λ / v ) . where x ∈ P . Unless otherwise specified, we will consider the restriction to λ ∈ R > ⊂ C ∗ . Note that we have S ∗ λ r = λ r . We define a smooth K¨ahler metric on O ( − ⊕ by ω sm = √− ∂∂r + p ∗ ω F S . We will consider also the cone metric ω co, = 32 √− ∂∂r . It will be important for us to compare ω co, and ω sm . Lemma A.1. There is a uniform constant C > such that following esti-mates hold ( i ) If < r < , then we have r C − ω co, ω sm Cr − ω co, ( ii ) If < r < then C − r ω co, ω sm ∧ ω co, r − ω co, Proof. First, since ω co, and ω sm define smooth K¨ahler metrics on the com-pact set { r } we can fix a constant A such that A − ω co, ω sm Aω co, . Consider S λ : { r } → { λ r λ } . From the homogeneity of r we have S ∗ λ ω co, = λ ω co, . On the other hand, S ∗ λ ω sm = λ √− ∂∂r + p ∗ ω F S , and so S ∗ λ ω sm = λ ( √− ∂∂r ) + 2 λ √− ∂∂r ∧ p ∗ ω F S using that p ∗ ω F S = 0. If λ λ ω sm S ∗ λ ω sm λ ω sm From the definition of A we have S ∗ λ ω sm λ ω sm λ A ω co, λ − A S ∗ λ ω co, and similarly S ∗ λ ω sm > λ ω sm > λ A − S ∗ λ ω co, . Since λ r λ , proves ( i ). The proof of ( ii ) is similar. (cid:3) Consider the four formΩ := √− ∂∂ ( χ ( r ) √− ∂∂r )where χ ( · ) is some smooth function to be determined. We computeΩ = χ ′′ ( r ) √− ∂r ∧ ∂r ∧ √− ∂∂r + χ ′ ( r ) ω co, . To understand the first term write √− ∂∂r = 4 √− ∂r ∧ ∂r + r √− ∂∂ log r . from which it follows that( √− ∂∂r ) = 8 r √− ∂r ∧ ∂r ∧ √− ∂∂ log r + ( r √− ∂∂ log r ) and also √− ∂r ∧ ∂r ∧ √− ∂∂r = r √− ∂r ∧ ∂r ∧ √− ∂∂ log r . Since √− ∂∂ log r > Lemma A.2. The following estimate holds everywhere on O P ( − ⊕ √− ∂r ∧ ∂r ∧ √− ∂∂r r √− ∂∂r ) . In particular, whenever χ ′′ < we have the lower bound Ω > (cid:18) χ ′ ( r ) + r χ ′′ ( r ) (cid:19) ω co, Fix R ≫ 1. Our goal is to find R ≫ 1, and a constant C R > R := C R S ∗ R √− ∂∂ ( χ ( r ) √− ∂∂r )to the Calabi-Yau metric ω CY . This will require carefully choosing χ , andthe constant C R . We are going to assume that χ ( s ) = s for s ∈ [0 , R agrees with a rescaling of ω co, on { r R } .More precisely, we have C R S ∗ R √− ∂∂ ( χ ( r ) √− ∂∂r ) = C R R ω co, { < r < R − } . In the remainder of the appendix we will determine conditions on χ, C R , R for this to be possible.We now consider Calabi-Yau metric ω CY . Consider the set U = { r < } ⊂ X . Since U is contractible onto P , we can write ω CY = λp ∗ ω F S + ∂β + ∂β for some (1 , 0) form β on U , and some λ > 0. To simplify notation, let usassume λ = 1. By solving the ∂ -equation we can write ω CY = p ∗ ω + √− ∂∂ϕ where ω is a K¨ahler form on P , with [ ω ] = [ ω F S ] ∈ H , ( P , R ) and ϕ : U → R a smooth function with ϕ | P = 0; see [35, Lemma 2.4]. Let h denote thedegree part of ϕ under S λ (recall that S λ corresponds to scaling withweight along the fibers), so that | ϕ − h | ∼ O ( | u | + | v | ), or in otherwords | ϕ − h | Cr . Let σ ( x ) be a positive cut-off function with 0 σ ( x ) σ ( x ) = 0for x ∈ [0 , 1] and σ ( x ) = 0 for x > 8. DefineΨ R = ω CY − √− ∂∂ Γ R Γ R = √− ∂∂ (cid:0) σ ( R r ) (cid:0) ( ϕ − h )(2 p ∗ ω + √− ∂∂ ( ϕ + h )) + h √− ∂∂h (cid:1)(cid:1) where, as before, R ≫ σ we have Ψ R = ω CY if { r > R − } We claim that Ψ R = 0 if { r < R − } . To see this, note that if x, y, w are commutative variables satisfying yw = 0,then( x − y )(2 w + ( x + y )) = x (2 w + x ) + xy − wy − xy − y = x (2 w + x ) − y . We apply this formula with x = √− ∂∂ϕ, y = √− ∂∂h , w = p ∗ ω andproduct being the wedge product. We only need to check that √− ∂∂h ∧ p ∗ ω F S = √− ∂∂h = 0, but this is clear since √− ∂∂h is linear along thefibers of O P ( − ⊕ → P .The main task is to find a lower bound for Ψ R in terms of ω co, in thetransition region { R − < r < R − } . To do this we expandΨ R = (1 − σ ) ω CY − (I) − (II) − (III) where(I) = 2Re (cid:0) ( √− σ ′ ( R r ) R ∂r ∧ ∂ ( ϕ − h ) ∧ (2 p ∗ ω + √− ∂∂ ( ϕ + h )) (cid:1) + 2Re (cid:0) σ ′ ( R r ) R ∂r ∧ ∂h ∧ √− ∂∂h (cid:1) (II) = σ ′′ ( L r ) L √− ∂r ∧ ∂r ∧ (cid:0) ( ϕ − h )(2 p ∗ ω + √− ∂∂ ( ϕ + h )) + h √− ∂∂h (cid:1) (III) = σ ′ ( L r ) L √− ∂∂r ∧ (cid:0) ( ϕ − h )(2 p ∗ ω + √− ∂∂ ( ϕ + h )) + h √− ∂∂h (cid:1) . Our goal is to estimate each term from below by ω co, . Each term will betreated differently, depending on whether it is homogeneous or not. • Term (I). Observe that ∂r ∧ ∂ ( ϕ − h ) ∼ r ω sm To see this recall that, in coordinates we have r = | u | h F S + | v | h F S so that, in coordinates where ∂h F S = 0, we have ∂r = ¯ udu + ¯ vdv. On the other hand, since h is linear along the fibers of p : O P ( − ⊕ → P we have ∂ ( ϕ − h ) = O ( u, v )( d ¯ u + d ¯ v )and so ∂r ∧ ∂ ( ϕ − h ) Cr ω sm . Thus, by Lemma A.1, the first term in (I) can be controlled by r ω sm r ω co, .To analyze the second term in (I) we observe that S ∗ λ ( ∂r ∧ ∂h ∧ √− ∂∂h ) = λ ∂r ∧ ∂h ∧ √− ∂∂h , which, by the homogeneity of ω co, , implies ∂r ∧ ∂h ∧ √− ∂∂h Cr ω co, . In total, we have (I) Cr ω co, • Term (II). Again, by homogeneity we have0 √− ∂r ∧ ∂r Cr ω co, , while the bound | ϕ − h | Cr yields a bound for the first term in (II) √− ∂r ∧ ∂r ∧ (cid:0) ( ϕ − h )(2 p ∗ ω + √− ∂∂ ( ϕ + h ) (cid:1) r ω co, ∧ r ω sm Cr ω co, where we used Lemma A.1, ( ii ). The second term in (II) can be treateddirectly by scaling. We have √− ∂r ∧ ∂r ∧ h √− ∂∂h Cr ω co, . In total, we have (II) Cr ω co, • Term (III) can be treated similarly to term (II). The first term can beestimated as √− ∂∂r ∧ ( ϕ − h )(2 p ∗ ω + √− ∂∂ ( ϕ + h )) Cr ω sm Cr ω co, , thanks to Lemma A.1 ( i ) again. The homogeneous term is easily estimatedas √− ∂∂r ∧ h √− ∂∂h Cr ω co, . In summation, we have proved Lemma A.3. There is a constant C > so that on the region { R − < r < R − } we have Ψ R > − CR r ω co, > − CRω co, . At this point we consider Ψ R + C C R S ∗ R Ω . In order for this form to be positive we need to show that C , C R can bechosen consistently. To do this we consider the conditions for positivityin four different regions. In the following C will denote a uniform constantwhich can change from line to line, but depends only on the fixed backgrounddata and is, in particular, independent of R, C , C R . • Region { < r < R − } .In this region we have σ = 0 and S ∗ R χ ( r ) = R r , so thatΨ R + C C R S ∗ R Ω = 49 C C R R ω co, > • Region { R − < r < R − } .Thanks to Lemma A.3, and the fact that χ ( s ) = s for s ∈ [0 , 4] we haveΨ R > − CRω co, C C R S ∗ R Ω = 49 C C R R ω co, . Thus, in order to ensure positivity we need C > C , C R R = R , so wemust take C R = R − . • Region { R − < r < } .By definition we have Ψ R = ω CY > Cω sm . On the other hand whenever χ ′′ < 0, Lemma A.2 gives the estimate C C R S ∗ R Ω > C R (2 χ ′ ( R r ) + ( Rr ) χ ′′ ( R r )) ω co, . Now, from Lemma A.1 we have ω co, Cr − ω sm and so we can ensurepositivity provided49 C Rr − (2 χ ′ ( R r ) + ( Rr ) χ ′′ ( R r )) > − C. If we can constant χ so that χ ′ > 0, then when χ ′′ > S ∗ R Ω > • Region { < r } .In this region we take χ ( s ) = const . , and so, by definitionΨ R + C C R S ∗ R Ω = ω CY In summary, Ψ R + C C R S ∗ R Ω will be positive definite provided we canconstruct a cut-off function χ ( s ) (upon defining s = R r ) with the followingproperties: for 0 < ε ≪ • χ ( s ) = s for s ∈ [0 , χ ′ ( s ) > • χ ( s ) is constant for s > R , • χ ( s ) satisfies 1 s (cid:0) χ ′ ( s ) + sχ ′′ ( s ) (cid:1) > − εR for s ∈ [4 , R ]A cut-off function with these properties is constructed in [35, Lemma 2.2],but for the readers convenience we give a slightly different proof here. Lemma A.4. For R ≫ sufficiently large there exists a smooth function χ ( s ) with the following properties ( i ) χ ( s ) = s for s ∈ [0 , , and χ ′ ( s ) > . ( ii ) χ ( s ) is constant for s > R , ( iii ) there is a uniform constant C ′ such that s (cid:0) χ ′ ( s ) + sχ ′′ ( s ) (cid:1) > − C ′ R Proof. Let v = χ ′ . Then we need to find v such that v ( s ) = 1 for s ∈ [0 , v ( s ) = 0 for s > R , and 1 s dds ( s v ) > − C ′ R . Consider v ( s ) = , s ∈ [0 , as − + bs − + cs + d, s ∈ [5 , R − , s > R − . Demanding that v ( s ) is C gives a system of 4 equations in the unknowns a, b, c, d , whose solutions are given by a = − O ( R − ) , b = 75+ O ( R − ) , c = 75 R − + O ( R − ) , d = − R − + O ( R − ) . On the other hand, if w ( s ) = C α s α , then1 s dds ( s w ) = ( α + 2) C α s α − = > α > − , and C α > 00 if α = − > α < − , and C α . Since the terms corresponding to a, b, c fall into one of these cases, it followsthat 1 s dds ( s v ) > − R + O ( R − ) . It only remains to check that v ( s ) > s ∈ [4 , R ]. To see this, observethat 0 = v ′ ( λR ) ⇐⇒ λ − λ = O ( R − ) . Since the equation λ − λ = 0 has only three real solutions λ = ± , 0, itfollows that for R sufficiently large v ′ ( s ) = 0 has only two real solutions on[4 , R ]. Since these solutions are given by s = 4 and s = R − 1, it followsthat v ′ ( s ) = 0 in (4 , R − v does not have an interior minimumin [4 , R ]. This immediately implies v ( s ) > v be the result of convolving v with a positive, symmetricmollifier with sufficiently small support. Then ˆ v is smooth and has thesame properties as v . Integrating ˆ v yields χ . (cid:3) Now let us explain the extension of this construction to the metrics ω co,a on the small resolution. For this, recall that ω co,a = √− ∂∂f a ( k z k ) + 4 a π ∗ ω F S where f a ( x ) = a f ( xa ) and f satisfies( xf ′ ) + 6( xf ′ ) = x . Rewriting this equation in terms of the variable z defined by x = z − / andapplying standard ODE techniques we obtain the following result, whoseproof we leave to the reader Lemma A.5. For x ≫ , the function f ( x ) has a convergent expansion f ( x ) = 32 x / − x ) + ∞ X n =0 c n x − n/ . In particular, f a ( x ) → x / smoothly and uniformly on any compact set as a → . We now describe how to glue ω co,a to the Calabi-Yau metric ω CY to obtaina balanced metric. We first recall thatΩ R = C R − √− ∂∂ ( χ ( R r ) R √− ∂∂r ) = C R − √− ∂∂ ( χ ( R r ) √− ∂∂r )On the other hand, we have ω co,a = ( √− ∂∂f a ( k z k )) + 8 a √− ∂∂f a ( k z k ) ∧ π ∗ ω F S = √− ∂∂ (cid:0) f a ( k z k ) (cid:0) √− ∂∂f a ( k z k ) + 8 a π ∗ ω F S (cid:1)(cid:1) This suggests that we defineΩ R,a = C R − √− ∂∂ (cid:18) χ (cid:18) R f a ( k z k ) (cid:19) (cid:0) √− ∂∂f a ( k z k ) + 8 a π ∗ ω F S (cid:1)(cid:19) Lemma A.6. For a ≪ sufficiently small there is an open set U containing the ( − , − rational curve such that Ω R,a = C Rω co,a Proof. We only need to observe that the formula holds whenever2 R f a ( k z k ) . Now since f a ( k z k ) converges uniformly to r on compact sets away fromthe ( − , − 1) rational curve, this inequality will hold for a sufficiently smallprovided r < R − . (cid:3) Next, recall that the gluing of ω co, and ω CY depending on only twoestimates. • The boundsΨ R > − CRω co, , C R − S ∗ R Ω = 4 C Rω co, in the region { R − < r < R − } . Since Ω R,a converges uniformly to ω co, onthis region, the same bound holds with ω co, replaced by ω co,a , after possiblychanging the constants. • The bound ω co, Cr − ω sm in the region { R − < r < } . Again, from the uniform convergence of ω co,a to ω co, this bound holds, up to possibly increasing C for ω co,a as well, fromthe uniform convergence.It follows that the gluing procedure used to glue ω co, to ω CY carries overin exactly the same way to glue ω co,a to ω CY for 0 < a ≪ 1. Furthermore,from Lemma A.5 we obtain the smooth, uniform convergence of ω a to ω oncompact sets away from the ( − , − 1) rational curves. References [1] B. Andreas and M. Garcia-Fernandez Solutions of the Strominger system via stablebundles on Calabi-Yau threefolds , Comm. Math. Phys. (2012), no. 1, 153–168.[2] B. Andreas and M. Garcia-Fernandez Heterotic non-K¨ahler geometries via polystablebundles on Calabi-Yau threefolds , J. Geom. Phys. (2012), no. 2, 183–188.[3] S. Bando, and Y.-T. Siu Stable sheaves and Einstein-Hermitian metrics , Geometryand analysis on complex manifolds, 39–50, World. Sci. Publ., River Edge, NJ, 1994.[4] L. Bedulli and L. Vezzoni On the stability of the anomaly flow , arXiv:2005.05670.[5] Y. Bozhkov, Specific complex geometry of certain complex surfaces and three-folds ,PhD Thesis, University of Warwick, England (1992).[6] E. Calabi, Improper affine hyperspheres of convex type and a generalization of atheorem by K. Jorgens , Michigan Math. J. 5 (1958) 105-–126[7] P. Candelas, P. Green, and T. H¨ubsch Finite distance between distinct Calabi-Yaumanifolds , Phys. Rev. Lett. (1989), no. 17, 1956–1959.[8] P. Candelas, P. Green, and T. H¨ubsch Rolling among Calabi-Yau vacua , NuclearPhys. B (1990), no. 1, 49–102.[9] P. Candelas, G. Horowitz, A. Strominger, and E. Witten Vacuum configurations forsuperstrings , Nuclear Phys. B (1985), no. 1, 46–74.[10] P. Candelas, and X. de la Ossa Comments on conifolds , Nuclear Phys. B (1990),no. 1, 246–268.[11] Y. M. Chan Desingularizations of Calabi–Yau 3-folds with conical singularities. II.The obstructed case Quarterly journal of mathematics (2009), 60(1), 1–44. [12] X. Chen and S. Sun Algebraic Tangent Cones of Reflexive Sheaves , Int. Math. Res.Not. IMRN 2020, no. 24, 10042–10063.[13] X. Chen and S. Sun Analytic tangent cones of admissible Hermitian-Yang-Mills con-nections , arXiv:1806.11247[14] X. Chen and S. Sun Singularities of Hermitian-Yang-Mills connections and Harder-Narasimhan-Seshadri filtrations , Duke Math. J. (2020), no. 14, 2629–2695.[15] X. Chen, and S. Sun Reflexive sheaves, Hermitian-Yang-Mills connections, and tan-gent cones , arXiv:1912.09561[16] M.-T. Chuan Existence of Hermitian-Yang-Mills metrics under conifold transitions ,Comm. Anal. Geom. (2012), no. 4, 677–749.[17] J. Chu, L. Huang, and X. Zhu, The Fu-Yau equation in higher dimensions , PekingMathematical Journal 2.1 (2019), 71-97.[18] R. Conlon, and H.-J. Hein Asymptotically conical Calabi-Yau manifolds, I. , DukeMath. J. (2013), no. 15, 2855–2902.[19] X. de la Ossa and E. Svanes, Holomorphic bundles and the moduli spaces of N=1supersymmetric heterotic compactifications , Journal of High Energy Physics 10 (2014)p.123.[20] J.-P. Demailly, Complex Analytic and Differential Geometry , available on the author’swebpage, 2012[21] S.K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfacesand stable vector bundles , Proc. London Math. Soc. (3) 50 (1985), no.1, 1-26.[22] T. Fei, A construction of non-K¨ahler Calabi-Yau manifolds and new solutions to theStrominger system , Adv. Math. (2016), 529–550[23] T. Fei, Some torsional local models for heterotic strings , Comm. Anal. Geom. (2017), no. 5, 941–968.[24] T. Fei, Z. Huang, S. Picard, A construction of infinitely many solutions to the Stro-minger system , Journal of Differential Geometry 117(1), 23–39.[25] T. Fei and D.H. Phong, Unification of the K¨ahler-Ricci and Anomaly flows , Differ-ential Geometry, Calabi-Yau Theory, and General Relativity, 89-104, Surv. Differ.Geom., 22, International Press, 2018.[26] T. Fei, D.H. Phong, S. Picard and X.-W. Zhang, Estimates for a geometric flow forthe Type IIB string , arXiv:2004.14529.[27] T. Fei and S.-T. Yau Invariant solutions to the Strominger system on complex Liegroups and their quotients , Comm. Math. Phys. (2015), no. 3, 1183–1195.[28] M. Fern´andez, S. Ivanov, L. Ugarte, and D. Vassilev Non-Kaehler heterotic stringsolutions with non-zero fluxes and non-constant dilaton , J. High Energy Phys. 2014,no. 6, 073.[29] M. Fern´andez, S. Ivanov, L. Ugarte, and R. Villacampa Non-Kaehler heterotic stringcompactifications with non-zero fluxes and constant dilaton , Comm. Math. Phys. (2009), no. 2, 677-697.[30] A. Fino, G. Grantcharov and L. Vezzoni, Solutions to the Hull-Strominger systemwith torus symmetry , arXiv:1901.10322.[31] R. Friedman On threefolds with trivial canonical bundle , Complex geometry and Lietheory (Sundance, UT, 1989), 103–134. Proc. Sympos. Pure Math., , Amer. Math.Soc., Providence, RI, 1991.[32] R. Friedman Simultaneous resolution of threefold double points , Math. Ann. (1986), no. 4, 671–689.[33] J. Fu, L.-S. Tseng, and S.-T. Yau Local heterotic torsional models , Comm. Math.Phys. (2009), no. 3, 1151–1169.[34] J.-X. Fu, and S.-T Yau The theory of superstring with flux on non-K¨ahler manifoldsand the complex Monge-Amp`ere equation J. Differential Geom. (2008), no. 3, 369–428. [35] J. Fu, J. Li, and S.-T. Yau Balanced metrics on non-K¨ahler Calabi-Yau threefolds , J.Differential Geom. (2012), 81–129.[36] M. Garcia-Fernandex Lectures on the Strominger system , Travaux math´ematiques.Vol XXIV, 7–61, Trav. Math., , Fac. Sci. Technol. Commun. Univ. Luxemb., Lux-embourg, 2016[37] M. Garcia-Fernandez T-dual solutions of the Hull-Strominger system on non-Kahlerthreefolds , J. Reine Angew. 766 (2020), 137–150.[38] M. Garcia-Fernandez, R. Rubio, C. Tipler Gauge theory for string algebroids ,arXiv:2004.11399.[39] M. Garcia-Fernandez, R. Rubio, C. Shahbazi, C. Tipler Canonical metrics on holo-morphic Courant algebroids , arXiv:1803.01873.[40] H. Grauert ¨Uber Modifikationen und exzeptionelle analytische Mengen , Math. Ann. (1962), 331–368.[41] P. Green and T. H¨ubsch Possible phase transitions among Calabi-Yau compactifica-tions , Phys Rev. Lett. (1988), no. 10, 1163–1166.[42] P. Green and T. H¨ubsch Connecting moduli spaces of Calabi-Yau threefolds , Comm.Math. Phys. (1988), no. 3, 431–441.[43] P. Lu and G. Tian Complex structures on connected sums of S3 × S3 , Manifolds andgeometry, 284-293, Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge,1996.[44] Q. Han and F. Lin, Elliptic partial differential equations . Vol. 1. American Mathe-matical Soc., 2011.[45] H.J. Hein and S. Sun Calabi-Yau manifolds with isolated conical singularities Publi-cations math´ematiques de l’IHES 126.1 (2017), 73–130.[46] L. Huang, The adiabatic limit of Fu-Yau equations , arXiv:2010.14667.[47] C. M. Hull, Compactifications of the heterotic superstring , Phys. Lett. B (1986),no. 4, 357–364.[48] A. Jacob, H. S´a Earp, and T. Walpuski Tangent cones of Hemitian Yang-Mills con-nections with isolated singularities , Math. Res. Lett. (2018), no. 5, 1429–1445.[49] A. Jacob, T. Walpuski Hermitian-Yang-Mills metrics on reflexive sheaves overasymptotically cylindrical K¨ahler manifolds , Comm. Partial Differential Equations (2018), no. 11, 1566–1598[50] S. Karigiannis Desingularization of G2 manifolds with isolated conical singularities Geometry and Topology 13 (2009), 1583–1655.[51] A. Kas, and M. Schlessinger On the versal deformation of a complex space with anisolated singularity , Math. Ann. (1972), 23–29.[52] Y. Kawamata Unobstructed deformations. A remark on a paper of Z. Ran: “Defor-mations of manifolds with torsion of negative canonical bundle” J. Algebraic Geom. (1992), no. 2, 183–190.[53] E. E. Levi, Studii sur punti singolari essenziali dell funzioni analitiche di due o piuvariabli complesse , Annali di Mat. Pura ed. Appl. (1910), no. 3, 61–87.[54] J. Li and S.T. Yau, Hermitian-Yang-Mills connections on non-Kahler manifolds ,Mathematical aspects of string theory, Adv. Ser. Math. Phys., World Sci. Publishing(1986), 560-573.[55] J. Li and S.T. Yau, The existence of supersymmetric string theory with torsion , J.Diff. Geom. no. 1 (2005), 143-181.[56] M.L. Michelsohn, On the existence of special metrics in complex geometry , Acta Math. , (1982) 261–295.[57] A. Otal, L. Ugarte, R. Villacampa, Invariant solutions to the Strominger system andthe heterotic equations of motion on solvmanifolds , Nuclear Physics B 920 (2017),442–474.[58] D. H. Phong Geometric partial differential equations from unified string theories ,preprint, arXiv:1906.03693 [59] D. H. Phong, S. Picard, and X. Zhang A second order estimate for general complexHessian equations , Anal. PDE (2016), no. 7, 1693–1709.[60] D. H. Phong, S. Picard, and X. Zhang The Fu-Yau equation with negative slopeparameter , Invent. Math. (2017), no. 2, 541–576.[61] D. H. Phong, S. Picard, and X. Zhang The Anomaly flow and the Fu-Yau equation ,Ann. PDE (2018), no. 2, Paper No. 13, 60 pp.[62] D.H. Phong, S. Picard, and X.W. Zhang, Geometric flows and Strominger systems ,Mathematische Zeitschrift (2018), 101–113.[63] D.H. Phong, S. Picard, and X.W. Zhang, Anomaly flows , Comm. Anal. Geom. (2018), No. 4, 955–1008.[64] D.H. Phong, S. Picard, and X.W. Zhang, A flow of conformally balanced metrics withK¨ahler fixed points , Math. Ann. 374 (2019), no. 3-4, 2005-2040.[65] D. H. Phong, S. Picard, and X. Zhang On estimates for the Fu-Yau generalization ofa Strominger system , J. Reine Angew. Math. (2019), 243–274.[66] D. H. Phong, S. Picard, and X. Zhang Fu-Yau Hessian equations , arXiv:1801.09842.[67] D.H. Phong, N. Sesum and J. Sturm, Multiplier ideal sheaves and the Kahler-Ricciflow , Communications in Analysis and Geometry 15(3), 613–632.[68] D. Popovici A simple proof of a theorem by Uhlenbeck and Yau , Math. Z. (2005),no. 4, 855–872.[69] X-C. Rong and Y-G. Zhang Continuity of extremal transitions and flops for Calabi-Yau manifolds , J. Differential Geom. 89 (2011), 233–269.[70] M. Sherman, B. Weinkove, Interior derivative estimates for the Kahler-Ricci flow ,Pacific J. Math. 257 (2012), no. 2, 491-–501.[71] J. Song On a conjecture of Candelas and de la Ossa , Comm. Math. Phys. 334 (2015),697—717.[72] C. Spotti, Deformations of nodal K¨ahler-Einstein del Pezzo surfaces with discreteautomorphism groups , Journal of the London Mathematical Society (2014), no. 2,539–558.[73] Z. Ran Deformations of manifolds with torsion or negative canonical bundle , J. Al-gebraic Geom. (1992), no. 2, 279–291.[74] M. Reid The moduli space of -folds with K = 0 may nevertheless be irreducible ,Math. Ann. (1987), no. 1-4, 329–334.[75] M. Stenzel Ricci-flat metrics on the complexification of a compact rank one symmetricspace , Manuscripta Math. (1993), no. 2, 151–163.[76] A. Strominger, Superstrings with torsion , Nuclear Phys. B (1986), no. 2, 253–284.[77] Y.-T. Siu Lectures on Hermitian-Einstein metrics for stable bundles and K¨ahler-Einstein metrics , DMV Seminar, . Birkh¨auser Verlag, Basel, 1987.[78] G. Tian On stability of the tangent bundles of Fano varieties , Internat. J. Math. (1992), no. 3, 401–413.[79] G. Tian Smooth -folds with trivial canonical bundle and ordinary double points ,Essays on mirror manifolds, 458–479, Int. Press, Hong Kong, 1992.[80] V. Tosatti Limits of Calabi-Yau metrics when the K¨ahler class degenerates J. Eur.Math. Soc. (JEMS) 11 (2009), no.4, 755–776.[81] V. Tosatti Adiabatic limits of Ricci-flat Kahler metrics , J. Differential Geom. Volume84, Number 2 (2010), 427-–453[82] L.-S. Tseng and S.-T. Yau, Non-Kaehler Calabi-Yau manifolds , in Strings Math 2011,241-254, Proceedings of Symposia in Pure Mathematics, 85, Amer. Math. Soc., Prov-idence, RI (2012).[83] K. Uhlenbeck and S.T. Yau On the existence of Hermitian-Yang-Mills connectionsin stable vector bundles , Comm. Pure Appl. Math. (1986), 257–293; (1989),703–707.[84] B. Yang, The uniqueness of tangent cones for Yang–Mills connections with isolatedsingularities , Adv. Math., 180(2) (2003), 648—691. [85] S.-T. Yau, Metrics on complex manifolds , Sci. China Math. (2010), no. 3, 565–572.[86] S.-T. Yau, On the Ricci curvature of a compact K¨ahler manifold and the complexMonge-Amp`ere equation. I, Comm. Pure Appl. Math. 31 (1978) 339-411.[87] S.-T. Yau, A splitting theorem and an algebraic geometric characterization of locallyhermitian symmetric spaces , Communications in Analysis and Geometry (1993),no. 3, 473–486. Email address : [email protected] Department of Mathematics, Massachusetts Institute of Technology, 77Massachusetts Avenue, Cambridge, MA 02139 Email address : [email protected] Department of Mathematics, The University of British Columbia, 1984 Math-ematics Road, Vancouver BC Canada V6T 1Z2 Email address : [email protected]@math.harvard.edu