On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties
aa r X i v : . [ m a t h . AG ] A p r ON THE HYPERBOLICITY OF BASE SPACES FOR MAXIMALLYVARIATIONAL FAMILIES OF SMOOTH PROJECTIVE VARIETIES
YA DENG, WITH AN APPENDIX BY DAN ABRAMOVICHAbstract.
For maximal variational smooth families of projective manifolds whose general fibers havesemi-ample canonical bundle, the Viehweg hyperbolicity conjecture states that the base spaces of suchfamilies are of log general type. This deep conjecture was recently proved by Campana-Păun and waslater generalized by Popa-Schnell. In this paper we prove that those base spaces are pseudo Kobayashihyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudoKobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbolicityof moduli spaces of polarized manifolds with semi-ample canonical bundle. This proves a conjectureby Viehweg-Zuo in 2003. We also prove the Kobayashi hyperbolicity of base spaces for effectivelyparametrized families of minimal projective manifolds of general type. This generalizes previous workby To-Yeung, in which they further assumed that these families are canonically polarized.
Contents
0. Introduction 20.1. Main theorems 20.2. Previous related results 20.3. Strategy of the proof 3Acknowledgments 6Notations and conventions. 61. Brody hyperbolicity of the base 61.1. Abstract Viehweg-Zuo Higgs bundles 71.2. A quick tour on Viehweg-Zuo’s construction 71.3. Proper metrics for logarithmic Higgs bundles 91.4. A generic local Torelli for VZ Higgs bundle 112. Pseudo Kobayashi hyperbolicity of the base 132.1. Finsler metric and (pseudo) Kobayashi hyperbolicity 132.2. Curvature formula 152.3. Construction of the Finsler metric 182.4. Proof of Theorem B 203. Kobayashi hyperbolicity of the base 213.1. Preliminary for positivity of direct images 213.2. Positivity of direct images 223.3. Sufficiently many “moving” hypersurfaces 273.4. Kobayashi hyperbolicity of the moduli spaces 28Appendix A. Q -mild reductions (by Dan Abramovich) 30References 32 Date : Friday 1 st May, 2020.2010
Mathematics Subject Classification.
Key words and phrases. pseudo Kobayashi hyperbolicity, Brody hyperbolicity, moduli spaces, Viehweg-Zuo question,polarized variation of Hodge structures, Viehweg-Zuo Higgs bundles, Finsler metric, positivity of direct images, Griffithscurvature formula of Hodge bundles.
0. Introduction
Main theorems.
A complex space X is Brody hyperbolic if there is no non-constant holo-morphic map γ : C → X . The first result in this paper is the affirmative answer to a conjectureby by Viehweg-Zuo [VZ03, Question 0.2] on the Brody hyperbolicity of moduli spaces for polarizedmanifolds with semi-ample canonical sheaf. Theorem A (Brody hyperbolicity of moduli spaces) . Consider the moduli functor P h of polarizedmanifolds with semi-ample canonical sheaf introduced by Viehweg [Vie95, §7.6], where h is the Hilbertpolynomial associated to the polarization H . Assume that for some quasi-projective manifold V thereexists a smooth family ( f U : U → V , H ) ∈ P h ( V ) for which the induced moduli map φ U : V → P h isquasi-finite over its image, where P h denotes to be the quasi-projective coarse moduli scheme for P h .Then the base space V is Brody hyperbolic. A complex space X is called pseudo Kobayashi hyperbolic , if X is hyperbolic modulo a properZariski closed subset ∆ ( X , that is, the Kobayashi pseudo distance d X : X × X → [ , + ∞[ of X satisfies that d X ( p , q ) > p , q ∈ X not both contained in ∆ . Inparticular, X is pseudo Brody hyperbolic : any non-constant holomorphic map γ : C → X has image γ ( C ) ⊂ ∆ . When such ∆ is an empty set, this definition reduces to the usual definition of Kobayashihyperbolicity , and the Kobayashi pseudo distance d X is a distance.In this paper we indeed prove a stronger result than Theorem A. Theorem B.
Let f U : U → V be a smooth projective morphism between complex quasi-projectivemanifolds with connected fibers. Assume that the general fiber of f U has semi-ample canonical bundle,and f U is of maximal variation, that is, the general fiber of f U can only be birational to at most countablymany other fibers. Then the base space V is pseudo Kobayashi hyperbolic. As a byproduct, we reduce the pseudo Kobayashi hyperbolicity of varieties to the existence ofcertain negatively curved Higgs bundles (which we call
Viehweg-Zuo Higgs bundles in Definition 1.1).This provides a main building block for our recent work [Den19] on the hyperbolicity of bases oflog Calabi-Yau pairs.Another aim of the paper is to prove affirmatively a folklore conjecture on the
Kobayashi hyper-bolicity for moduli spaces of minimal projective manifolds of general type, which can be thought ofas an analytic refinement of Theorem A in the case that fibers have big and nef canonical bundle.
Theorem C.
Let f U : U → V be a smooth projective family of minimal projective manifolds ofgeneral type over a quasi-projective manifold V . Assume that f U is effectively parametrized , that is,the Kodaira-Spencer map ρ y : T V , y → H ( U y , T U y ) (0.1.1) is injective for each point y ∈ V , where T U y denotes the tangent bundle of the fiber U y : = f − U ( y ) . Thenthe base space V is Kobayashi hyperbolic. Previous related results.
Theorem B is closely related to the
Viehweg hyperbolicity conjecture :let f U : U → V be a maximally variational smooth projective family of projective manifolds withsemi-ample canonical bundle over a quasi-projective manifold V , then the base V must be of log-general type. In the series of works [VZ01, VZ02, VZ03], Viehweg-Zuo constructed in a first step abig subsheaf of symmetric log differential forms of the base (so-called Viehweg-Zuo sheaves ). Built onthis result, Viehweg hyperbolicity conjecture was shown by Kebekus-Kovács [KK08a, KK08b, KK10]when V is a surface or threefold, by Patakfalvi [Pat12] when V is compact or admits a non-uniruledcompactification, and it was completely solved by Campana-Păun [CP19], in which they proved avast generalization of the famous generic semipositivity result of Miyaoka (see also [CP15, CP16,Schn17] for other different proofs). More recently, using deep theory of Hodge modules, Popa-Schnell [PS17] constructed Viehweg-Zuo sheaves on the base space V of the smooth family f U : U → V of projective manifolds whose geometric generic fiber admits a good minimal model. Combining The quasi-projectivity of P h was proved by Viehweg in [Vie95]. NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 3 this with the aforementioned theorem of Campana-Păun, they proved that such base space V is of loggeneral type. Therefore, Theorem B is predicted by a famous conjecture of Lang (cf. [Lan91, ChapterVIII. Conjecture 1.4]), which stipulates that a complex quasi-projective manifold is pseudo Kobayashihyperbolic if and only if it is of log general type. To our knowledge, Lang’s conjecture is by nowknown for the trivial case of curves, for general hypersurface X in the complex projective space C P n of high degrees [Bro17, Dem20, Siu15] as well as their complements C P n \ X [BD19], for projectivemanifolds whose universal cover carries a bounded strictly plurisubharmonic function [BD18], forquotients of bounded (symmetric) domains [Rou16, CRT19, CDG19], and for subvarieties on abelianvarieties [Yam19]. Theorem B therefore provides some new evidences for Lang’s conjecture.Theorem A was first proved by Viehweg-Zuo [VZ03, Theorem 0.1] for moduli spaces of canonic-ally polarized manifolds. Combining the approaches by Viehweg-Zuo [VZ03] with those by Popa-Schnell [PS17], very recently, Popa-Taji-Wu [PTW19, Theorem 1.1] proved Theorem A for modulispaces of polarized manifolds with big and semi-ample canonical bundles. As we will see below, ourwork owes a lot to the general strategies and techniques in their work [VZ03, PTW19].The Kobayashi hyperbolicity of moduli spaces M д of compact Riemann surfaces of genus д > V considered in Theorem C when the canonical bundle K U y of each fiber U y : = f − U ( y ) of f U : U → V is further assumed to be ample (see also [BPW17,Sch18] for alternative proofs). Differently from the approaches in [VZ03, PTW19], their strategy isto study the curvature of the generalized Weil-Petersson metric for families of canonically polarizedmanifolds, along the approaches initiated by Siu [Siu86] and later developed by Schumacher [Sch12].For the smooth family of Calabi-Yau manifolds (resp. orbifolds), Berndtsson-Păun-Wang [BPW17]and Schumacher [Sch18] (resp. To-Yeung [TY18]) proved the Kobayashi hyperbolicity of the baseonce this family is assumed to be effectively parametrized.Recently, Lu, Sun, Zuo and the author [DLSZ19] proved a big Picard type theorem for modulispaces of polarized manifolds with semi-ample canonical sheaf. A crucial step of the proof relieson the “generic local Torelli-type theorem” in Theorem D. Theorem D also inspired us a lot in ourmore recent work [Den20] on the big Picard theorem for varieties admitting variation of Hodgestructures.0.3. Strategy of the proof.
For the smooth family f U : U → V of canonically polarized manifoldswith maximal variation, Viehweg-Zuo [VZ03] constructed certain negatively twisted Higgs bundles(which we call Viehweg-Zuo Higgs bundles in Definition 1.1) ( ˜ E , ˜ θ ) : = ( É nq = L − ⊗ E n − q , q , É nq = ⊗ θ n − q , q ) , over some smooth projective compactification Y of a certain birational model ˜ V of V , where L is some big and nef line bundle on Y , and (cid:0) É nq = E n − q , q , É nq = θ n − q , q (cid:1) is a Higgs bundle in-duced by a polarized variation of Hodge structure defined over a Zariski open set of ˜ V . In a recentpaper [PTW19], Popa-Taji-Wu introduced several new inputs to develop Viehweg-Zuo’s strategyin [VZ03], which enables them to construct those Higgs bundles on base spaces of smooth familieswhose geometric generic fiber admits a good minimal model (see also Theorem 1.2 for a weakerstatement as well as a slightly different proof following the original construction by Viehweg-Zuo).As we will see in the main content, the Viehweg-Zuo Higgs bundles (VZ Higgs bundles for short)are the crucial tools in proving our main results.When each fibers U y : = f − U ( y ) of the smooth family f U : U → V considered in Theorem B haveample or big and nef canonical bundles, let us briefly recall the general strategies in proving the pseudo Brody hyperbolicity of V in [VZ03, PTW19]. A certain sub-Higgs bundle ( F , η ) of ( ˜ E , ˜ θ ) withlog poles contained in the divisor D : = Y \ ˜ V gives rise to a morphism τ γ , k : T ⊗ k C → γ ∗ ( L − ⊗ E n − k , k ) (0.3.1)for any entire curve γ : C → ˜ V . If γ : C → ˜ V is Zariski dense, by the Kodaira-Nakano vanishing(when K U y is ample) and Bogomolov-Sommese vanishing theorems (when K U y is big and nef), onecan verify that τ γ , ( C ) .
0. Hence there is some m > γ ) so that τ γ , m factors through YA DENG γ ∗ ( L − ⊗ N n − m , m ) , where N n − m , m is the kernel of the Higgs field θ m : E n − m , m → E n − m − , m + ⊗ Ω Y ( log D ) . Applying Zuo’s theorem [Zuo00] on the negativity of N n − m , m , a certain positively curvedmetric for L can produce a singular hermitian metric on T C with the Gaussian curvature boundedfrom above by a negative constant, which contradicts with the (Demailly’s) Ahlfors-Schwarz lemma[Dem97, Lemma 3.2]. However, this approach did not provide enough information for the Kobayashipseudo distance of the base V . Moreover, the use of vanishing theorem cannot show τ γ , ( C ) . f U : U → V is not minimal manifolds of general type.One of the main results in the present paper is to apply the VZ Higgs bundle to construct a(possibly degenerate) Finsler metric F on some birational model ˜ V of the base V , whose holomorphicsectional curvature is bounded above by a negative constant (say negatively curved Finsler metric inDefinition 2.3.(ii)). A bimeromorphic criteria for pseudo Kobayashi hyperbolicity in Lemma 2.4 statesthat, the base is pseudo Kobayashi hyperbolic if F is positively definite over a Zariski dense open set.Let us now briefly explain our idea of the constructions. By factorizing through some sub-Higgssheaf ( F , η ) ⊆ ( ˜ E , ˜ θ ) with logarithmic poles only along the boundary divisor D : = Y \ ˜ V , one candefine a morphism for any k = , . . . , n : τ k : Sym k T Y (− log D ) → L − ⊗ E n − k , k , (0.3.2)where L is some big line bundle over Y equipped with a positively curved singular hermitian metric h L . Then for each k , the hermitian metric h k on ˜ E k : = L − ⊗ E n − k , k induced by the Hodge metricas well as h L (see Proposition 1.3 for details) will give rise to a Finsler metric F k on T Y (− log D ) by taking the k -th root of the pull-back τ ∗ k h k . However, the holomorphic sectional curvature of F k might not be negatively curved. Inspired by the aforementioned work of Schumacher, To-Yeung andBerndtsson-Păun-Wang [Sch12, Sch18, TY15, BPW17] on the curvature computations of generalizedWeil-Petersson metric for families of canonically polarized manifolds, we define a convex sum ofFinsler metrics F : = ( n Õ k = α k F k ) / with α , . . . , α n ∈ R + (0.3.3)on T Y (− log D ) , to offset the unwanted positive terms in the curvature Θ ˜ E k by negative contribu-tions from the Θ ˜ E k + (the last order term was Θ ˜ E n is always semi-negative by the Griffiths curvatureformula). We proved in Proposition 2.14 that for proper α , . . . , α n >
0, the holomorphic sectionalcurvature of F is negative and bounded away from zero. To summarize, we establish an algorithm for the construction of Finsler metrics via VZ Higgs bundles.To prove Theorem B, we first note that the VZ Higgs bundles over some birational model ˜ V of thebase space V were constructed by Popa-Taji-Wu in their elaborate work [PTW19]. Let Y be somesmooth projective compactification ˜ V with simple normal crossing boundary D : = Y \ ˜ V . By ourconstruction of negatively curved Finsler metric F defined in (0.3.3) via VZ Higgs bundles, to showthat F is positively definite over some Zariski open set, it suffices to prove that τ : T Y (− log D ) → L − ⊗ E n − , defined in (0.3.2) is generically injective (which we call generic local Torelli for VZ Higgsbundles in § 1.1). This was proved in Theorem D, by using the degeneration of Hodge metric andthe curvature properties of Hodge bundles. In particular, we show that the generic injectivity of τ is indeed an intrinsic feature of all VZ Higgs bundles (not related to the Kodaira dimension of fibersof f !). By a standard inductive argument in [VZ03, PTW19], one can easily show that Theorem Bimplies Theorem A.Now we will explain the strategy to prove Theorem C. Note that the VZ Higgs bundles are onlyconstructed over some birational model ˜ V of V , which is not Kobayashi hyperbolic in general. Thismotivates us first to establish a bimeromorphic criteria for Kobayashi hyperbolicity in Lemma 2.5.Based on this criteria, in order to apply the VZ Higgs bundles to prove the Kobayashi hyperbolicityof the base V in Theorem C, it suffices to show that( ♠ ) for any given point y on the base V , there exists a VZ Higgs bundle ( ˜ E , ˜ θ ) constructed over somebirational model ν : ˜ V → V , such that ν − : V d ˜ V is defined at y . NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 5 ( ♣ ) The negatively curved Finsler metric F on ˜ V defined in (0.3.3) induced by the above VZ Higgsbundle ( ˜ E , ˜ θ ) is positively definite at the point ν − ( y ) .Roughly speaking, the idea is to produce an abundant supply of fine VZ Higgs bundles to constructsufficiently many negatively curved Finsler metrics, which are obstructions to the degeneracy ofKobayashi pseudo distance d V of V . This is much more demanding than the Brody hyperbolicityand Viewheg hyperbolicity of V , which can be shown by the existence of only one VZ Higgs bundleon an arbitrary birational model of V , as mentioned in [VZ02, VZ03, PS17, PTW19].Let us briefly explain how we achieve both ( ♠ ) and ( ♣ ).As far as we see in [VZ03, PTW19], in their construction of VZ Higgs bundles, one has to blow-up the base for several times (indeed twice). Recall that the basic setup in [VZ03, PTW19] is thefollowing: after passing to some smooth birational model f ˜ U : ˜ U = U × V ˜ V → ˜ V of f U : U → V , onecan find a smooth projective compactification f : X → Y of ˜ U r → ˜ VU r (cid:15) (cid:15) ˜ U r bir ∼ o o (cid:15) (cid:15) ⊆ / / X f (cid:15) (cid:15) V ˜ V ν bir ∼ o o ⊆ / / Y (0.3.4)so that there exists (at least) one hypersurface H ∈ (cid:12)(cid:12) ℓ K X / Y − ℓ f ∗ L (cid:12)(cid:12) for some ℓ ≫ transverse to the general fibers of f . Here L is some big and nef line bundle over Y , and U r : = U × V × · · · × V U (resp. ˜ U r ) is the r -fold fiber product of f U : U → V (resp. f ˜ U : ˜ U → ˜ V ).The VZ Higgs bundle is indeed the logarithmic Higgs bundles associated to the Hodge filtration ofan auxiliary variation of polarized Hodge structures constructed by taking the middle dimensionalrelative de Rham cohomlogy on the cyclic cover of X ramified along H .In order to find such H in (0.3.5), a crucial step in [VZ03, PTW19] is the use of weakly semi-stable reduction by Abramovich-Karu [AK00] so that, after changing the birational model U → V by performing certain (uncontrollable) base change ˜ U : = U × V ˜ V → ˜ V , one can find a “good"compactification X → Y of ˜ U r → ˜ V and a finite dominant morphism W → Y from a smoothprojective manifold W such that the base change X × Y W → W is birational to a mild morphism Z → W , which is in particular flat with reduced fibers (even fonctorial under fiber products). For our goal( ♠ ), we need a more refined control of the alteration for the base in the weakly semistable reduction[AK00, Theorem 0.3], which remains unknown at the moment. Fortunately, as was suggested to usand proved in Appendix A by Abramovich, using moduli of Alexeev stable maps one can establish a Q -mild reduction for the family U → V in place of the mild reduction in [VZ03], so that we can alsofind a “good" compactification X → Y of U r → V without passing the birational models ˜ V → V asin (0.3.4). This is the main theme of Appendix A.Even if we can apply Q -mild reduction to avoid the first blow-up of the base as in [VZ03,PTW19],the second blow-up is in general inevitable. Indeed, the discriminant of the new family Z H → Y ⊃ V obtained by taking the cyclic cover along H in (0.3.5) is in general not normal crossing. One thus hasto blow-up this discriminant locus of Z H → Y to make it normal crossing as in [PTW19]. Therefore,to assure ( ♠ ), it then suffices to show that there exists a compactification f : X → Y of the smoothfamily U r → V so that for some sufficiently ample line bundle A over Y ,( ∗ ) f ∗ ( mK X / Y ) ⊗ A − m is globally generated over V for some m ≫ y ∈ V , by ( ∗ ) one can find H transverse to the fiber X y : = f − ( y ) , andthus the new family Z H → Y will be smooth over an open set containing y . To the bests of ourknowledge, ( ∗ ) was only known to us when the moduli is canonically polarized [VZ02, Proposition3.4]. § 3.2 is devoted to the proof of ( ∗ ) for the family U → V in Theorem C (see Theorem 3.7.(iii)below). This in turn achieves ( ♠ ).To achieve ( ♣ ), our idea is to take different cyclic coverings by “moving” H in (0.3.5), to producedifferent “fine” VZ Higgs bundles. For any given point y ∈ V , by ( ♠ ), one can take a birational model YA DENG ν : ˜ V → V so that ν is isomorphic at y , and there exists a VZ Higgs bundle ( ˜ E , ˜ θ ) on the normalcrossing compactification Y ⊃ ˜ V . To prove that the induced negatively curved Finsler metric F ispositively definite at ˜ y : = ν − ( y ) , by our definition of F in (0.3.3), it suffices to show that τ definedin (0.3.2) is injective at ˜ y in the sense of C -linear map between complex vector spaces τ , ˜ y : T ˜ V , ˜ y ≃ −→ T Y (− log D ) ˜ y ρ ˜ y −−→ H ( X ˜ y , T X ˜ y ) φ ˜ y −−→ ˜ E , ˜ y . As we will see in § 3.4, when H in (0.3.5) is properly chosen (indeed being transverse to the fiber X y ) which is ensured by ( ∗ ), φ ˜ y is injective at ˜ y . Hence τ , ˜ y is injective by our assumption of effectiveparametrization (hence ρ ˜ y is injective) in Theorem C. This is our strategy to prove Theorem C. Acknowledgments.
This paper is the merger of my previous two articles [Den18a, Den18b] withslight improvements. It owes a lot to the celebrated work [VZ02, VZ03, PTW19], to which I expressmy gratitudes. I would like to sincerely thank Professors Dan Abramovich, Sébastien Boucksom,Håkan Samuelsson Kalm, Kalle Karu, Mihai Păun, Mihnea Popa, Georg Schumacher, Jörg Winkel-mann, Chenyang Xu, Xiaokui Yang, Kang Zuo, and Olivier Benoists, Junyan Cao, Chen Jiang, RuiranSun, Lei Wu, Jian Xiao for answering my questions and very fruitful discussions. I thank in partic-ular Junyan Cao and Lei Wu for their careful reading of the early draft of the paper and numeroussuggestions. I am particularly grateful to Professor Dan Abramovich for suggesting the Q -mildreduction, and writing Appendix A which provides a crucial step for the present paper. I thank Pro-fessors Damian Brotbek and Jean-Pierre Demailly for their encouragements and supports. Lastly,I thank the referee for his careful reading of the paper and his suggestions on rewriting the papercompletely. Notations and conventions.
Throughout this article we will work over the complex number field C . • An algebraic fiber space (or fibration for short) f : X → Y is a surjective projective morphismbetween projective manifolds with connected geometric fibers. Any Q -divisor E in X is said to be f -exceptional if f ( E ) is an algebraic variety of codimension at least two in Y . • We say that a morphism f U : U → V is a smooth family if f U is a surjective smooth projectivemorphism with connected fibers between quasi-projective varieties. • For any surjective morphism Y ′ → Y , and the algebraic fiber space f : X → Y , we denoteby ( X × Y Y ′ ) ~ the (unique) irreducible component (say the main component ) of X × Y Y ′ whichdominates Y ′ . • Let µ : X ′ → X be a birational morphism from a projective manifold X ′ to a singular variety X . µ is called a strong desingularization if µ − ( X reg ) → X reg is an isomorphism. Here X reg denotes tobe the smooth locus of X . • For any birational morphism µ : X ′ → X , the exceptional locus is the inverse image of the smallestclosed set of X outside of which µ is an isomorphism, and denoted by Ex ( µ ) . • Denote by X r : = X × Y · · · × Y X the r -fold fiber product of the fibration f : X → Y , ( X r ) ~ the maincomponent of X r dominating Y , and X ( r ) a strong desingularization of ( X r ) ~ . • For any quasi-projective manifold Y , a Zariski open subset Y ⊂ Y is called a big open set of Y ifand only if codim Y \ Y ( Y ) > • A singular hermitian metric h on the line bundle L is said to be positively curved if the curvaturecurrent Θ h ( L ) >
1. Brody hyperbolicity of the base
To begin with, let us introduce the definition of
Viehweg-Zuo Higgs bundles over quasi-projectivemanifolds in an abstract way following [VZ03,PTW19]. Then we prove a generic local Torelli for VZHiggs bundles. We will show that based on the previous work by Viehweg-Zuo and Popa-Taji-Wu,this generic local Torelli theorem suffices to prove Theorem A. Here we follow the definition in [Mor87].
NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 7
Abstract Viehweg-Zuo Higgs bundles.
The definition we present below follows from theformulation in [VZ02, VZ03] and [PTW19, Proposition 2.7].
Definition 1.1 (Abstract Viehweg-Zuo Higgs bundles) . Let V be a quasi-projective manifold, andlet Y ⊃ V be a projective compactification of V with the boundary D : = Y \ V simple normalcrossing. A Viehweg-Zuo Higgs bundle on V is a logarithmic Higgs bundle ( ˜ E , ˜ θ ) over Y consistingof the following data:(i) a divisor S on Y so that D + S is simple normal crossing,(ii) a big and nef line bundle L over Y with B + ( L ) ⊂ D ∪ S ,(iii) a Higgs bundle ( E , θ ) : = (cid:0) É nq = E n − q , q , É nq = θ n − q , q (cid:1) induced by the lower canonical extensionof a polarized VHS defined over Y \ ( D ∪ S ) ,(iv) a sub-Higgs sheaf ( F , η ) ⊂ ( ˜ E , ˜ θ ) ,which satisfy the following properties.(1) The Higgs bundle ( ˜ E , ˜ θ ) : = ( L − ⊗ E , ⊗ θ ) . In particular, ˜ θ : ˜ E → ˜ E ⊗ Ω Y (cid:0) log ( D + S ) (cid:1) , and˜ θ ∧ ˜ θ = ( F , η ) has log poles only on the boundary D , that is, η : F → F ⊗ Ω Y ( log D ) .(3) Write ˜ E k : = L − ⊗ E n − k , k , and denote by F k : = ˜ E k ∩ F . Then the first stage F of F is an effective line bundle . In other words, there exists a non-trivial morphism O Y → F .As shown in [VZ02], by iterating η for k -times, we obtain F k times z }| { η ◦ · · · ◦ η −−−−−−−−−→ F k ⊗ (cid:0) Ω Y ( log D ) (cid:1) ⊗ k . Since η ∧ η =
0, the above morphism factors through F k ⊗ Sym k Ω Y ( log D ) , and by (3) one thusobtains O Y → F → F k ⊗ Sym k Ω Y ( log D ) → L − ⊗ E n − k , k ⊗ Sym k Ω Y ( log D ) . Equivalently, we have a morphism τ k : Sym k T Y (− log D ) → L − ⊗ E n − k , k . (1.1.1)It was proven in [VZ02, Corollary 4.5] that τ is always non-trivial. We say that a VZ Higgs bundlesatisfies the generic local Torelli if τ : T Y (− log D ) → L − ⊗ E n − , in (1.1.1) is generically injective.As we will see in § 1.4, in Theorem D we prove that the generic local Torelli holds for any VZ Higgsbundles.1.2. A quick tour on Viehweg-Zuo’s construction.
For the smooth family U → V in The-orems A and B, it was shown in [VZ02] and [PTW19, Proposition 2.7] that there is a VZ Higgsbundle over some birational model ˜ V of V . Indeed, using the deep theory of mixed Hodge modules,Popa-Taji-Wu [PTW19] can even construct VZ Higgs bundles over the bases of maximal variationalsmooth families whose geometric generic fiber admits a good minimal model. Since we need tostudy the precise loci where τ is injective in the proof of Theorem C, in this subsection we recollectViehweg-Zuo’s construction on VZ Higgs bundles over the base space V (up to a birational modeland a projective compactification) in Theorem B. We refer the readers to see [VZ02] and [PTW19]for more details. In § 3.4, we show how to refine this construction to prove Theorem C. Let usmention that we do not clarify any originality for this subsection. Theorem 1.2.
Let U → V be the smooth family in Theorem B. Then after replacing V by a birationalmodel ˜ V , there is a smooth compactification Y ⊃ ˜ V and a VZ Higgs bundle over ˜ V .Proof. By [VZ03, PTW19], one can take a birational morphism ν : ˜ V → V and a smooth compacti-fication f : X → Y of U r × V ˜ V → ˜ V so that there exists a hypersurface H ∈ | ℓ Ω nX / Y ( log ∆ ) − ℓ f ∗ L + ℓ E | , n : = dim X − dim Y (1.2.1)with L a big and nef line bundle over Y satisfying that YA DENG (1) the complement D : = Y \ ˜ V is simple normal crossing.(2) The hypersurface H is smooth over some Zariski open set V ⊂ ˜ V with D + S : = Y \ V simplenormal crossing.(3) The divisor E is effective and f -exceptional divisor with f ( E ) ∩ V = ∅ .(4) The augmented base locus B + ( L ) ∩ V = ∅ .Here we denote by ∆ : = f − ( D ) so that ( X , ∆ ) → ( Y , D ) is a log morphism. Within this basic setup,let us first recall two Higgs bundles in the theorem following [VZ02, §4]. Leaving out a codimensiontwo subvariety of Y supported on D + S , we assume that • the morphism f is flat, and E in (1.2.1) disappears. • The divisor D + S is smooth. Moreover, both ∆ and Σ = f − S are relative normal crossing.Set L : = Ω nX / Y ( log ∆ ) . Let δ : W → X be a blow-up of X with centers in ∆ + Σ such that δ ∗ ( H + ∆ + Σ ) is a normal crossing divisor. One thus obtains a cyclic covering of δ ∗ H , by taking the ℓ -th root outof δ ∗ H . Let Z to be a strong desingularization of this covering, which is smooth over V by (2). Wedenote the compositions by h : W → Y and д : Z → Y , whose restrictions to V are both smooth.Write Π : = д − ( S ∪ D ) which can be assumed to be normal crossing. Leaving out codimension twosubvariety supported D + S further, we assume that h and д are also flat, and both δ ∗ ( H + ∆ + Σ ) and Π are relative normal crossing. Set F n − q , q : = R q h ∗ (cid:16) δ ∗ (cid:0) Ω n − qX / Y ( log ∆ ) (cid:1) ⊗ δ ∗ L − ⊗ O W (cid:0) ⌊ δ ∗ H ℓ ⌋ (cid:1) (cid:17) / torsion . It was shown in [VZ02, §4] that there exists a natural edge morphism τ n − q , q : F n − q , q → F n − q − , q + ⊗ Ω Y ( log D ) , (1.2.2)which gives rise to the first Higgs bundle (cid:0) É nq = F n − q , q , É nq = τ n − q , q (cid:1) defined over a big open set of Y containing V .Write Z : = Z \ Π . Then the local system R n д ∗ C ↾ Z extends to a locally free sheaf V on Y (here Y is projective rather than the big open set!) equipped with the logarithmic connection ∇ : V → V ⊗ Ω Y (cid:0) log ( D + S ) (cid:1) , whose eigenvalues of the residues lie in [ , ) (the so-called lower canonical extension ). By Schmid’s nilpotent orbit theorem [Sch73], the Hodge filtration of R n д ∗ C ↾ Z extends to a filtration V : = F ⊃F ⊃ · · · ⊃ F n of subbundles so that their graded sheaves E n − q , q : = F n − q /F n − q + are also locallyfree, and there exists θ n − q , q : E n − q , q → E n − q − , q + ⊗ Ω Y ( log D + S ) . This defines the second Higgs bundle (cid:0) É nq = E n − q , q , θ n − q , q (cid:1) . As observed in [VZ02, VZ03], E n − q , q = R q д ∗ Ω n − qZ / Y ( log Π ) over a big open set of Y by the theorem of Steenbrink [Ste77, Zuc84]. By the con-struction of the cyclic cover Z , this in turn implies the following commutative diagram over a bigopen set of Y : L − ⊗ E n − q , q ⊗ θ n − q , q / / L − ⊗ E n − q − , q + ⊗ Ω Y (cid:0) log ( D + S ) (cid:1) F n − q , qρ n − q , q O O τ n − q , q / / F n − q − , q + ⊗ Ω Y ( log D ) ρ n − q − , q + ⊗ ι O O (1.2.3)as shown in [VZ03, Lemma 6.2] (cf. also [VZ02, Lemma 4.4]).Note that all the objects are defined on a big open set of Y except for (cid:0) É nq = E n − q , q , θ n − q , q (cid:1) , whichare defined on the whole Y . Following [VZ03, §6], for every q = , . . . , n , we define F n − q , q to be thereflexive hull, and the morphisms τ n − q , q and ρ n − q , q extend naturally.To conclude that (cid:0) É nq = L − ⊗ E n − q , q , É nq = ⊗ θ n − q , q (cid:1) is a VZ Higgs bundle as in Definition 1.1,we have to introduce a sub-Higgs sheaf with log poles supported on D . Write ˜ θ n − q , q : = ⊗ θ n − q , q for short. Following [VZ02, Corollary 4.5] (cf. also [PTW19]), for each q = , . . . , n , we define a NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 9 coherent torsion-free sheaf F q : = ρ n − q , q ( F n − q , q ) ⊂ E n − q , q . By F n , ⊃ O Y , F ⊃ O Y . By (1.2.2) and(1.2.3), one has ˜ θ n − q , q : F q → F q + ⊗ Ω Y ( log D ) , and let us by η q the restriction of ˜ θ n − q , q to F q . Then ( F , η ) : = (cid:0) É nq = F q , É nq = η q (cid:1) is a sub-Higgsbundle of ( ˜ E , ˜ θ ) : = (cid:0) É nq = L − ⊗ E n − q , q , É nq = ˜ θ n − q , q (cid:1) . (cid:3) Proper metrics for logarithmic Higgs bundles.
We adopt the same notations as Defini-tion 1.1 in the rest of § 1. As is well-known, E can be endowed with the Hodge metric h induced bythe polarization, which may blow-up around the simple normal crossing boundary D + S . However,according to the work of Schmid, Cattani-Schmid-Kaplan and Kashiwara [Sch73, CKS86, Kas85], h has mild singularities (at most logarithmic singularities), and as proved in [VZ03, §7] (for unipotentmonodromies) and [PTW19, §3] (for quasi-unipotent monodromies), one can take a proper singu-lar metric д α on L such that the induced singular hermitian metric д − α ⊗ h on ˜ E : = L − ⊗ E islocally bounded from above. Before we summarize the above-mentioned results in [PTW19, §3], weintroduce some notations in loc. cit. Write the simple normal crossing divisor D = D + · · · + D k and S = S + · · · + S ℓ . Let f D i ∈ H (cid:0) Y , O Y ( D i ) (cid:1) and f S i ∈ H (cid:0) Y , O Y ( S i ) (cid:1) be the canonical section defining D i and S i . We fix smoothhermitian metrics д D i and д S i on O Y ( D i ) and O Y ( S i ) . Set r D i : = − log k f D i k д Di , r S i : = − log k f S i k д Si , and define r D : = k Ö i = r D i , r S : = ℓ Ö i = r S i . Let д be a singular hermitian metric with analytic singularities of the big and nef line bundle L such that д is smooth on Y \ B + ( L ) ⊃ Y \ D ∪ S , and the curvature current √− Θ д ( L ) > ω forsome smooth Kähler form ω on Y . For α ∈ N , define д α : = д · ( r D · r S ) α The following proposition is a slight variant of [PTW19, Lemma 3.1, Corollary 3.4].
Proposition 1.3 ( [PTW19]) . When α ≫ , after rescaling f D i and f S i , there exists a continuous,positively definite hermitian form ω α on T Y (− log D ) such that (i) over V : = Y \ D ∪ S , the curvature form √− Θ д α ( L ) ↾ V > r − D · ω α ↾ V . (ii) The singular hermitian metric h αд : = д − α ⊗ h on L − ⊗ E is locally bounded on Y , and smoothoutside ( D + S ) . Moreover, h αд is degenerate on D + S . (iii) The singular hermitian metric r D h αд on L − ⊗ E is also locally bounded on Y . (cid:3) Remark . It follows from Proposition 1.3 that both h αд and r D h αд can be seen as Finsler metrics on L − ⊗ E which are degenerate on Supp ( D + S ) , and positively definite on V .Although the last statement of Proposition 1.3.(ii) is not explicitly stated in [PTW19], it can beeasily seen from the proof of [PTW19, Corollary 3.4]. Proposition 1.3 mainly relies on the asymptoticbehavior of the Hodge metric for lower canonical extension of a variation of Hodge structure (cf.Theorem 1.5 below) when the monodromy around the boundaries are only quasi-unipotent. Theorem 1.5 ( [PTW19, Lemma 3.2]) . Let H = F ⊃ F ⊃ · · · ⊃ F N ⊃ be a variation of Hodgestructures defined over ( ∆ ∗ ) p × ∆ q , where ∆ (resp. ∆ ∗ ) is the (resp. punctured) unit disk. Considerthe lower canonical extension l F • over ∆ p + q ⊃ ( ∆ ∗ ) p × ∆ q , and denote by ( E , θ ) the associated Higgsbundle. Then for any holomorphic section s ∈ Γ ( U , E ) , where U ( ∆ p + q is a relatively compact openset containing the origin, one has the following norm estimate | s | hod C (cid:0) (− log | t |) · (− log | t |) · · · (− log | t p |) (cid:1) α , (1.3.1) where α is some positive constant independent of s , and t = ( t , . . . , t p + q ) denotes to be the coordinatesof ∆ p + q . Let us mention that the estimates of Hodge metric for upper canonical extension were obtained byPeters [Pet84] in one variable, and by Catanese-Kawamata [CK17] in several variables, based on thework [Sch73, CKS86]. We provide a slightly different proof of Theorem 1.5 for completeness sake,following closely the approaches in [Pet84, CK17].
Proof of Theorem 1.5.
The fundamental group π (cid:0) ( ∆ ∗ ) p × ∆ q (cid:1) is generated by elements γ , . . . , γ p ,where γ j may be identified with the counter-clockwise generator of the fundamental group of the j -th copy of ∆ ∗ in ( ∆ ∗ ) p . Set T j to be the monodromy transformation with respect to γ j , which pairwisecommute and are known to be quasi-unipotent; that is, for any multivalued section v ( t , . . . , t p + q ) of H , one has v ( t , . . . , e πi t j , . . . , t p + q ) = T j · v ( t , . . . , t p + q ) and [ T j , T k ] = j , k = , . . . , p . Set T j = D j · U j to be the (unique) Jordan-Chevally decomposi-tion, so that D j diagonalizable and U j is unipotent with [ D j , U j ] =
0. Since T j is quasi-unipotent by thetheorem of Borel, all the eigenvalues of D j are thus the roots of unity. Set N j : = πi Í k > ( I − U j ) k / k . If D j = diag . ( d j ℓ ) then we set S j = diag . ( λ j ℓ ) with λ j ℓ ∈ (− πi , ] and exp ( λ j ℓ ) = d j ℓ . Since [ T j , T k ] = [ S j , S k ] = [ S j , N k ] = [ N j , N k ] = . (1.3.2)Fix a point t ∈ ( ∆ ∗ ) p × ∆ q , and take a basis v , . . . , v r ∈ V t so that S , . . . , S p are simultaneouslydiagonal, that is, one has S j ( v ℓ ) = λ j ℓ . (1.3.3)Let us define v ( t ) , . . . , v r ( t ) to be the induced multivalued flat sections. Then e j ( t ) : = exp (cid:0) − πi p Õ i = ( S i + N i ) · log t i (cid:1) v j ( t ) is single-valued and forms a basis of holomorphic sections for the lower canonical extension l H .Recall that d j ℓ are all roots of unity. One thus can take a positive integer m so that m j ℓ : = − mλ j ℓ / πi are all non-negative integers . Equivalently, each T mj is unipotent. Define a ramified cover π : ∆ p + q → ∆ p + q ( w , . . . , w p + q ) 7→ ( w m , . . . , w mp , w p + , . . . , w p + q ) and set π ′ to be the restriction of π to ( ∆ ∗ ) p × ∆ q . Then π ′∗ F • is a variation of Hodge structure on ( ∆ ∗ ) p × ∆ q with unipotent monodromy, and we define c π ′∗ H the canonical extension of π ′∗ H . Set u j ( w ) = π ′∗ v j which are multivalued sections for the local system π ′∗ H . Then u j ( w , . . . , e πi w j , . . . , w p + q ) = T mj · u j ( w , . . . , w p + q ) . Define ˜ e j ( w ) : = exp (cid:0) − πi p Õ i = mN i · log w i (cid:1) u j ( w ) (1.3.4)which forms a basis of c π ′∗ H . Based on the work of [Sch73, CKS86], it was shown in [VZ03, Claim7.8] that one has the upper bound of norms | ˜ e j ( w )| hod C (cid:0) (− log | w |) · (− log | w |) · · · (− log | w p |) (cid:1) α (1.3.5) NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 11 for some positive constants C and α . One the other hand, we have π ′∗ e j ( w ) = exp (cid:0) − πi p Õ i = ( S i + N i ) · log w mi (cid:1) π ′∗ v j ( w ) (1.3.2) = exp (cid:0) − πi p Õ i = mN i · log w i (cid:1) · exp (cid:0) − πi p Õ i = mS i log w i (cid:1) π ′∗ v j ( w ) (1.3.3) = exp (cid:0) − πi p Õ i = mN i · log w i (cid:1) · exp (cid:0) − πi p Õ i = mλ ij log w i (cid:1) π ′∗ v j ( w ) = p Ö i = w m ij i · exp (cid:0) − πi p Õ i = mN i · log w i (cid:1) · u j ( w ) (1.3.4) = p Ö i = w m ij i · ˜ e j ( w ) . By the definition of lower canonical extension, m ij are all non-negative integers, and thus π ′∗ | e j | hod ( w ) = | π ′∗ e j ( w )| hod = p Ö i = | w i | m ij | ˜ e j ( w )| hod (1.3.5) C (cid:0) (− log | w |) · (− log | w |) · · · (− log | w p |) (cid:1) α . Hence | e j | hod ( t ) C m p (cid:0) (− log | t |) · (− log | t |) · · · (− log | t p |) (cid:1) α . Note that l H C ∞ ≃ E . Therefore, for any holomorphic section s ∈ Γ ( U , E ) , there exist smooth func-tions f , . . . , f r ∈ O ( U ) so that s = Í rj = f j e j . This shows the estimate (1.3.1). (cid:3) Remark . For the Hodge metric of upper canonical extension, one makes the choice that λ j ℓ ∈[ , πi ) instead of λ j ℓ ∈ (− πi , ] in the proof of Theorem 1.5. Then the same computation as abovecan easily show that | e j | hod ( t ) p Ö i = | t i | − λij πi Cm p (cid:0) (− log | t |) · (− log | t |) · · · (− log | t p |) (cid:1) α , which were obtained in [CK17].1.4. A generic local Torelli for VZ Higgs bundle.
In this section we prove that the generic localTorelli holds for any VZ Higgs bundle, which is a crucial step in the proofs of Theorems A and B.
Theorem D (Generic local Torelli) . For the abstract Viehweg-Zuo Higgs bundles defined in Defini-tion 1.1, the morphism τ : T Y (− log D ) → L − ⊗ E n − , defined in (1.1.1) is generically injective.Proof of Theorem D. By Definition 1.1, the non-zero morphism O Y → F → L − ⊗ E n , induces aglobal section s ∈ H ( Y , L − ⊗ E n , ) , which is generically non-vanishing over V : = Y \ D ∪ S . Set V : = { y ∈ V | s ( y ) , } (1.4.1)which is a non-empty Zariski open set of V . For the first stage of VZ Higgs bundle L − ⊗ E n , ,we equip it with a singular metric h αд : = д − α ⊗ h as in Proposition 1.3, so that Propositions 1.3.(i)and 1.3.(ii) are satisfied. Note that h αд is smooth over V . Let us denote D ′ to be the ( , ) -part of itsChern connection over V , and Θ to be its curvature form. Then by the Griffiths curvature formulaof Hodge bundles (see [GT84]), over V we have Θ = − Θ L , д α ⊗ + ⊗ Θ h ( E n , ) = − Θ L , д α ⊗ − ⊗ ( θ ∗ n , ∧ θ n , ) = − Θ L , д α ⊗ − ˜ θ ∗ n , ∧ ˜ θ n , , (1.4.2) where we set ˜ θ n − k , k : = ⊗ θ n − k , k : L − ⊗ E n − k , k → L − ⊗ E n − k − , k + ⊗ Ω Y (cid:0) log ( D + S ) (cid:1) , and define˜ θ ∗ n , to be the adjoint of ˜ θ n , with respect to the metric h αд . Hence over V one has −√− ∂ ¯ ∂ log | s | h αд = (cid:8) √− Θ ( s ) , s (cid:9) h αд | s | h αд + √− { D ′ s , s } h αд ∧ { s , D ′ s } h αд | s | h αд − √− { D ′ s , D ′ s } h αд | s | h αд (cid:8) √− Θ ( s ) , s (cid:9) h αд | s | h αд (1.4.3)thanks to the Lagrange’s inequality √− | s | h αд · { D ′ s , D ′ s } h αд > √− { D ′ s , s } h αд ∧ { s , D ′ s } h αд . Putting (1.4.2) to (1.4.3), over V one has √− Θ L , д α − √− ∂ ¯ ∂ log | s | h αд − (cid:8) √− θ ∗ n , ∧ ˜ θ n , ( s ) , s (cid:9) h αд | s | h αд = √− (cid:8) ˜ θ n , ( s ) , ˜ θ n , ( s ) (cid:9) h αд | s | h αд (1.4.4)where ˜ θ n , ( s ) ∈ H (cid:16) Y , L − ⊗ E n − , ⊗ Ω Y (cid:0) log ( D + S ) (cid:1) (cid:17) . By Proposition 1.3.(ii), for any y ∈ D ∪ S ,one has lim y ′ ∈ V , y ′ → y | s | h αд ( y ′ ) = . Therefore, it follows from the compactness of Y that there exists y ∈ V so that | s | h αд ( y ) > | s | h αд ( y ) for any y ∈ V . Hence | s | h αд ( y ) >
0, and by (1.4.1), y ∈ V . Since | s | h αд is smooth over V , √− ∂ ¯ ∂ log | s | h αд ( y ) is semi-negative. By Proposition 1.3.(i), √− Θ L , д α is strictly positive at y . By(1.4.4) and | s | h ( y ) >
0, we conclude that √− (cid:8) ˜ θ n , ( s ) , ˜ θ n , ( s ) (cid:9) h αд is strictly positive at y . In particular,for any non-zero ξ ∈ T Y , y , ˜ θ n , ( s )( ξ ) ,
0. For τ : T Y (− log D ) → L − ⊗ E n − , in (1.1.1), over V it is defined by τ ( ξ ) : = ˜ θ n , ( s )( ξ ) , which is thus injective at y ∈ V . Hence τ is generically injective . The theorem is thus proved. (cid:3) Remark . Viehweg-Zuo [VZ02] showed that τ : T Y (− log D ) → L − ⊗ E n − , defined in (1.1.1)does not vanishing on V using a global argument relying on the Griffiths curvature computation forHodge metric and the bigness of direct image sheaves due to Kawamata and Viehweg. Moreover,by the work of Viehweg-Zuo [VZ03] and Popa-Taji-Wu [PTW19], it has already been known to usthat, when fibers in Theorem A have big and semi-ample canonical bundle, the VZ Higgs bundlesconstructed in Theorem 1.2 over the base always satisfy Theorem D.Though Theorem A follows from our more general result in Theorem B, we are able to proveTheorem A by directly applying the results by Viehweg-Zuo [VZ03] and Popa-Taji-Wu [PTW19].Since we need some efforts to prove Theorem B, let us quickly show how to combine their workwith Theorem D to prove Theorem A. Proof of Theorem A.
By the stratified arguments of Viehweg-Zuo [VZ03], it suffices to prove thatthere cannot exists a Zariski dense entire curve. Assume by contradiction that there exists such γ : C → V . The existence of VZ Higgs bundle on some birational model ˜ V of V is known to us byTheorem 1.2. Let ˜ γ : C → ˜ V is the lift of γ which is also Zariski dense. In [VZ03,PTW19], the authorsproved that the restriction of τ defined in (1.1.1) on C , say τ | C : T C → ˜ γ ∗ ( L − ⊗ E n − , ) , has tovanish identically, or else, they can construct a pseudo hermitian metric on C with strictly negativeGaussian curvature, which violates the Ahlfors-Schwarz lemma. By Theorem D, this cannot happensince ˜ γ : C → ˜ V is Zariski dense. The theorem is proved. (cid:3) NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 13
2. Pseudo Kobayashi hyperbolicity of the base
In this section we first establish an algorithm to construct Finsler metrics whose holomorphicsectional curvatures are bounded above by a negative constant via VZ Higgs bundles. By our con-struction and generic local Torelli Theorem D, those Finsler metrics are positively definite over aZariski open set, and by the Ahlfors-Schwarz lemma, we prove that a quasi-projective manifold ispseudo Kobayashi hyperbolic once it is equipped with a VZ Higgs bundle, and thus prove Theorem B.2.1.
Finsler metric and (pseudo) Kobayashi hyperbolicity.
Throughout this subsection X willdenote to be a complex manifold of dimension n . Definition 2.1 (Finsler metric) . Let E be a holomorphic vector bundle on X . A Finsler metric on E is a real non-negative continuous function F : E →[ , + ∞[ such that F ( av ) = | a | F ( v ) for any a ∈ C and v ∈ E . The Finsler metric F is positively definite at some subset S ⊂ X if for any x ∈ S and any non-zero vector v ∈ E x , F ( v ) > F is a Finsler metric on T X , we also say that F is a Finsler metric on X .Let E and G be two locally free sheaves on X , and suppose that there is a morphism φ : Sym m E → G Then for any Finsler metric F on G , φ induces a pseudo metric ( φ ∗ F ) m on E defined by ( φ ∗ F ) m ( e ) : = F (cid:0) φ ( e ⊗ m ) (cid:1) m (2.1.1)for any e ∈ E . It is easy to verify that ( φ ∗ F ) m is also a Finsler metric on E . Moreover, if over someopen set U , φ is an injection as a morphism between vector bundles, and F is positively definite over U , then ( φ ∗ F ) m is also positively definite over U . Definition 2.2. (i) The
Kobayashi-Royden infinitesimal pseudo-metric of X is a length function κ X : T X → [ , + ∞[ , defined by κ X ( ξ ) = inf γ (cid:8) λ > | ∃ γ : D → X , γ ( ) = x , λ · γ ′ ( ) = ξ (cid:9) (2.1.2)for any x ∈ X and ξ ∈ T X , where D denotes the unit disk in C .(ii) The Kobayashi pseudo distance of X , denoted by d X : X × X → [ , + ∞[ , is d X ( p , q ) = inf ℓ ∫ κ X (cid:0) ℓ ′ ( τ ) (cid:1) dτ for every pair of points p , q ∈ X , where the infimum is taken over all differentiable curves ℓ : [ , ] → X joining p to q .(iii) Let ∆ ( X be a closed subset. A complex manifold X is Kobayashi hyperbolic modulo ∆ if d X ( p , q ) > p , q ∈ X not both contained in ∆ . When ∆ is anempty set, the manifold X is Kobayashi hyperbolic ; when ∆ is proper and Zariski closed, themanifold X is pseudo Kobayashi hyperbolic .By definition it is easy to show that if X is Kobayashi hyperbolic (resp. pseudo Kobayashi hyper-bolic), then X is Brody hyperbolic (resp. algebraically degenerate). Brody’s theorem says that when X is compact, X is Kobayashi hyperbolic if it is Brody hyperbolic. However unlike the case of Kobay-ashi hyperbolicity, no criteria is known for pseudo Kobayashi hyperbolicity of a compact complexspace in terms of entire curves. Moreover, there are many examples of complex (quasi-projective)manifolds which are Brody hyperbolic but not Kobayashi hyperbolic.For any holomorphic map γ : D → X , the Finsler metric F induces a continuous Hermitianpseudo-metric on D γ ∗ F = √− λ ( t ) dt ∧ d ¯ t , This definition is a bit different from the definition in [Kob98], which requires convexity or triangle inequality , andthe Finsler metric there can be upper-semi continuous. where λ ( t ) is a non-negative continuous function on D . The Gaussian curvature K γ ∗ F of the pseudo-metric γ ∗ F is defined to be K γ ∗ F : = − λ ∂ log λ ∂ t ∂ ¯ t . (2.1.3) Definition 2.3.
Let X be a complex manifold endowed with a Finsler metric F .(i) For any x ∈ X , and v ∈ T X , x , let [ v ] denote the complex line spanned by v . We define theholomorphic sectional curvature K F , [ v ] in the direction of [ v ] by K F , [ v ] : = sup K γ ∗ F ( ) where the supremum is taken over all γ : D → X such that γ ( ) = x and [ v ] is tangent to γ ′ ( ) .(ii) We say that F is negatively curved if there is a positive constant c such that K F , [ v ] − c for all v ∈ T X , x for which F ( v ) > x ∈ X is a degeneracy point of F if F ( v ) = v ∈ T X , x , and the set ofsuch points is denoted by ∆ F .As mentioned in § 0, our negatively curved Finsler metrics are only constructed on birationalmodels of the base spaces in Theorems B and C, we thus have to establish bimeromorphic criteriafor (pseudo) Kobayashi hyperbolicity to prove the main theorems. Lemma 2.4 (Bimeromorphic criteria for pseudo Kobayashi hyperbolicity) . Let µ : X → Y be abimeromorphic morphism between complex manifolds. If there exists a Finsler metric F on X which isnegatively curved in the sense of Definition 2.3.(ii), then X is Kobayashi hyperbolic modulo ∆ F , and Y is Kobayashi hyperbolic modulo µ (cid:0) Ex ( µ ) ∪ ∆ F (cid:1) , where Ex ( µ ) is the exceptional locus of µ . In particular,when ∆ F is a proper analytic subvariety of X , both X and Y are pseudo Kobayashi hyperbolic.Proof. The first statement is a slight variant of [Kob98, Theorem 3.7.4]. By normalizing F we mayassume that K F −
1. By the Ahlfors-Schwarz lemma, one has F κ X . Let δ F : X × X → [ , + ∞[ be the distance function on X defined by F in a similar way as d X : δ F ( p , q ) : = inf ℓ ∫ F (cid:0) ℓ ′ ( τ ) (cid:1) dτ for every pair of points p , q ∈ X , where the infimum is taken over all differentiable curves ℓ : [ , ] → X joining p to q . Since F is continuous and positively definite over X \ ∆ F , for any p ∈ X \ ∆ F , onehas d X ( p , q ) > δ F ( p , q ) > q , p , which proves the first statement.Let us denote by Hol ( Y , y ) to be the set of holomorphic maps γ : D → Y with γ ( ) = y . Pick anypoint y ∈ U : = Y \ µ (cid:0) Ex ( µ ) (cid:1) , then there is a unique point x ∈ X with µ ( x ) = y . Hence µ induces abijection between the sets Hol ( X , x ) ≃ → Hol ( Y , y ) defined by ˜ γ µ ◦ ˜ γ . Indeed, observe that µ − : Y d X is a meromorphic map, so is µ − ◦ γ for any γ ∈ Hol ( Y , y ) . Since dim D =
1, the map µ − ◦ γ is moreover holomorphic. It follows from (2.1.2)that κ X ( ξ ) = κ Y (cid:0) µ ∗ ( ξ ) (cid:1) for any ξ ∈ T X , x . Hence one has µ ∗ κ Y | µ − ( U ) = κ X | µ − ( U ) > F | µ − ( U ) . Let G : T U → [ , + ∞[ be the Finsler metric on U so that µ ∗ G = F | µ − ( U ) . Then G is continuous andpositively definite over U \ µ ( ∆ F ) , and one has κ Y | U > G . Therefore, for any y ∈ Y \ µ (cid:0) ∆ F ∪ Ex ( µ ) (cid:1) , one has d Y ( y , z ) > z , y , which proves the secondstatement. (cid:3) NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 15
The above criteria can be refined further to show the Kobayashi hyperbolicity of the complexmanifold.
Lemma 2.5 (Bimeromorphic criteria for Kobayashi hyperbolicity) . Let X be a complex manifold.Assume that for each point p ∈ X , there is a bimeromorphic morphism µ : ˜ X → X with ˜ X equippedwith a negatively curved Finsler metric F such that p < µ (cid:0) ∆ F ∪ Ex ( µ ) (cid:1) . Then X is Kobayashi hyperbolic.Proof. It suffices to show that d X ( p , q ) > p , q ∈ X . We takethe bimeromorphic morphism µ : ˜ X → X in the lemma with respect to p . By Lemma 2.4, X isKobayashi hyperbolic modulo µ (cid:0) ∆ F ∪ Ex ( µ ) (cid:1) , which shows that d X ( p , q ) > q , p . Thelemma follows. (cid:3) Curvature formula.
Let ( ˜ E , ˜ θ ) be the VZ Higgs bundles on a quasi-projective manifold V defined in § 1.1. In the next two subsections, we will construct a negatively curved Finsler metricon V via ( ˜ E , ˜ θ ) . Our main result is the following. Theorem 2.6 (Existence of negatively curved Finsler metrics) . Same notations as Definition 1.1.Assume that τ is injective over a non-empty Zariski open set V ⊆ Y \ D ∪ S . Then there exists a Finslermetric F (see (2.3.6) below) on T Y (− log D ) such that (i) it is positively definite over V . (ii) When F is seen as a Finsler metric on V = Y \ D , it is negatively curved in the sense of Defini-tion 2.3.(ii). Let us first construct the desired Finsler metric F , and we then proved the curvature property. By(1.1.1), for each k = , . . . , n , there exists τ k : Sym k T Y (− log D ) → L − ⊗ E n − k , k . (2.2.1)Then it follows from Proposition 1.3.(ii) that the Finsler metric h αд on L − ⊗ E n − k , k induces a Finslermetric F k on T Y (− log D ) defined as follows: for any e ∈ T Y (− log D ) y , F k ( e ) : = ( τ ∗ k h αд ) k ( e ) = h αд (cid:0) τ k ( e ⊗ k ) (cid:1) k (2.2.2)For any γ : D → V , one has dγ : T D → γ ∗ T V ֒ → γ ∗ T Y (− log D ) and thus the Finsler metric F k induces a continuous Hermitian pseudo-metric on D , denoted by γ ∗ F k : = √− G k ( t ) dt ∧ d ¯ t . (2.2.3)In general, G k ( t ) may be identically equal to zero for all k . However, if we further assume that γ ( D ) ∩ V , ∅ , by the assumption in Theorem 2.6 that the restriction of τ to V is injective, onehas G ( t ) .
0. Denote by ∂ t : = ∂∂ t the canonical vector fields in D , and ¯ ∂ t : = ∂∂ ¯ t its conjugate. Set C : = γ − ( V ) , and note that D \ C is a discrete set in D . Lemma 2.7.
Assume that G k ( t ) . for some k > . Then the Gaussian curvature K k of the continuouspseudo-hermitian metric γ ∗ F k on C satisfies that K k : = − ∂ log G k ∂ t ∂ ¯ t / G k k (cid:16) − (cid:0) G k G k − (cid:1) k − + (cid:0) G k + G k (cid:1) k + (cid:17) (2.2.4) over C ⊂ D .Proof. For i = , . . . , n , let us write e i : = τ i (cid:0) dγ ( ∂ t ) ⊗ i (cid:1) , which can be seen as a section of γ ∗ ( L − ⊗ E n − i , i ) . Then by (2.2.2) one observes that G i ( t ) = k e i k / ih αд . (2.2.5) Let R k = Θ h αд ( L − ⊗ E n − k , k ) be the curvature form of L − ⊗ E n − k , k on V : = Y \ D ∪ S inducedby the metric h αд = д − α · h defined in Proposition 1.3.(ii), and let D ′ be the ( , ) -part of the Chernconnection D of ( L − ⊗ E n − k , k , h αд ) . Then for k = , . . . , n , one has −√− ∂ ¯ ∂ log G k = k (cid:16) (cid:8) √− R k ( e k ) , e k (cid:9) h αд k e k k h αд + √− { D ′ e k , e k } h αд ∧ { e k , D ′ e k } h αд k e k k h αд − √− { D ′ e k , D ′ e k } h αд k e k k h αд (cid:17) k (cid:8) √− R k ( e k ) , e k (cid:9) h αд k e k k h αд thanks to the Lagrange’s inequality √− k e k k h αд · { D ′ e k , D ′ e k } h αд > √− { D ′ e k , e k } h αд ∧ { e k , D ′ e k } h αд . Hence − ∂ log G k ∂ t ∂ ¯ t k · (cid:10) R k ( e k )( ∂ t , ¯ ∂ t ) , e k (cid:11) h αд k e k k h αд . (2.2.6)Recall that for the logarithmic Higgs bundle ( É nk = E n − k , k , É nk = θ n − k , k ) , the curvature Θ k on E n − k , k ↾ V induced by the Hodge metric h is given by Θ k = − θ ∗ n − k , k ∧ θ n − k , k − θ n − k + , k − ∧ θ ∗ n − k + , k − , where we recall that θ n − k , k : E n − k , k → E n − k − , k + ⊗ Ω Y (cid:0) log ( D + S ) (cid:1) . Set ˜ θ n − k , k : = ⊗ θ n − k , k : L − ⊗ E n − k , k → L − ⊗ E n − k − , k + ⊗ Ω Y (cid:0) log ( D + S ) (cid:1) , and one has L − ⊗ E n − k + , k − θ n − k + , k − ( ∂ t ) , , L − ⊗ E n − k , k ˜ θ ∗ n − k + , k − ( ¯ ∂ t ) l l ˜ θ n − k , k ( ∂ t ) - - L − ⊗ E n − k − , k + θ ∗ n − k , k ( ¯ ∂ t ) l l where ˜ θ ∗ n − k , k is the adjoint of ˜ θ n − k , k with respect to the metric h αд over Y \ D ∪ S . Here we also write ∂ t (resp. ¯ ∂ t ) for dγ ( ∂ t ) (resp. dγ ( ¯ ∂ t ) ) abusively. Then over V , we have R k = − Θ L , д α ⊗ + ⊗ Θ k = − Θ L , д α ⊗ − ˜ θ ∗ n − k , k ∧ ˜ θ n − k , k − ˜ θ n − k + , k − ∧ ˜ θ ∗ n − k + , k − . (2.2.7)By the definition of τ k in (1.1.1), for any k = , . . . , n one has e k = ˜ θ n − k + , k − ( ∂ t )( e k − ) , (2.2.8) NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 17 and we can derive the following curvature formula h R k ( e k )( ∂ t , ¯ ∂ t ) , e k (cid:11) h αд = − Θ L , д α ( ∂ t , ¯ ∂ t )k e k k h αд + (cid:10) ˜ θ ∗ n − k , k ( ¯ ∂ t ) ◦ ˜ θ n − k , k ( ∂ t )( e k ) − ˜ θ n − k + , k − ( ∂ t ) ◦ ˜ θ ∗ n − k + , k − ( ¯ ∂ t )( e k ) , e k (cid:11) h αд (cid:10) ˜ θ ∗ n − k , k ( ¯ ∂ t ) ◦ ˜ θ n − k , k ( ∂ t )( e k ) , e k (cid:11) h αд − (cid:10) ˜ θ n − k + , k − ( ∂ t ) ◦ ˜ θ ∗ n − k + , k − ( ¯ ∂ t )( e k ) , e k (cid:11) h αд (2.2.8) = k e k + k h αд − k ˜ θ ∗ n − k + , k − ( ¯ ∂ t )( e k )k h αд k e k + k h αд − | (cid:10) ˜ θ ∗ n − k + , k − ( ¯ ∂ t )( e k ) , e k − (cid:11) h αд | k e k − k h αд ( Cauchy-Schwarz inequality ) = k e k + k h αд − | (cid:10) e k , ˜ θ n − k + , k − ( ∂ t )( e k − ) (cid:11) h αд | k e k − k h αд (2.2.8) = k e k + k h αд − k e k k h αд k e k − k h αд (2.2.5) = G k + k + − G kk G k − k − Putting this into (2.2.6), we obtain (2.2.4). (cid:3)
Remark . For the final stage E , n of the Higgs bundle ( É nq = E n − q , q , É nq = θ n − q , q ) . We make theconvention that G n + ≡
0. Then the Gaussian curvature for G n in (2.2.6) is always semi-negative,which is similar as the Griffiths curvature formula for Hodge bundles in [GT84].When k =
1, by (2.2.6) one has − ∂ log G ∂ t ∂ ¯ t / G (cid:10) R ( e )( ∂ t , ¯ ∂ t ) , e (cid:11) h αд k e k h αд (2.2.7) = − Θ L , д α ( ∂ t , ¯ ∂ t )k e k h αд + (cid:10) ˜ θ ∗ n − , ( ¯ ∂ t ) ◦ ˜ θ n − , ( ∂ t )( e ) − ˜ θ n , ( ∂ t ) ◦ ˜ θ ∗ n , ( ¯ ∂ t )( e ) , e (cid:11) h αд k e k h αд (2.2.8) − Θ L , д α ( ∂ t , ¯ ∂ t )k e k h αд + k e k h αд k e k h αд = − Θ L , д α ( ∂ t , ¯ ∂ t )k e k h αд + (cid:0) G G (cid:1) We need the following lemma to control the negative term in the above inequality.
Lemma 2.9.
When α ≫ , there exists a universal constant c > , such that for any γ : D → V with γ ( D ) ∩ V , ∅ , one has Θ L , д α ( ∂ t , ¯ ∂ t )k e k h αд > c . In particular, − ∂ log G ∂ t ∂ ¯ t / G − c + (cid:0) G G (cid:1) Proof.
By Proposition 1.3.(ii), it suffices to prove that γ ∗ (cid:0) r − D · ω α (cid:1) ( ∂ t , ¯ ∂ t )k e k h αд > c . (2.2.9)Note that γ ∗ (cid:0) r − D · ω α (cid:1) ( ∂ t , ¯ ∂ t )k e k h αд = γ ∗ (cid:0) ω α (cid:1) ( ∂ t , ¯ ∂ t ) γ ∗ ( r D ) · k e k h αд = γ ∗ ω α ( ∂ t , ¯ ∂ t ) γ ∗ τ ∗ ( r D · h αд )( ∂ t , ¯ ∂ t ) , where τ ∗ ( r D · h αд ) is the Finsler metric on T Y (− log D ) defined by (2.1.1). By Proposition 1.3.(iii), ω α is a positively definite Hermitian metric on T Y (− log D ) . Since Y is compact, there exists a uniformconstant c > ω α > cτ ∗ ( r D · h αд ) . We thus obtained the desired inequality (2.2.9). (cid:3)
In summary, we have the following curvature estimate for the Finsler metrics F , . . . , F n definedin (2.2.2), which is similar as [Sch18, Lemma 9] for the Weil-Petersson metric. Proposition 2.10.
For any γ : D → V such that γ ( D ) ∩ V , ∅ . Assume that G k . for k = , . . . , q ,and G q + ≡ (thus G j ≡ for all j > q + ). Then q > , and over C : = γ − ( V ) , which is a complementof a discrete set in D , one has − ∂ log G ∂ t ∂ ¯ t / G − c + (cid:0) G G (cid:1) (2.2.10) − ∂ log G k ∂ t ∂ ¯ t / G k k (cid:16) − (cid:0) G k G k − (cid:1) k − + (cid:0) G k + G k (cid:1) k + (cid:17) ∀ < k q . (2.2.11) Here the constant c > does not depend on the choice of γ . Construction of the Finsler metric.
By Proposition 2.10, we observe that none of the Finslermetrics F , . . . , F n defined in (2.2.2) is negatively curved. Following the similar strategies in [TY15,Sch18, BPW17], we construct a new Finsler metric F (see (2.3.6) below) by defining a convex sum ofall F , . . . , F n , to cancel the positive terms in (2.2.10) and (2.2.11) by negative terms in the next stage.By Remark 2.8, we observe that the highest last order term is always semi-negative. We mainlyfollow the computations in [Sch18], and try to make this subsection as self-contained as possible.Let us first recall the following basic inequalities by Schumacher. Lemma 2.11 ( [Sch12, Lemma 8]) . Let V be a complex manifold, and let G , . . . , G n be non-negative C functions on V . Then √− ∂ ¯ ∂ log ( n Õ i = G i ) > Í nj = G j √− ∂ ¯ ∂ log G j Í ni = G i (2.3.1) Lemma 2.12 ([Sch18, Lemma 17]) . Let α j > for j = , . . . , n . Then for all x j > n Õ j = ( α j x j + j − α j − x jj ) x j − · . . . · x > − α α x + α n − n − α n − n x n · . . . · x + n − Õ j = (cid:18) α j − j − α j − j − α j + j α j + j + (cid:19) x j · . . . · x ! (2.3.2) NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 19
Set x j = G j G j − for j = , . . . , n and x : = G where G j > j = , . . . , n . Put them into (2.3.2) andwe obtain n Õ j = (cid:16) α j G j + j G j − j − − α j − G jj G j − j − (cid:17) > − α α G + α n − n − α n − n G n + n − Õ j = (cid:18) α j − j − α j − j − α j + j α j + j + (cid:19) G j ! (2.3.3)The following technical lemma is crucial in constructing our negatively curved Finsler metric F . Lemma 2.13 ( [Sch18, Lemma 10]) . Let F , . . . , F n be Finsler metrics on a complex space X , with theholomorphic sectional curvatures denoted by K , . . . , K n . Then for the Finsler metric F : = ( F + . . . + F n ) / , its holomorphic sectional curvature K F Í nj = K j F j F . (2.3.4) Proof.
For any holomorphic map γ : D → X , we denote by G , . . . , G n the semi-positive functionson D such that γ ∗ F i = √− G i dt ∧ d ¯ t for i = , . . . , n . Then γ ∗ F = √− ( n Õ i = G i ) dt ∧ d ¯ t , and it follows from (2.1.3) that the Gaussian curvature of γ ∗ F K γ ∗ F = − Í ni = G i ∂ log ( Í ni = G i ) ∂ t ∂ ¯ t (2.3.1) − ( Í ni = G i ) n Õ j = G j ∂ log G j ∂ t ∂ ¯ t Í nj = K j G j ( Í ni = G i ) . The lemma follows from Definition 2.3.(i). (cid:3)
For any γ : D → V with C : = γ − ( V ) , ∅ , we define a Hermitian pseudo-metric σ : = √− H ( t ) dt ∧ d ¯ t on D by taking convex sum in the following form H ( t ) : = n Õ k = kα k G k ( t ) , where G k is defined in (2.2.3), and α , . . . , α n ∈ R + are some universal constants which will be fixedlater. Following the similar estimate in [Sch18, Proposition 11], one can choose those constantsproperly such that the Gaussian curvature K σ of σ is uniformly bounded. Proposition 2.14.
There exists universal constants < α . . . α n and K > (independent of γ : D → V ) such that the Gaussian curvature K σ − K . on C .Proof. It follows from (2.3.4) that K σ H n Õ j = jα j K j G j and K j : = − ∂ log G j ∂ t ∂ ¯ t / G j . By Proposition 2.10, one has K σ α G H (cid:18) − c + (cid:16) G G (cid:17) (cid:19) + H n Õ j = α j G j (cid:18) − (cid:16) G j G j − (cid:17) j − + (cid:16) G j + G j (cid:17) j + (cid:19) H (cid:18) − cα G − n Õ j = (cid:16) α j G j + j G j − j − − α j − G jj G j − j − (cid:17) (cid:19) (2.3.3) H (cid:18) (cid:16) − c + α α (cid:17) α G + n − Õ j = (cid:16) α j + j α j + j + − α j − j − α j − j (cid:17) G j − α n − n − α n − n G n (cid:19) = : − H n Õ j = β j G j One can take α =
1, and choose the further α j > α j − inductively such that min j β j >
0. Set β : = min j β j ( jα j ) . Then K σ − H β n Õ j = ( jα j G j ) − β nH ( n Õ j = jα j G j ) = − β n = : − K . Note that α , . . . , α n and K is universal. The lemma is thus proved. (cid:3) It follows from Proposition 2.14 and (2.1.3) that one has the following estimate ∂ log H ( t ) ∂ t ∂ ¯ t > KH ( t ) > C ⊆ D , and in particular log H ( t ) is a subharmonic function over C .Since H ( t ) ∈[ , + ∞[ is continuous (in particular locally bounded from above) over D , log H ( t ) is asubharmonic function over D , and the estimate (2.3.5) holds over the whole D .In summary, we construct a negatively curved Finsler metric F on Y \ D , defined by F : = ( n Õ k = kα k F k ) / , (2.3.6)where F k is defined in (2.2.2), such that γ ∗ F = √− H ( t ) dt ∧ d ¯ t for any γ : D → V . Since weassume that τ is injective over V , the Finsler metric F is positively definite on V , and a fortiori F .Therefore, we finish the proof of Theorem 2.6.2.4. Proof of Theorem B.
Proof of Theorem B.
By Theorem 1.2, there is a VZ Higgs bundle over some birational model ˜ V of V .By Theorem D and Theorem 2.6, we can associate this VZ Higgs bundle with a negatively curvedFinsler metric which is positively definite over some Zariski dense open set of ˜ V . The theoremfollows directly from the bimeromorphic criteria for pseudo Kobayashi hyperbolicity in Lemma 2.4. (cid:3) Remark . Let me mention that Sun and Zuo also have the similar idea in constructing Finslermetric over the base using Viehweg-Zuo Higgs bundles combining with To-Yeung’s method [TY15].
NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 21
3. Kobayashi hyperbolicity of the base
In this section we will prove Theorem C. We first refine Viehweg-Zuo’s result on the positivity ofdirect images. We then apply this result to take different branch covering in the construction of VZHiggs bundles to prove the Kobayashi hyperbolicity of the base in Theorem C.3.1.
Preliminary for positivity of direct images.
We first recall a pluricanonical extension the-orem due to Cao [Cao19, Theorem 2.10]. Its proof is a combination of the Ohsawa-Takegoshi-Manivel L -extension theorem, with the semi-positivity of m -relative Bergman metric studied by Berndtsson-Păun [BP08, BP10] and Păun-Takayama [PT18]. Theorem 3.1 (Pluricanonical L -extension) . Let f : X → Y be an algebraic fiber space so that theKodaira dimension of the general fiber is non-negative. Assume that f is smooth over a dense Zariskiopen set of Y ⊂ Y so that both B : = Y \ Y and f ∗ B are normal crossing. Let L be any pseudo-effectiveline bundle L on X equipped with a positively curved singular metric h L with algebraic singularitiessatisfying the following property (i) There exists some regular value z ∈ Y of f , such that for some m ∈ N , all the sections H (cid:0) X z , ( mK X + L ) ↾ X z (cid:1) extends locally near z . (ii) H (cid:0) X z , ( mK X z + L ↾ X z ) ⊗ J ( h m L ↾ X z ) (cid:1) , ∅ .Then for any regular value y of f satisfying that (i) all sections H (cid:0) X y , mK X y + L ↾ X y (cid:1) extends locally near y , (ii) the metric h L ↾ X y is not identically equal to + ∞ ,the following restriction map H ( X , mK X / Y − m ∆ f + L + f ∗ A Y ) H (cid:0) X y , ( mK X y + L ↾ X y ) ⊗ J ( h m L ↾ X y ) (cid:1) is surjective. Here A Y is a universal ample line bundle on Y which does not depend on L , f and m , and ∆ f : = Õ j ( a j − ) V j . (3.1.1) where the sum is taken over all prime divisors V j of f ∗ B with multiplicity a j and its image f ( V j ) adivisor in Y . We will apply a technical lemma in [CaoP17, Claim 3.5] to prove Theorem 3.7.(i). Let us first recallsome definitions of singularities of divisors in [Vie95, Chapter 5.3] in a slightly different language.
Definition 3.2.
Let X be a smooth projective variety, and let L be a line bundle such that H ( X , L ) , ∅ . One defines e ( L ) = sup (cid:8) c ( D ) | D ∈ | L | is an effective divisor (cid:9) (3.1.2)where c ( D ) : = sup { c > | ( X , c · D ) is a klt divisor } is the log canonical threshold of D .Viehweg showed that one can control the lower bound of e ( L ) . Lemma 3.3 ([Vie95, Corollary 5.11]) . Let X be a smooth projective variety equipped with a very ampleline bundle H , and let L be a line bundle such that H ( X , L ) , ∅ . (i) Then there is a uniform estimate e ( L ) c ( H ) dim X − · c ( L ) + . (3.1.3)(ii) Let Z : = X × · · · × X be the r -fold product. Then for M : = Ë ri = pr ∗ i L , one has e ( M ) = e ( L ) . Let us recall the following result by Cao-Păun [CaoP17].
Lemma 3.4 (Cao-Păun) . Let f : X → Y be an algebraic fiber space so that the Kodaira dimension ofthe general fiber is non-negative. Assume that f is smooth over a dense Zariski open set of Y ⊂ Y sothat both B : = Y \ Y and f ∗ B are normal crossing. Then there exists some positive integer C > sothat for any m > m and a ∈ N , any y ∈ Y and any section σ ∈ H ( X y , amCK X y ) , there exists a section Σ ∈ H (cid:0) X , f ∗ A Y − a f ∗ det f ∗ ( mK X / Y ) + amr m CK X / Y + a ( P m + F m ) (cid:1) (3.1.4) whose restriction to the fiber X y is equal to σ ⊗ r m . Here F m and P m are effective divisors on X (inde-pendent of a ) such that F m is f -exceptional with f ( F m ) ⊂ Supp ( B ) , Supp ( P m ) ⊂ Supp ( ∆ f ) , r m : = rank f ∗ ( mK X / Y ) , and A Y is the universal ample line bundle on Y defined in Theorem 3.1. We recall the definition of
Kollár family of varieties with semi-log canonical singularities ( slc family for short). Definition 3.5 (slc family) . An slc family is a flat proper morphism f : X → B such that:(i) each fiber X b : = f − ( b ) is a projective variety with slc singularities.(ii) ω [ m ] X / B is flat.(iii) The family f : X → B satisfies the Kollár condition , which means that, for any m ∈ N , thereflexive power ω [ m ] X / B commutes with arbitrary base change.To make Definition 3.5.(iii) precise, for every base change τ : B ′ → B , given the induced morphism ρ : X ′ = X × B B ′ → X we have that the natural homomorphism ρ ∗ ω [ m ] X / B → ω [ m ] X ′ / B ′ is an isomorphism.Let us collect the basic properties of slc families, as is well-known to the experts. Lemma 3.6.
Let д : Z → W be a surjective morphism between quasi-projective manifolds with connec-ted fibers, which is birational to an slc family д ′ : Z ′ → W whose generic fiber has at most Gorensteincanonical singularities. Then (i) the total space Z ′ is normal and has only canonical singularities at worst. (ii) If ν : W ′ → W is a dominant morphism with W ′ smooth quasi-projective, then Z ′ × W W ′ → W ′ is still an slc family whose generic fiber has at most Gorenstein canonical singularities, and isbirational to ( Z × W W ′ ) ~ → W ′ . (iii) Denote by Z ′ r the r -fold fiber product Z ′ × W · · · × W Z ′ . Then д ′ r : Z ′ r → W is also an slc familywhose generic fiber has at most Gorenstein canonical singularities. Moreover, Z ′ r is birational tothe main component ( Z r ) ~ of Z r dominating W . (iv) Let Z ( r ) be a desingularization of ( Z r ) ~ . Then ( д ( r ) ) ∗ ( ℓ K Z ( r ) / W ) ≃ ( д ′ r ) ∗ ( ℓ K Z ′ r / W ) is reflexive forevery sufficiently divisible ℓ > . Positivity of direct images.
This section is devoted to prove Theorem 3.7 on positivity ofdirect images, which refines results by Viehweg-Zuo [VZ02, Proposition 3.4] and [VZ03, Proposition4.3]. It will be crucially used to proved Theorem C.
Theorem 3.7.
Let f : X → Y be a smooth family of projective manifolds of general type. Assumethat for any y ∈ Y , the set of z ∈ Y with X z bir ∼ X y is finite. (i) For any smooth projective compactification f : X → Y of f : X → Y and any sufficientlyample line bundle A Y over Y , f ∗ ( ℓ K X / Y ) ⋆⋆ ⊗ A − Y is globally generated over Y for any ℓ ≫ . Inparticular, f ∗ ( ℓ K X / Y ) is ample with respect to Y . (ii) In the same setting as (i), det f ∗ ( ℓ K X / Y ) ⊗ A − r ℓ Y is also globally generated over Y for any ℓ ≫ ,where r ℓ = rank f ∗ ( ℓ K X / Y ) . In particular, B + (cid:0) det f ∗ ( ℓ K X / Y ) (cid:1) ⊂ Y \ Y . (iii) For some r ≫ , there exists an algebraic fiber space f : X → Y compactifying X r → Y , so that f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y is globally generated over Y for ℓ large and divisible enough. Here X r denotes tobe the r -fold fiber product of X → Y , and A Y is some sufficiently ample line bundle over Y . NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 23
Proof.
Let us first show that, to prove Claims (i) and (ii), one can assume that both B : = Y \ Y and f ∗ B are normal crossing.For the arbitrary smooth projective compactification f ′ : X ′ → Y ′ of f : X → Y , we take alog resolution ν : Y → Y ′ with centers supported on Y ′ \ Y so that B : = ν − ( Y ′ \ Y ) is a simplenormal crossing divisor. Define X to be strong desingularization of the main component ( X ′ × Y ′ Y ) ~ dominant over Y X / / f ❍❍❍❍❍❍❍❍❍❍❍ X ′ × Y ′ Y (cid:15) (cid:15) / / X ′ f ′ (cid:15) (cid:15) Y ν / / Y ′ (3.2.1)so that f ∗ B is normal crossing. By [Vie90, Lemma 2.5.a], there is the inclusion ν ∗ f ∗ ( mK X / Y ) ֒ → f ′∗ ( mK X ′ / Y ′ ) (3.2.2)which is an isomorphism over Y for each m ∈ N . Hence for any ample line bundle A over Y ′ , once f ∗ ( mK X / Y ) ⋆⋆ ⊗ ( ν ∗ A ) − is globally generated over ν − ( Y ) ≃ Y for some m > f ′∗ ( mK X ′ / Y ′ ) ⋆⋆ ⊗ A − will be also globally generated over Y . As we will see, Claim (ii) is a direct consequence of Claim(i). This proves the above statement.(i) Let us fix a sufficiently ample line bundle A Y on Y . Assume that both B : = Y \ Y and f ∗ B are normal crossing. It follows from [Vie90, Theorem 5.2] that one can take some b ≫ a ≫ µ ≫ m ≫ s ≫ L : = det f ∗ ( µmK X / Y ) ⊗ a ⊗ det f ∗ ( mK X / Y ) ⊗ b is ample over Y . In otherwords, B + ( L ) ⊂ Supp ( B ) . By the definition of augmented base locus, one can even arrange a , b ≫ h of L − A Y which is smooth over Y , and thecurvature current √− Θ h L ( L ) > ω for some Kähler form ω in Y . Denote by r : = rank f ∗ ( µmK X / Y ) and r : = rank f ∗ ( mK X / Y ) . It follows from Lemma 3.4 that for any sections σ ∈ H ( X y , aµmCK X y ) , σ ∈ H ( X y , bmCK X y ) , there exists effective divisors Σ and Σ such that Σ + a f ∗ det f ∗ ( mµK X / Y ) − f ∗ A Y linear ∼ amµr CK X / Y + P + F Σ + b f ∗ det f ∗ ( mK X / Y ) − f ∗ A Y linear ∼ bmr CK X / Y + P + F and Σ ↾ X y = σ ⊗ r , Σ ↾ X y = σ ⊗ r . Here F i is f -exceptional with f ( F i ) ⊂ Supp ( B ) , Supp ( P i ) ⊂ Supp ( ∆ f ) for i = , N : = amµr C + bmr C , P : = P + P and F : = F + F . Fix any y ∈ Y . Then the effective divisor Σ + Σ induces a singular hermitian metric h for the line bundle L : = N K X / Y − f ∗ L + f ∗ A Y + P + F such that h | X y is not identically equal to + ∞ , and so is the singular hermitian metric h : = f ∗ h · h over L : = L + f ∗ L − f ∗ A Y = N K X / Y − f ∗ A Y + P + F . In particular, when ℓ sufficiently large,the multiplier ideal sheaf J ( h ℓ ↾ X y ) = O X y . By Siu’s invariance of plurigenera, all the global sections H (cid:0) X y , ( ℓ K X + L ) ↾ X y (cid:1) ≃ H (cid:0) X y , ( ℓ + N ) K X y (cid:1) extends locally, and we thus can apply Theorem 3.1 toobtain the desired surjectivity H (cid:0) X , ℓ K X / Y + L − ℓ ∆ f + f ∗ A Y (cid:1) ։ H (cid:0) X y , ( ℓ + N ) K X y (cid:1) , (3.2.3)Recall that Supp ( P ) ⊂ Supp ( ∆ f ) . Then ℓ f ∗ B > P for ℓ ≫
0, and one has the inclusion of sheaves ℓ K X / Y + L − ℓ ∆ f + f ∗ A Y ֒ → ( N + ℓ ) K X / Y − f ∗ A Y + F . which is an isomorphism over X . By (3.2.3) this implies that the direct image sheaves f ∗ ( ℓ K X / Y − f ∗ A Y + F ) are globally generated over some Zariski open set U y ⊂ Y containing y for ℓ ≫
0. Since y is an arbitrary point in Y , the direct image f ∗ ( ℓ K X / Y + F ) ⊗ A − Y is globally generated over Y for ℓ ≫ F is f -exceptional with f ( F ) ⊂ Supp ( B ) . Then there is aninjection f ∗ ( ℓ K X / Y + F ) ⊗ A − Y ֒ → f ∗ ( ℓ K X / Y ) ⋆⋆ ⊗ A − Y which is an isomorphism over Y . Hence f ∗ ( ℓ K X / Y ) ⋆⋆ ⊗ A − Y is also globally generated over Y . Hence f ∗ ( ℓ K X / Y ) is ample with respect to Y for ℓ ≫
0. The first claim follows.(ii) The trick to prove the second claim has already appeared in [Den17] in proving a conjecture byDemailly-Peternell-Schneider. We first recall that f ∗ ( ℓ K X / Y ) is locally free outside a codimension 2analytic subset of Y . By the proof of Theorem 3.7.(i), for ℓ sufficiently large and divisible, f ∗ ( ℓ K X / Y + F ) ⊗ A − Y is locally free and generated by global sections over Y , where F is some f -exceptionaleffective divisor. Therefore, its determinant det f ∗ ( ℓ K X / Y + F ) ⊗ A − r ℓ Y is also globally generated over Y , where r ℓ : = rank f ∗ ( ℓ K X / Y ) . Since F is f -exceptional and effective, one hasdet f ∗ ( ℓ K X / Y + F ) ⊗ A − r ℓ Y = det f ∗ ( ℓ K X / Y ) ⊗ A − r ℓ Y , and therefore, det f ∗ ( ℓ K X / Y ) ⊗ A − r ℓ Y is also globally generated over Y . By the very definition of theaugmented base locus B + (•) we conclude that B + (cid:0) det f ∗ ( ℓ K X / Y ) (cid:1) ⊂ Supp ( B ) . The second claim is proved.(iii) We combine the ideas in [VZ03, Proposition 4.1] as well as the pluricanonical extension tech-niques in Theorem 3.1 to prove the result. By Corollary A.2, there exists a smooth projective compac-tification Y of Y with B : = Y \ Y simple normal crossing, a non-singular finite covering ψ : W → Y ,and an slc family д ′ : Z ′ → W , which extends the family X × Y W . By Lemma 3.6.(iii) for any r ∈ Z > , the r -fold fiber product д ′ r : Z ′ r → W is still an slc family, which compactifies the smoothfamily X r × Y W → W , where W : = ψ − ( Y ) . Note that Z ′ r has canonical singularities.Take a smooth projective compactification f : X → Y of X r → Y so that f ∗ B is normal crossing.Let Z → Z ′ r be a strong desingularization of Z ′ r , which also resolves this birational map Z ′ r d ( X × Y W ) ~ . Then д : Z → W is smooth over W : = ψ − ( Y ) . Z Z ′ r X ( X × Y W ) ~ W W Y W д д ′ r fψ ψ Let ˜ Z be a strong desingularization of Z ′ , which is thus smooth over W : = ψ − ( Y ) . For the newfamily ˜ д : ˜ Z → W , we denote by ˜ Z : = ˜ д − ( W ) . Then ˜ Z → W is also a smooth family, and anyfiber of Z w with w ∈ W is a projective manifold of general type. By our assumption in the theorem,for any w ∈ W , the set of w ′ ∈ W with ˜ Z w ′ bir ∼ ˜ Z w is finite as ψ : W → Y is a finite morphism. Wethus can apply Theorems 3.7.(i) and 3.7.(ii) to our new family ˜ д : ˜ Z → W .From now on, we will always assume that ℓ ≫ ℓ K Z ′ is Cartier. Let A Y be a sufficiently ample line bundle over Y , so that A W : = ψ ∗ A Y is also sufficiently ample . Since Z ′ has canonical singularity, ˜ д ∗ ( ℓ K ˜ Z / W ) = д ′∗ ( ℓ K Z ′ / W ) . It follows from Theorem 3.7.(ii) that, for any ℓ ≫
0, the line bundle det ˜ д ∗ ( ℓ K ˜ Z / W ) ⊗ A − rW = det д ′∗ ( ℓ K Z ′ / W ) ⊗ A − rW (3.2.4)is globally generated over W , where r : = rank д ′∗ ( ℓ K Z ′ / W ) depending on ℓ . Then there exists apositively-curved singular hermitian metric h det on the line bundle det д ′∗ ( ℓ K Z ′ / W ) ⊗ A − rW such that h det is smooth over W .By the base change properties of slc families (see [BHPS13, Proposition 2.12] and [KP17, Lemma2.6]), one has ω [ ℓ ] Z ′ r / W ≃ r Ì i = pr ∗ i ω [ ℓ ] Z ′ / W , д ′ r ∗ ( ℓ K Z ′ r / W ) ≃ r Ì д ′∗ ( ℓ K Z ′ / W ) , where pr i : Z ′ r → Z ′ is the i -th directional projection map. Hence ℓ K Z ′ r is Cartier as well, and wehave r Ì д ′∗ ( ℓ K Z ′ / W ) ≃ д ′ r ∗ ( ℓ K Z ′ r / W ) = д ∗ ( ℓ K Z / W ) . NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 25
By Lemma 3.6.(iv), д ∗ ( ℓ K Z / W ) is reflexive, and we thus havedet д ′∗ ( ℓ K Z ′ / W ) → r Ì д ′∗ ( ℓ K Z ′ / W ) ≃ д ∗ ( ℓ K Z / W ) , which induces a natural effective divisor Γ ∈ | ℓ K Z / W − д ∗ det д ′∗ ( ℓ K Z ′ / W )| such that Γ ↾ Z w , Z w with w ∈ W . By Lemma 3.3 for any w ∈ W the logcanonical threshold c ( Γ ↾ Z w ) > e ( ℓ K Z w ) = e (cid:0) Ë ri = pr ∗ i ( K ⊗ ℓ Z ′ w ) (cid:1) = e (cid:0) ℓ K Z ′ w (cid:1) > ℓ · c ( A ) d − · c ( K Z ′ w ) + > ( C − ) ℓ (3.2.5)for some C ∈ N which does not depend on ℓ and w ∈ W . Denote by h the singular hermitian metricon ℓ K Z / W − д ∗ det д ′∗ ( ℓ K Z ′ / W ) induced by Γ . By the definition of log canonical threshold, the multiplier ideal sheaf J (cid:0) h ( C − ) ℓ ↾ Z w (cid:1) = O Z w for any fiber Z w with w ∈ W . Let us define a positively-curved singular metric h F for the linebundle F : = ℓ K Z / W − rд ∗ A W by setting h F : = h · д ∗ h det . Then J (cid:0) h ( C − ) ℓ F ↾ Z w (cid:1) = O Z w for any w ∈ W .For any n ∈ N ∗ , applying Theorem 3.1 to n F we obtain the surjectivity H (cid:0) Z , ( C − ) n ℓ K Z / W + n F + д ∗ A W (cid:1) ։ H (cid:0) Z w , Cn ℓ K Z w (cid:1) (3.2.6)for all w ∈ W . In other words, д ∗ (cid:0) C ℓ nK Z / W ) ⊗ A −( nr − ) W is globally generated over W for any ℓ ≫ n > K X y is big, one thus has r = r ℓ ∼ ℓ d as ℓ → + ∞ where d : = dim Z w > f are curves, one can take a fiber product to replace theoriginal family). Recall that C is a constant which does not depend on ℓ . One thus can take an apriori ℓ ≫ r ≫ C ℓ . In conclusion, for sufficiently large and divisible m , д ∗ (cid:0) mK Z / W ) ⊗ A − mW = д ∗ (cid:0) mK Z / W ) ⊗ ψ ∗ A − mY is globally generated over W . Therefore, we have a morphism N Ê i = ψ ∗ A mY → д ∗ (cid:0) mK Z / W (cid:1) ⊗ ψ ∗ A − mY , (3.2.7)which is surjective over W . On the other hand, by [Vie90, Lemma 2.5.b], one has the inclusion д ∗ (cid:0) mK Z / W (cid:1) ֒ → ψ ∗ f ∗ ( mK X / Y ) , which is an isomorphism over W . (3.2.7) thus induces a morphism N Ê i = ψ ∗ O W ⊗ A mY → ψ ∗ д ∗ (cid:0) mK Z / W (cid:1) ⊗ A − mY → ψ ∗ ψ ∗ (cid:0) f ∗ ( mK X / Y ) (cid:1) ⊗ A − mY , (3.2.8)which is surjective over Y . Note that that even if f ∗ ( mK X / Y ) is merely a coherent sheaf, the projec-tion formula ψ ∗ ψ ∗ (cid:0) f ∗ ( mK X / Y ) (cid:1) = f ∗ ( mK X / Y ) (cid:1) ⊗ ψ ∗ O W still holds for ψ is finite (see [Ara04, Lemma5.7]). The trace map ψ ∗ O W → O Y splits the natural inclusion O Y → ψ ∗ O W , and is thus surjective. Hence (3.2.8) gives rise to a morph-ism N Ê i = ψ ∗ O W ⊗ A mY → ψ ∗ д ∗ (cid:0) mK Z / W (cid:1) ⊗ A − mY Φ −→ f ∗ ( mK X / Y ) ⊗ A − mY , (3.2.9)which is surjective over Y . By taking m sufficiently large, we may assume that ψ ∗ O W ⊗ A mY isgenerated by its global sections. Then f ∗ ( mK X / Y ) ⊗ A − mY is globally generated over Y . We completethe proof. (cid:3) Let us prove the following Bertini-type result, which will be used in the proof of Theorem 3.11.
Lemma 3.8 (A Bertini-type result) . Let f : X → Y be the projective family in Theorem 3.7.(iii). Thenfor any given smooth fiber X y with y ∈ Y , there is H ∈ | ℓ K X / Y − ℓ f ∗ A Y | so that H ↾ X y is smooth. Inparticular, there is a Zariski open neighborhood V ⊃ y so that H is smooth over V .Proof. By Siu’s invariance of plurigenera and Grauert-Grothedieck’s “cohomology and base change”,we know that f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y is locally free on Y , and the natural map ( f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y ) y → H ( X y , ℓ K X y ) is an isomorphism for any y ∈ Y . Since K X y is assumed to be semi-ample, one can take ℓ ≫ | ℓ K X y | is base point free. By the Bertini theorem, one can take a section s ∈ H ( X y , ℓ K X y ) whosezero locus is a smooth hypersurface on X y . By Theorem 3.7.(iii), one has the surjection H (cid:0) Y , f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y (cid:1) ։ (cid:0) f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y (cid:1) y ≃ −→ H ( X y , ℓ K X y ) . Hence there is σ ∈ H ( X , ℓ K X / Y − ℓ f ∗ A Y ) = H (cid:0) Y , f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y (cid:1) which extends the section s . In other words, for the zero divisor H = ( σ = ) , its restriction to X y is smooth. Hence there is a Zariski open neighborhood V ⊃ y so that H is smooth over V . Thelemma is proved. (cid:3) Remark . Note that we do not know how to find a hypersurface H ∈ | ℓ K X / Y − ℓ f ∗ A Y | so that itsdiscriminant locus in Y is normal crossing. We have to blow-up the base ν : Y ′ → Y to achieve this.As f : X → Y is not flat in general, for the new family X ′ ( X × Y Y ′ ) ~ X × Y Y ′ XY ′ Y ′ Y ′ Y f ′ fν where X ′ is a desingularization of ( X × Y Y ′ ) ~ , in general ν ∗ f ∗ ( ℓ K X / Y ) f ′∗ ( ℓ K X ′ / Y ′ ) . In other words, although the discriminant locus of ν ∗ H → Y ′ is a simple normal crossing divisor in Y ′ , ν ∗ H might not lie at | ℓ K X ′ / Y ′ − ℓ f ′∗ ν ∗ A Y | . We will overcome this problem in Theorem 3.11 at thecost of the appearance of some f ′ -exceptional divisors.Since the Q -mild reduction in Corollary A.2 holds for any smooth surjective projective morphismwith connected fibers and smooth base, it follows from our proof in Theorem 3.7.(iii) and Kawamata’stheorem [Kaw85], one still has the generic global generation as follows. Theorem 3.10.
Let f U : U → V be a smooth projective morphism between quasi-projective varietieswith connected fibers. Assume that the general fiber F of f U has semi-ample canonical bundle, and f U is of maximal variation. Then there exists a positive integer r ≫ and a smooth projective compacti-fication f : X → Y of U r → V so that f ∗ ( mK X / Y ) ⊗ A − m is globally generated over some Zariski opensubset of V . Here U r → V is the r -fold fiber product of U → V , and A is some ample line bundle on Y . (cid:3) NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 27
Sufficiently many “moving” hypersurfaces.
As we have seen in § 1.2 on the constructionof VZ Higgs bundles, one has to apply branch cover trick to construct a negatively twisted Hodgebundle on the compactification of the base, which is well-defined outside a simple normal crossingdivisor. This means that the hypersurface H ∈ | ℓ K X / Y − ℓ f ∗ A Y | in constructing the cyclic coveris smooth over the complement of an SNC divisor of the base. As we discussed in Lemma 3.8, ingeneral we cannot perform a simple blow-up of the base to achieve this. In this subsection we willovercome this difficulty in applying the methods in [PTW19, Proposition 4.4]. It will be our basicsetup in constructing refined VZ Higgs bundles in § 3.4. Theorem 3.11.
Let X → Y be a smooth family of minimal projective manifolds of general type overa quasi-projective manifold Y . Suppose that for any y ∈ Y , the set of z ∈ Y with X z bir ∼ X y is finite. Let Y ⊃ Y be the smooth compactification in Corollary A.2. Fix any y ∈ Y and some sufficiently ampleline bundle A Y on Y . Then there exist a birational morphism ν : Y ′ → Y and a new algebraic fiberspace f ′ : X ′ → Y ′ which is smooth over ν − ( Y ) , so that for any sufficiently large and divisible ℓ , onecan find a hypersurface H ∈ | ℓ K X ′ / Y ′ − ℓ ( ν ◦ f ′ ) ∗ A Y + ℓ E | (3.3.1) satisfying that • the divisor D : = ν − ( Y \ Y ) is simple normal crossing. • There exists a reduced divisor S in Y ′ , so that D + S is simple normal crossing, and H → Y ′ is smoothover Y ′ \ D ∪ S . • The exceptional locus Ex ( ν ) ⊂ Supp ( D + S ) , and y < ν ( D ∪ S ) . • The divisor E is effective and f ′ -exceptional with f ′ ( E ) ⊂ Supp ( D + S ) .Moreover, when X → Y is effectively parametrized over some open set containing y , so is the newfamily X ′ → Y ′ .Proof. The proof is a continuation of that of Theorem 3.7.(iii), and we adopt the same notationstherein. By (3.2.9) and the isomorphism H ( Z , ℓ K Z / W − ℓ д ∗ A W ) ≃ H (cid:0) Z ′ r , ℓ K Z ′ r / W − ℓ ( д ′ r ) ∗ A W (cid:1) , the morphism Φ : ψ ∗ д ∗ (cid:0) ℓ K Z / W (cid:1) ⊗ A − ℓ Y → f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y in (3.2.9) gives rise to a natural map ϒ : H ( Z ′ r , ℓ K Z ′ r / W − ℓ ( д ′ r ) ∗ A W ) → H (cid:0) Y , f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y (cid:1) (3.3.2)whose image I generates f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y over Y . Note that ϒ is fonctorial in the sense that it doesnot depend on the choice of the birational model Z → Z ′ r . By the base point free theorem, for any y ∈ Y , K X y is semi-ample, and we can assume that ℓ ≫ ℓ K X / Y is relatively semi-ample over Y . Hence we can take a section σ ∈ H (cid:0) Z ′ r , ℓ K Z ′ r / W − ℓ ( д ′ r ) ∗ A W (cid:1) (3.3.3)so that the zero divisor of ϒ ( σ ) ∈ H (cid:0) X , ℓ K X / Y − ℓ f ∗ A Y (cid:1) = H (cid:0) Y , f ∗ ( ℓ K X / Y ) ⊗ A − ℓ Y (cid:1) , denoted by H ∈ | ℓ K X ( r ) / Y − ℓ ( f ( r ) ) ∗ A Y | , is transverse to the fiber X y . Denote by T the discriminantlocus of H → Y , and B : = Y \ Y . Then y < T ∪ B . Take a log-resolution ν : Y ′ → Y withcenters in T ∪ B so that both D : = ν − ( B ) and D + S : = ν − ( T ∪ B ) are simple normal crossing.Let X ′ be a strong desingularization of ( X × Y Y ′ ) ~ , and write f ′ : X ′ → Y ′ , which is smooth over Y ′ : = ν − ( Y ) . Set X ′ : = f ′− ( Y ′ ) . It suffices to show that, there exists a hypersurface H in (3.3.1)with H ↾ ( ν ◦ f ′ ) − ( V ) = H ↾ ( f ( r ) ) − ( V ) , where V : = Y \ S ′ ∪ B ⊂ Y . Since the birational morphism ν isisomorphic at y , we can write y as ν − ( y ) abusively.Now we follow the similar arguments in [PTW19, Proposition 4.4] to prove the existence of H (in which they apply their methods for mild morphisms ). Let W ′ be a strong desingularization of W × Y Y ′ which is finite at y ∈ Y ′ . Write W ′ : = ν ′− ( W ) . By Lemma 3.6.(ii), the new family Z ′′ : = Z ′ r × W W ′ → W ′ is still an slc family, which compactifies the smooth family X ′ × Y ′ W ′ → W ′ .Let M ′ be a desingularization of Z ′′ so that it resolves the rational maps to X ′ as well as Z . X f (cid:15) (cid:15) Z / / o o д (cid:15) (cid:15) Z ′ rд ′ r (cid:15) (cid:15) X ′ µ ✎✎✎✎ G G ✎✎✎✎ f ′ (cid:15) (cid:15) M ′ o o G G ✎✎✎✎✎✎✎✎ h ′ (cid:15) (cid:15) / / Z ′′ д ′′ (cid:15) (cid:15) µ ′ ✎✎✎ G G ✎✎✎ Y W ψ o o WY ′ ν ✎✎✎ G G ✎✎✎✎ W ′ ψ ′ o o ν ′ ✎✎✎ G G ✎✎✎✎ W ′ ν ′ ✎✎✎ G G ✎✎✎✎ By the properties of slc families, µ ′∗ ω [ ℓ ] Z ′ r / W = ω [ ℓ ] Z ′′ / W ′ , which induces a natural map µ ∗ : H (cid:0) Z ′ r , ℓ K Z ′ r / W − ℓ ( д ′ r ) ∗ A W (cid:1) → H (cid:0) Z ′′ , ℓ K Z ′′ / W ′ − ℓ ( ν ′ ◦ д ′′ ) ∗ A W (cid:1) . (3.3.4)Since both Z ′ r and Z ′′ have canonical singularities, one has the following natural morphisms д ∗ ( ℓ K Z / W ) ≃ ( д ′ r ) ∗ ( ℓ K Z ′ r / W ) , h ′∗ ( ℓ K M ′ / W ′ ) = д ′′∗ ( ℓ K Z ′′ / W ′ ) . We can leave out a subvariety of codimension at least two in Y ′ supported on D + S (which thusavoids y by our construction) so that ψ ′ : W ′ → Y ′ becomes a flat finite morphism. As discussed atthe beginning of the proof, there is also a natural map ϒ ′ : H ( Z ′′ , ℓ K Z ′′ / W ′ − ℓ ( ν ′ ◦ д ′′ ) ∗ A W ) → H (cid:0) X ′ , ℓ K X ′ / Y ′ − ℓ ( ν ◦ f ′ ) ∗ A Y (cid:1) (3.3.5)as (3.3.2) by factorizing through M ′ .Note that for V : = Y \ T ∪ B , ν : ν − ( V ) ≃ −→ V is also an isomorphism, and thus the restriction of X → Y to V is isomorphic to that of X ′ → Y ′ to ν − ( V ) . Hence by our construction,the restriction of Z ′ r → W to ψ − ( V ) is isomorphic to that of Z ′′ → W ′ to ( ν ◦ ψ ′ ) − ( V ) = ( ν ′ ◦ ψ ) − ( V ) . In particular,under the above isomorphism, for the section σ ∈ H (cid:0) Z ′ r , ℓ K Z ′ r / W − ℓ ( д ′ r ) ∗ A W (cid:1) in (3.3.3) with ϒ ( σ ) defining H , one has ϒ ( σ ) ↾ f − ( V ) ≃ ϒ ′ ( µ ∗ σ ) ↾ ( ν ◦ f ′ ) − ( V ) . where µ ∗ and ϒ ′ are defined in (3.3.4) and (3.3.5). Denote by ˜ H the zero divisor defined by ϒ ′ ( µ ∗ σ ) ∈ H (cid:0) X ′ , ℓ K X ′ / Y ′ − ℓ ( ν ◦ f ′ ) ∗ A Y (cid:1) . Recall that H is smooth over V , then ˜ H is also smooth over ν − ( V ) .Note that ϒ ′ ( µ ∗ σ ) ∈ H (cid:0) Y ′ , f ′∗ ( ℓ K X ′ / Y ′ )⊗ ν ∗ A − ℓ Y (cid:1) is only defined over a big open set of Y ′ containing ν − ( V ) . Hence it extends to a global section s ∈ H ( X ′ , ℓ K X ′ / Y ′ − ℓ ( ν ◦ f ′ ) ∗ A Y + ℓ E ) , where E is an f ′ -exceptional effective divisor with f ′ ( E ) ⊂ Supp ( D + S ) . Denote by H the hypersur-face in X ′ defined by s . Hence H ↾ ( ν ◦ f ′ ) − ( V ) = ˜ H ↾ ( ν ◦ f ′ ) − ( V ) , which is smooth over ν − ( V ) = Y ′ \ D ∪ S ≃ V ∋ y . Note that the property of effective parametrization is invariant under fiber product. Thetheorem follows. (cid:3) Kobayashi hyperbolicity of the moduli spaces.
In this subsection, for effectively paramet-rized smooth family of minimal projective manifolds of general type, we refine the Viehweg-ZuoHiggs bundles in Theorem 1.2 so that we can apply Theorem 2.6 and the bimeromorphic criteria forKobayashi hyperbolicity in Lemma 2.5 to prove Theorem C.
Theorem 3.12.
Let U → V be an effectively parametrized smooth family of minimal projective man-ifolds of general type over the quasi-projective manifold V . Then for any given point y ∈ V , there existsa smooth projective compactification Y for a birational model ν : ˜ V → V , and a VZ Higgs bundle ( ˜ E , ˜ θ ) ⊃ ( F , η ) over Y satisfying the following properties: NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 29 (i) there is a Zariski open set V of V containing y so that ν : ν − ( V ) → V is an isomorphism. (ii) Both D : = Y \ ˜ V and D + S : = Y \ ν − ( V ) are simple normal crossing divisors in Y . (iii) The Higgs bundle ( ˜ E , ˜ θ ) has log poles supported on D ∪ S , that is, ˜ θ : ˜ E → ˜ E ⊗ (cid:0) log ( D + S ) (cid:1) . (iv) The morphism τ : T Y (− log D ) → L − ⊗ E n − , (3.4.1) induced by the sub-Higgs sheaf ( F , η ) is injective over V .Proof. The proof is a continuation of that of Theorem 1.2, and we will adopt the same notations.We first prove that for any y ∈ V , the set of z ∈ V with X z bir ∼ X y is finite. Take a polarization H for U → V with the Hilbert polynomial h . Denote by P h ( V ) the set of such pairs ( U → V , H ) , up toisomorphisms and up to fiberwise numerical equivalence for H . By [Vie95, Section 7.6], there existsa coarse quasi-projective moduli scheme P h for P h , and thus the family induces a morphism V → P h .By the assumption that the family U → V is effectively parametrized, the induced morphism V → P h is quasi-finite, which in turn shows that the set of z ∈ V with X z isomorphic to X y is finite. Notethat a projective manifold of general type has finitely many minimal models. Hence the set of z ∈ V ′ with X z bir ∼ X y is finite as well.Now we will choose the hypersurface in (1.2.1) carefully so that the cyclic cover construction inTheorem 1.2 can provide the desired refined VZ Higgs bundle. Let Y ′ ⊃ V be the smooth compac-tification in Corollary A.2. By Theorem 3.11, for any given point y ∈ V and any sufficiently ampleline bundle A on Y ′ , there exists a birational morphism ν : Y → Y ′ and a new algebraic fiber space f : X → Y so that one can find a hypersurface H ∈ | ℓ Ω nX / Y ( log ∆ ) − ℓ ( ν ◦ f ) ∗ A + ℓ E | , n : = dim X − dim Y (3.4.2)satisfying that • the inverse image D : = ν − ( Y ′ \ V ) is a simple normal crossing divisor. • There exists a reduced divisor S so that D + S is simple normal crossing, and H → Y is smoothover V : = Y \ ( D ∪ S ) . • The restriction ν : ν − ( V ) → V is an isomorphism. • The given point y is contained in V . • The divisor E is effective and f -exceptional with f ( E ) ⊂ Supp ( D + S ) . • For any z ∈ V : = ν − ( V ′ ) , the canonical bundle of the fiber X z : = f − ( z ) is big and nef. • The restricted family f − ( V ) → V is smooth and effectively parametrized.Here we set ∆ : = f ∗ D and Σ : = f ∗ S . Write L : = ν ∗ A . Now we take the cyclic cover with respect to H in (3.4.2) instead of that in (1.2.1), and perform the same construction of VZ Higgs ( ˜ E , ˜ θ ) ⊃ ( F , τ ) bundle as in Theorem 1.2. Theorems 3.12.(i) to 3.12.(iii) can be seen directly from the properties of H and the cyclic construction.Theorem 3.12.(iv) has already appeared in [PTW19, Proposition 2.11] implicitly, and we give aproof here for the sake of completeness. Recall that both Z and H are smooth over V . Denote by H : = H ∩ f − ( V ) , f : X = f − ( V ) → V , and д : Z = д − ( V ) → V . We have F n , ↾ V = f ∗ (cid:0) Ω nX / Y ( log ∆ ) ⊗ L − (cid:1) ↾ V = O V E n − , ↾ V = R ( д ) ∗ ( Ω n − Z / V ) = R ( f ) ∗ (cid:0) Ω n − X / V ⊕ ℓ − Ê i = Ω n − X / V ( log H ) ⊗ ( K X / V ⊗ f ∗ L − ) − i (cid:1) F n − , ↾ V = R f ∗ (cid:0) Ω n − X / Y ( log ∆ ) ⊗ L − (cid:1) ↾ V = R ( f ) ∗ (cid:0) Ω n − X / V ⊗ K − X / V (cid:1) ≃ R ( f ) ∗ ( T X / V ) . (3.4.3)Hence τ ↾ V factors through τ ↾ V : T V ρ −→ R ( f ) ∗ ( T X / V ) ≃ −→ R ( f ) ∗ (cid:0) Ω n − X / V ⊗ K − X / V (cid:1) → R ( f ) ∗ (cid:0) Ω n − X / V ( log H ) ⊗ K − X / V (cid:1) → R ( д ) ∗ ( Ω n − Z / V ) ⊗ L − , where ρ is the Kodaira-Spencer map. Although the intermediate objects in the above factorizationmight not be locally free, the induced C -linear map by the sheaf morphism τ ↾ V at the z ∈ V τ , z : T Y , z → ( L − ⊗ E n − , ) z coincides with the following composition of C -linear maps between finite dimensional complexvector spaces τ , z : T Y , z ρ z −−→ H ( X z , T X z ) ≃ −→ H ( X z , Ω n − X z ⊗ K − X z ) j z −→ (3.4.4) H (cid:0) X z , Ω n − X z ( log H z ) ⊗ K − X z (cid:1) → H (cid:0) Z z , Ω n − Z z (cid:1) . To prove Theorem 3.12.(iv), it then suffices to prove that each linear map in (3.4.4) is injective forany z ∈ V .By the effective parametrization assumption, ρ z is injective. The map j z in (3.4.4) is the same asthe H -cohomology map of the short exact sequence0 → K − X z ⊗ Ω n − X z → K − X z ⊗ Ω n − X z ( log H z ) → K − X z ↾ H z ⊗ Ω n − H z → . Observe that K X z ↾ H z is big. Indeed, this follows from thatvol ( K X z ↾ H z ) = c ( K X z ↾ H z ) n − = c ( K X z ) n − · H z = ℓ c ( K X z ) n = ℓ vol ( K X z ) > . Hence j z injective by the Bogomolov-Sommese vanishing theorem H (cid:0) H z , K − X z ↾ H z ⊗ Ω d − H z (cid:1) = , as observed in [PTW19]. Since ψ z : Z z → X z is the cyclic cover obtained by taking the ℓ -th roots outof the smooth hypersurface H z ∈ | ℓ K X z | , the morphism ψ is finite. It follows from the degenerationof the Leray spectral sequence that H ( Z z , Ω n − Z z ) ≃ H (cid:0) X z , ( ψ z ) ∗ Ω n − Z z (cid:1) = H (cid:0) X z , Ω n − X z (cid:1) ⊕ ℓ − Ê i = H (cid:0) X z , Ω n − X z ( log H z ) ⊗ K − iX z (cid:1) . (3.4.5)The last map in (3.4.4) is therefore injective, for the cohomology group H (cid:0) X z , Ω n − X z ( log H z ) ⊗ K − X z (cid:1) is a direct summand of H ( Z z , Ω n − Z z ) by (3.4.5). As a consequence, the composition τ , z in (3.4.4) isinjective at each point z ∈ V . Theorem 3.12.(iv) is thus proved. (cid:3) Let us explain how Lemma 2.5 and Theorems 2.6 and 3.12 imply Theorem C.
Proof of Theorem C.
We first take a smooth compactification Y ⊃ V as in Corollary A.2,. By The-orem 3.12, for any given point y ∈ V , there exists a birational morphism ν : Y ′ → Y which isisomorphic at y , so that D : = Y ′ \ ν − ( V ) is a simple normal crossing divisor, and there exists a VZHiggs bundle ( ˜ E , ˜ θ ) whose log pole D + S avoids y ′ : = ν − ( y ) . Moreover, by Theorem 3.12.(iv), τ isinjective at y ′ . Applying Theorem 2.6, we can associate ( ˜ E , ˜ θ ) a Finsler metric F on T Y ′ (− D ) whichis positively definite at y ′ . Moreover, if we think of F as a Finsler metric on ν − ( V ) , it is negativelycurved in the sense of Definition 2.3.(ii). Hence the base V satisfies the conditions in Lemma 2.5,and we conclude that V is Kobayashi hyperbolic. (cid:3) Appendix A. Q -mild reductions (by Dan Abramovich) Let us work over complex number field C .The main result in this appendix is the following: Theorem A.1.
Let f : S → T be a projective family of smooth varieties with T quasi-projective. (i) There are compactifications S ⊂ S and T ⊂ T , with S and T Deligne-Mumford stacks withprojective coarse moduli spaces, and a projective morphism f : S → T extending f which is aKollár family of slc varieties. (ii) Given a finite subset Z ⊂ T there is a projective variety W and finite surjective lci morphism ρ : W → T , unramified over Z , such that ρ − T sm = W sm . NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 31
Here the notion of Kollár family refers to the condition that the sheaf ω [ m ]S/T is flat and its formationcommutes with arbitrary base change for each m . We refer the readers to [AH11, Definition 5.2.1]for further details.Note that the pullback family S × T W → W is a Kollár family of slc varieties compactifying thepullback S × T W → W of the original family to W : = W × T T .This is applied in the present paper, where some mild regularity assumption on T and W isrequired: Corollary A.2 ( Q -mild reduction) . Assume further T is smooth. For any given finite subset Z ⊂ T ,there exist (i) a compactification T ⊂ T with T a regular projective scheme, (ii) a simple normal crossings divisor D ⊂ T containing T r T and disjoint from Z , (iii) a finite morphism W → T unramified outside D , and (iv) A Kollár family S W → W of slc varieties extending the given family S × T W . The significance of these extended families is through their Q -mildness property. Recall from[AK00] that a family S → T is Q -mild if whenever T → T is a dominant morphism with T having atmost Gorenstein canonical singularities, then the total space S = T × S T has canonical singularities.It was shown by Kollár–Shepherd-Barron [KSB88, Theorem 5.1] and Karu [Kar00, Theorem 2.5] thatKollár families of slc varieties whose generic fiber has at most Gorenstein canonical singularities are Q -mild.The main result is proved using moduli of Alexeev stable maps.Let V be a projective variety. A morphism ϕ : U → V is a stable map if U is slc and K U is ϕ -ample.More generally, given π : U → T , a morphism ϕ : U → V is a stable map over T or a family ofstable maps parametrized by T if π is a Kollár family of slc varieties and K U / T is ϕ × π -ample. Notethat this condition is very flexible and does not require the fibers to be of general type, although keyapplications in Theorems 3.7.(iii) and 3.11 require some positivity of the fibers. Theorem A.3 ([DR18, Theorem 1.5]) . Stable maps form an algebraic stack M ( V ) locally of finite typeover C , each of whose connected components is a proper global quotient stack with projective coarsemoduli space. The existence of an algebraic stack satisfying the valuative criterion for properness was known toAlexeev, and can also be deduced directly from the results of [AH11], which presents it as a globalquotient stack. The work [DR18] shows that the stack has bounded, hence proper components,admitting projective course moduli spaces. An algebraic approach for these statements is providedin [Kar00, Corollary 1.2].
Proof of Theorem A.1. (i) Let T ⊂ T and S ⊂ S be projective compactifications with π : S → T extending f . The family S → T with the injective morphism ϕ : S → S is a family of stable mapsinto S , providing a morphism T → M ( S ) which is in fact injective. Let T be the closure of T . Since M ( S ) is proper, T is proper. Let S be the pullback of the universal family along T → M ( S / T ) . Then S ⊃ S is a compactification as needed.(ii) The existence of W follows from the main result of [KV04]. (cid:3) Proof of Corollary A.2.
Consider the coarse moduli space T of the stack T provided by the firstpart of the main result. This might be singular, but by Hironaka’s theorem we may replace it bya resolution of singularities such that D ∞ : = T r T is a simple normal crossings divisor. Thuscondition (i) is satisfied.For each component D i ⊂ D ∞ denote by m i the ramification index of T → T . In particular anycovering W → T whose ramification indices over D i are divisible by m i lifts along the generic pointof D i to T .Choosing a Kawamata covering package [AK00] disjoint from Z we obtain a simple normal cross-ings divisor D as required by (ii), and finite covering W → T as required by (iii), such that W → T factors through T at every generic point of D i . By the Purity Lemma [AV02, Lemma 2.4.1] the morphism W → T extends over all of W , hencewe obtain a family S W → W as required by (iv). (cid:3) References [AH11] Dan Abramovich and Brendan Hassett. ‘Stable varieties with a twist.’ In ‘Classificationof algebraic varieties,’ EMS Ser. Congr. Rep., 1–38, (Eur. Math. Soc., Zürich2011). URL http://dx.doi.org/10.4171/007-1/1 . ↑ Invent. Math. (2000) vol.139 (2): 241–273. URL https://doi.org/10.1007/s002229900024 . ↑
5, 31[Ara04] Donu Arapura. ‘Frobenius amplitude and strong vanishing theorems for vector bundles.’
Duke Math. J. (2004)vol. 121 (2): 231–267. URL http://dx.doi.org/10.1215/S0012-7094-04-12122-0 . Withan appendix by Dennis S. Keeler. ↑ J. Amer. Math. Soc. (2002)vol. 15 (1): 27–75. URL http://dx.doi.org/10.1090/S0894-0347-01-00380-0 . ↑ arXiv e-prints (2018) arXiv:1809.02398. ↑ Geometric and Functional Analysis (2019) URL http://dx.doi.org/10.1007/s00039-019-00496-2 . ↑ Compos. Math. (2013) vol. 149 (12): 2036–2070. URL http://dx.doi.org/10.1112/S0010437X13007288 . ↑ Duke Math. J. (2008) vol. 145 (2): 341–378. URL https://doi.org/10.1215/00127094-2008-054 . ↑ arXiv e-prints (2010) arXiv:1002.4145. ↑ arXiv e-prints (2017) arXiv:1704.02279. ↑
3, 4, 18[Bro17] Damian Brotbek. ‘On the hyperbolicity of general hypersurfaces.’
Publ. Math. Inst. Hautes Études Sci. (2017)vol. 126: 1–34. URL https://doi.org/10.1007/s10240-017-0090-3 . ↑ Ann. Sci. Éc. Norm. Supér.(4) (2019) vol. 52 (4): 1137–1154. ↑ arXiv e-prints (2019) arXiv:1905.04212. ↑ arXiv e-prints (2017) arXiv:1709.08065. ↑
10, 11[CKS86] Eduardo Cattani, Aroldo Kaplan, and Wilfried Schmid. ‘Degeneration of Hodge structures.’
Ann. of Math. (2) (1986) vol. 123 (3): 457–535. URL https://doi.org/10.2307/1971333 . ↑
9, 10[CP15] Frédéric Campana and Mihai Păun. ‘Orbifold generic semi-positivity: an application to families ofcanonically polarized manifolds.’
Ann. Inst. Fourier (Grenoble) (2015) vol. 65 (2): 835–861. URL http://aif.cedram.org/item?id=AIF_2015__65_2_835_0 . ↑ ———. ‘Positivity properties of the bundle of logarithmic tensors on compact Kähler manifolds.’ Compos. Math. (2016) vol. 152 (11): 2350–2370. URL http://dx.doi.org/10.1112/S0010437X16007442 . ↑ ———. ‘Foliations with positive slopes and birational stability of orbifold cotangent bundles.’ Publ. Math., Inst.Hautes Étud. Sci. (2019) vol. 129: 1–49. ↑ Invent. Math. (2017) vol. 207 (1): 345–387. URL https://doi.org/10.1007/s00222-016-0672-6 . ↑ J. Éc. polytech. Math. (2019) vol. 6: 1–18. URL http://dx.doi.org/10.5802/jep.85 . ↑ y—Santa Cruz 1995,’ vol. 62 of Proc. Sympos. Pure Math. , 285–360, (Amer. Math. Soc.,Providence, RI1997). ↑ Jpn. J. Math. (3) (2020)vol. 15 (1): 1–120. ↑ arXiv e-prints (2017) arXiv:1703.07279. To appear in Int. Math. Res. Not. IMRN . ↑ arXiv e-prints (2018) arXiv:1806.01666. ↑ arXiv e-prints (2018) arXiv:1809.05891. ↑ NALYTIC SHAFAREVICH HYPERBOLICITY CONJECTURE 33 [Den19] ———. ‘Hyperbolicity of bases of log Calabi-Yau families.’ arXiv e-prints (2019) arXiv:1901.04423. ↑ arXiv e-prints (2020) arXiv:2001.04426. ↑ arXiv e-prints (2019) arXiv:1911.02973. ↑ Mathematische Annalen (2018) URL http://dx.doi.org/10.1007/s00208-018-1706-8 . ↑ Ann. of Math. Stud. , 29–49, (Princeton Univ. Press,Princeton, NJ1984). ↑
11, 17[Kar00] Kalle Karu. ‘Minimal models and boundedness of stable varieties.’
J. Algebraic Geom. (2000) vol. 9 (1): 93–109. ↑ Publ. Res. Inst. Math.Sci. (1985) vol. 21 (4): 853–875. URL http://dx.doi.org/10.2977/prims/1195178935 . ↑ J. Reine Angew. Math. (1985) vol. 363: 1–46. URL http://dx.doi.org/10.1515/crll.1985.363.1 . ↑ Invent. Math. (2008) vol. 172 (3): 657–682. URL http://dx.doi.org/10.1007/s00222-008-0128-8 . ↑ Adv. Math. (2008) vol. 218 (3): 649–652. URL http://dx.doi.org/10.1016/j.aim.2008.01.005 . ↑ Duke Math. J. (2010) vol. 155 (1): 1–33. URL http://dx.doi.org/10.1215/00127094-2010-049 . ↑ Hyperbolic complex spaces , vol. 318 of
Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences] , (Springer-Verlag, Berlin1998). URL https://doi.org/10.1007/978-3-662-03582-5 . ↑
13, 14[KP17] Sándor J. Kovács and Zsolt Patakfalvi. ‘Projectivity of the moduli space of stable log-varieties andsubadditivity of log-Kodaira dimension.’
J. Amer. Math. Soc. (2017) vol. 30 (4): 959–1021. URL http://dx.doi.org/10.1090/jams/871 . ↑ Invent.Math. (1988) vol. 91 (2): 299–338. URL http://dx.doi.org/10.1007/BF01389370 . ↑ Bull. London Math. Soc. (2004) vol. 36 (2): 188–192. URL http://dx.doi.org/10.1112/S0024609303002728 . ↑ Number theory. III , vol. 60 of
Encyclopaedia of Mathematical Sciences , (Springer-Verlag, Berlin1991).URL http://dx.doi.org/10.1007/978-3-642-58227-1 . Diophantine geometry. ↑ Proc. Sympos. Pure Math. , 269–331, (Amer. Math. Soc., Providence, RI1987). ↑ Adv. Math. (2012) vol. 229 (3):1640–1642. URL http://dx.doi.org/10.1016/j.aim.2011.12.013 . ↑ Math. Ann. (1984) vol. 268 (1): 1–19. URL http://dx.doi.org/10.1007/BF01463870 . ↑ Invent. Math. (2017) vol. 208 (3): 677–713. URL https://doi.org/10.1007/s00222-016-0698-9 . ↑
2, 3, 5[PT18] Mihai Păun and Shigeharu Takayama. ‘Positivity of twisted relative pluricanonicalbundles and their direct images.’
J. Algebraic Geom. (2018) vol. 27 (2): 211–272. URL http://dx.doi.org/10.1090/jag/702 . ↑ Algebra and Number Theory (2019) vol. 13 (9): 2205–2242. URL http://dx.doi.org/10.2140/ant.2019.13.2205 . ↑
3, 4, 5, 6, 7, 8, 9, 12, 27, 29, 30[Rou16] Erwan Rousseau. ‘Hyperbolicity, automorphic forms and Siegel modular varieties.’
Ann. Sci. Éc. Norm. Supér.(4) (2016) vol. 49 (1): 249–255. URL http://dx.doi.org/10.24033/asens.2281 . ↑ Proceedings of the International Congress of Mathem-aticians (Vancouver, B. C., 1974), Vol. 2 (1975) 217–221. ↑ Invent. Math. (1973)vol. 22: 211–319. URL https://doi.org/10.1007/BF01389674 . ↑
8, 9, 10[Sch12] Georg Schumacher. ‘Positivity of relative canonical bundles and applications.’
Invent. Math. (2012) vol. 190 (1):1–56. URL https://doi.org/10.1007/s00222-012-0374-7 . ↑
3, 4, 18[Sch18] ———. ‘Moduli of canonically polarized manifolds, higher order Kodaira-Spencer maps, and an ana-logy to Calabi-Yau manifolds.’ In ‘Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds and Picard-Fuchs equations.’, 369–399, (Somerville, MA: International Press; Beijing: Higher EducationPress2018). ↑
3, 4, 18, 19[Schn17] Christian Schnell. ‘On a theorem of Campana and Păun.’
Épijournal de Géométrie Algébrique (2017) vol. Volume1. URL https://epiga.episciences.org/3871 . ↑ ↑ Invent. Math. (2015)vol. 202 (3): 1069–1166. URL http://dx.doi.org/10.1007/s00222-015-0584-x . ↑ ↑ Ann. of Math. (2) (2015) vol. 181 (2): 547–586. URL https://doi.org/10.4007/annals.2015.181.2.3 . ↑
3, 4, 18, 20[TY18] ———. ‘Augmented Weil-Petersson metrics on moduli spaces of polarized Ricci-flatKähler manifolds and orbifolds.’
Asian J. Math. (2018) vol. 22 (4): 705–727. URL http://dx.doi.org/10.4310/AJM.2018.v22.n4.a6 . ↑ Invent. Math. (1990) vol. 101 (1):191–223. URL https://doi.org/10.1007/BF01231501 . ↑
23, 25[Vie95] ———.
Quasi-projective moduli for polarized manifolds , vol. 30 of
Ergebnisse der Mathematik und ihrerGrenzgebiete (3) [Results in Mathematics and Related Areas (3)] , (Springer-Verlag, Berlin1995). URL https://doi.org/10.1007/978-3-642-79745-3 . ↑
2, 21, 29[VZ01] Eckart Viehweg and Kang Zuo. ‘On the isotriviality of families of projective manifolds over curves.’
J. Algeb-raic Geom. (2001) vol. 10 (4): 781–799. ↑ ↑
2, 5, 6, 7, 8, 12, 22[VZ03] ———. ‘On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds.’
Duke Math. J. (2003)vol. 118 (1): 103–150. URL https://doi.org/10.1215/S0012-7094-03-11815-3 . ↑
2, 3, 4,5, 6, 7, 8, 9, 10, 12, 22, 24[Wol86] Scott A. Wolpert. ‘Chern forms and the Riemann tensor for the moduli space of curves.’
Invent. Math. (1986)vol. 85 (1): 119–145. URL http://dx.doi.org/10.1007/BF01388794 . ↑ J. Math. Soc. Japan (2019) vol. 71 (1): 259–298. URL http://dx.doi.org/10.2969/jmsj/75817581 . ↑ Ann. of Math. Stud. , 121–141, (Princeton Univ. Press, Princeton,NJ1984). ↑ AsianJ. Math. (2000) vol. 4 (1): 279–301. URL https://doi.org/10.4310/AJM.2000.v4.n1.a17 .Kodaira’s issue. ↑ Université de Strasbourg, Institut de Recherche Mathématiqe Avancée, 7 Rue René-Descartes, 67084Strasbourg, France
Current address : Institut des Hautes Études Scientifiques, Université Paris-Saclay, 35 route de Chartres, 91440, Bures-sur-Yvette, France
E-mail address : [email protected] URL : Department of Mathematics, Box 1917, Brown University, Providence, RI, 02912, U.S.A
E-mail address : [email protected] URL ::