On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties

For smooth families with maximal variation, whose general fibers have semi-ample canonical bundle, the generalized Viehweg hyperbolicity conjecture states that the base spaces of such families are of log general type. This deep conjecture was recently proved by Popa-Schnell using the theory of Hodge modules and a theorem by Campana-P\u{a}un. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbolicity of moduli spaces of polarized manifolds with semi-ample canonical bundle. This proves a conjecture by Viehweg-Zuo in 2003. We also prove the Kobayashi hyperbolicity of base spaces of effectively parametrized families of minimal projective manifolds of general type. This generalizes previous work by To-Yeung, in which they further assumed that these families are canonically polarized.

this with the aforementioned theorem of Campana-Păun, they proved that such base space V is of log general type. Therefore, Theorem B is predicted by a famous conjecture of Lang (cf. [Lan91, Chapter VIII. Conjecture 1.4]), which stipulates that a complex quasi-projective manifold is pseudo Kobayashi hyperbolic if and only if it is of log general type. To our knowledge, Lang's conjecture is by now known for the trivial case of curves, for general hypersurface X in the complex projective space CP n of high degrees [Bro17,Dem20,Siu15] as well as their complements CP n \ X [BD19], for projective manifolds whose universal cover carries a bounded strictly plurisubharmonic function [BD18], for quotients of bounded (symmetric) domains [Rou16, CRT19,CDG19], and for subvarieties on abelian varieties [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 canonically polarized manifolds. Combining the approaches by  with those by Popa-Schnell [PS17], very recently, Popa-Taji-Wu [PTW19, Theorem 1.1] proved Theorem A for moduli spaces of polarized manifolds with big and semi-ample canonical bundles. As we will see below, our work owes a lot to the general strategies and techniques in their work [VZ03, PTW19].
The Kobayashi hyperbolicity of moduli spaces M of compact Riemann surfaces of genus 2 has long been known to us by the work of Royden and Wolpert [Roy75,Wol86]. The first important breakthrough on higher dimensional generalizations was made by To-Yeung [TY15], in which they proved Kobayashi hyperbolicity of the base V considered in Theorem C when the canonical bundle K U of each fiber U := f −1 U ( ) 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 is to study the curvature of the generalized Weil-Petersson metric for families of canonically polarized manifolds, 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 base once this family is assumed to be effectively parametrized.
Recently, Lu, Sun, Zuo and the author [DLSZ19] proved a big Picard type theorem for moduli spaces of polarized manifolds with semi-ample canonical sheaf. A crucial step of the proof relies on the "generic local Torelli-type theorem" in Theorem D. Theorem D also inspired us a lot in our more recent work [Den20] on the big Picard theorem for varieties admitting variation of Hodge structures. 0.3. Strategy of the proof. For the smooth family f U : U → V of canonically polarized manifolds with maximal variation, Viehweg-Zuo [VZ03] constructed certain negatively twisted Higgs bundles (which we call Viehweg-Zuo Higgs bundles in Definition 1.1) (Ẽ ,θ) := ( n q=0 L −1 ⊗E n−q,q , n q=0 1 ⊗ θ n−q,q ), over some smooth projective compactification Y of a certain birational modelṼ of V , where L is some big and nef line bundle on Y , and n q=0 E n−q,q , n q=0 θ n−q,q is a Higgs bundle induced by a polarized variation of Hodge structure defined over a Zariski open set ofṼ . In a recent paper [PTW19], Popa-Taji-Wu introduced several new inputs to develop Viehweg-Zuo's strategy in [VZ03], which enables them to construct those Higgs bundles on base spaces of smooth families whose geometric generic fiber admits a good minimal model (see also Theorem 1.2 for a weaker statement 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 := f −1 U ( ) of the smooth family f U : U → V considered in Theorem B have ample 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 (Ẽ ,θ ) with log poles contained in the divisor D := Y \Ṽ gives rise to a morphism τ γ ,k : T ⊗k C → γ * (L −1 ⊗ E n−k,k ) (0.3.1) for any entire curve γ : C →Ṽ . If γ : C →Ṽ is Zariski dense, by the Kodaira-Nakano vanishing (when K U is ample) and Bogomolov-Sommese vanishing theorems (when K U is big and nef), one can verify that τ γ ,1 (C) 0. Hence there is some m > 0 (depending on γ ) so that τ γ ,m factors through γ * (L −1 ⊗ 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−1,m+1 ⊗ Ω Y (log D). Applying Zuo's theorem [Zuo00] on the negativity of N n−m,m , a certain positively curved metric for L can produce a singular hermitian metric on T C with the Gaussian curvature bounded from 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 Kobayashi pseudo distance of the base V . Moreover, the use of vanishing theorem cannot show τ γ ,1 (C) 0 when fibers of 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 modelṼ of the base V , whose holomorphic sectional curvature is bounded above by a negative constant (say negatively curved Finsler metric in Definition 2.3.(ii)). A bimeromorphic criteria for pseudo Kobayashi hyperbolicity in Lemma 2.4 states that, 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-Higgs sheaf (F , η) ⊆ (Ẽ ,θ ) with logarithmic poles only along the boundary divisor D := Y \Ṽ , one can define a morphism for any k = 1, . . . , n: 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 onẼ k := L −1 ⊗ E n−k,k induced by the Hodge metric as 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 and Berndtsson-Păun-Wang [Sch12, Sch18, TY15, BPW17] on the curvature computations of generalized Weil-Petersson metric for families of canonically polarized manifolds, we define a convex sum of Finsler metrics on T Y (− log D), to offset the unwanted positive terms in the curvature ΘẼ k by negative contributions from the ΘẼ k +1 (the last order term was ΘẼ n is always semi-negative by the Griffiths curvature formula). We proved in Proposition 2.14 that for proper α 1 , . . . , α n > 0, the holomorphic sectional curvature 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 modelṼ of the base space V were constructed by Popa-Taji-Wu in their elaborate work [PTW19]. Let Y be some smooth projective compactificationṼ with simple normal crossing boundary D := Y \Ṽ . By our construction of negatively curved Finsler metric F defined in (0.3.3) via VZ Higgs bundles, to show that F is positively definite over some Zariski open set, it suffices to prove that τ 1 : is generically injective (which we call generic local Torelli for VZ Higgs bundles in § 1.1). This was proved in Theorem D, by using the degeneration of Hodge metric and the curvature properties of Hodge bundles. In particular, we show that the generic injectivity of τ 1 is indeed an intrinsic feature of all VZ Higgs bundles (not related to the Kodaira dimension of fibers of f !). By a standard inductive argument in [VZ03, PTW19], one can easily show that Theorem B implies Theorem A. Now we will explain the strategy to prove Theorem C. Note that the VZ Higgs bundles are only constructed over some birational modelṼ of V , which is not Kobayashi hyperbolic in general. This motivates 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 hyperbolicity of the base V in Theorem C, it suffices to show that (♠) for any given point on the base V , there exists a VZ Higgs bundle (Ẽ ,θ ) constructed over some birational model ν :Ṽ → V , such that ν −1 : V Ṽ is defined at .
(♣) The negatively curved Finsler metric F onṼ defined in (0.3.3) induced by the above VZ Higgs bundle (Ẽ ,θ ) is positively definite at the point ν −1 ( ). Roughly speaking, the idea is to produce an abundant supply of fine VZ Higgs bundles to construct sufficiently many negatively curved Finsler metrics, which are obstructions to the degeneracy of Kobayashi pseudo distance d V of V . This is much more demanding than the Brody hyperbolicity and Viewheg hyperbolicity of V , which can be shown by the existence of only one VZ Higgs bundle on 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 blowup the base for several times (indeed twice). Recall that the basic setup in [VZ03, PTW19] is the following: after passing to some smooth birational model fŨ : so that there exists (at least) one hypersurface which is 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.Ũ r ) is the r -fold fiber product of f U : U → V (resp. fŨ :Ũ →Ṽ ). The VZ Higgs bundle is indeed the logarithmic Higgs bundles associated to the Hodge filtration of an auxiliary variation of polarized Hodge structures constructed by taking the middle dimensional relative 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 semistable reduction by Abramovich-Karu [AK00] so that, after changing the birational model U → V by performing certain (uncontrollable) base changeŨ := U × VṼ →Ṽ , one can find a "good" compactification X → Y ofŨ r →Ṽ and a finite dominant morphism W → Y from a smooth projective 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 us and 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 also find a "good" compactification X → Y of U r → V without passing the birational modelsṼ → V as in (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 has to 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 smooth family U r → V so that for some sufficiently ample line bundle A over Y , Indeed, for any given point ∈ V , by ( * ) one can find H transverse to the fiber X := f −1 ( ), and thus the new family Z H → Y will be smooth over an open set containing . To the bests of our knowledge, ( * ) was only known to us when the moduli is canonically polarized [VZ02, Proposition 3.4]. § 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 produce different "fine" VZ Higgs bundles. For any given point ∈ V , by (♠), one can take a birational model ν :Ṽ → V so that ν is isomorphic at , and there exists a VZ Higgs bundle (Ẽ ,θ) on the normal crossing compactification Y ⊃Ṽ . To prove that the induced negatively curved Finsler metric F is positively definite at˜ := ν −1 ( ), by our definition of F in (0.3.3), it suffices to show that τ 1 defined in (0.3.2) is injective at˜ in the sense of C-linear map between complex vector spaces As we will see in § 3.4, when H in (0.3.5) is properly chosen (indeed being transverse to the fiber X ) which is ensured by ( * ), φ˜ is injective at˜ . Hence τ 1,˜ is injective by our assumption of effective parametrization (hence ρ˜ is injective) in Theorem C. This is our strategy to prove Theorem C. slight improvements. It owes a lot to the celebrated work [VZ02,VZ03,PTW19], to which I express my gratitudes. I would like to sincerely thank Professors Dan Abramovich, Sébastien Boucksom, Håkan Samuelsson Kalm, Kalle Karu, Mihai Păun, Mihnea Popa, Georg Schumacher, Jörg Winkelmann, Chenyang Xu, Xiaokui Yang, Kang Zuo, and Olivier Benoists, Junyan Cao, Chen Jiang, Ruiran Sun, Lei Wu, Jian Xiao for answering my questions and very fruitful discussions. I thank in particular Junyan Cao and Lei Wu for their careful reading of the early draft of the paper and numerous suggestions. I am particularly grateful to Professor Dan Abramovich for suggesting the Q-mild reduction, and writing Appendix A which provides a crucial step for the present paper. I thank Professors 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 paper completely.

N .
Throughout this article we will work over the complex number field C. • An algebraic fiber space 2 (or fibration for short) f : X → Y is a surjective projective morphism between 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 projective morphism with connected fibers between quasi-projective varieties. • For any surjective morphism Y ′ → Y , and the algebraic fiber space f : X → Y , we denote by (X × Y Y ′ )~the (unique) irreducible component (say the main component) of X × Y Y ′ which dominates 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 −1 (X reg ) → X reg is an isomorphism. Here X reg denotes to be the smooth locus of X . • For any birational morphism µ : X ′ → X , the exceptional locus is the inverse image of the smallest closed 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 main component of X r dominating Y , and X (r ) a strong desingularization of (X r )~.

B
To begin with, let us introduce the definition of Viehweg-Zuo Higgs bundles over quasi-projective manifolds in an abstract way following [VZ03, PTW19]. Then we prove a generic local Torelli for VZ Higgs 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. Definition 1.1 (Abstract Viehweg-Zuo Higgs bundles). Let V be a quasi-projective manifold, and let Y ⊃ V be a projective compactification of V with the boundary D := Y \ V simple normal crossing. A Viehweg-Zuo Higgs bundle on V is a logarithmic Higgs bundle (Ẽ ,θ ) over Y consisting of 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 ) := n q=0 E n−q,q , n q=0 θ n−q,q induced by the lower canonical extension of a polarized VHS defined over Y \ (D ∪ S), (iv) a sub-Higgs sheaf (F , η) ⊂ (Ẽ ,θ), which satisfy the following properties.
(3) WriteẼ k := L −1 ⊗ E n−k,k , and denote by F k :=Ẽ k ∩ F . Then the first stage F 0 of F is an effective line bundle. In other words, there exists a non-trivial morphism O Y → F 0 .
As shown in [VZ02], by iterating η for k-times, we obtain Since η ∧ η = 0, the above morphism factors through F k ⊗ Sym k Ω Y (log D), and by (3) one thus obtains . Equivalently, we have a morphism It was proven in [VZ02, Corollary 4.5] that τ 1 is always non-trivial. We say that a VZ Higgs bundle satisfies the generic local Torelli if τ 1 : T Y (− log D) → L −1 ⊗ E n−1,1 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 Higgs bundles.
1.2. A quick tour on Viehweg-Zuo's construction. For the smooth family U → V in Theorems A and B, it was shown in [VZ02] and [PTW19, Proposition 2.7] that there is a VZ Higgs bundle over some birational modelṼ 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 variational smooth families whose geometric generic fiber admits a good minimal model. Since we need to study the precise loci where τ 1 is injective in the proof of Theorem C, in this subsection we recollect Viehweg-Zuo's construction on VZ Higgs bundles over the base space V (up to a birational model and 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 us mention 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 birational modelṼ , there is a smooth compactification Y ⊃Ṽ and a VZ Higgs bundle overṼ .
Proof. By [VZ03, PTW19], one can take a birational morphism ν :Ṽ → V and a smooth compactification f : X → Y of U r × VṼ →Ṽ so that there exists a hypersurface with L a big and nef line bundle over Y satisfying that Here we denote by ∆ := f −1 (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 codimension two 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 −1 S are relative normal crossing. Set L := Ω n X /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 out of δ * H. Let Z to be a strong desingularization of this covering, which is smooth over V 0 by (2). We denote the compositions by h : W → Y and : Z → Y , whose restrictions to V 0 are both smooth. Write Π := −1 (S ∪ D) which can be assumed to be normal crossing. Leaving out codimension two subvariety supported D + S further, we assume that h and are also flat, and both δ * (H + ∆ + Σ) and Π are relative normal crossing. Set It was shown in [VZ02, §4] that there exists a natural edge morphism which gives rise to the first Higgs bundle n q=0 F n−q,q , n q=0 τ n−q,q defined over a big open set of Y containing V 0 .
Write Z 0 := Z \ Π. Then the local system R n * C ↾Z 0 extends to a locally free sheaf V on Y (here Y is projective rather than the big open set!) equipped with the logarithmic connection whose eigenvalues of the residues lie in [0, 1) (the so-called lower canonical extension). By Schmid's nilpotent orbit theorem [Sch73], the Hodge filtration of R n * C ↾Z 0 extends to a filtration V := F 0 ⊃ F 1 ⊃ · · · ⊃ F n of subbundles so that their graded sheaves E n−q,q := F n−q /F n−q+1 are also locally free, and there exists θ n−q,q : This defines the second Higgs bundle n q=0 E n−q,q , θ n−q,q . As observed in [VZ02,VZ03], E n−q,q = R q * Ω n−q Z /Y (log Π) over a big open set of Y by the theorem of Steenbrink [Ste77,Zuc84]. By the construction of the cyclic cover Z , this in turn implies the following commutative diagram over a big open set of Y : Note that all the objects are defined on a big open set of Y except for n q=0 E n−q,q , θ n−q,q , which are defined on the whole Y . Following [VZ03, §6], for every q = 0, . . . , n, we define F n−q,q to be the reflexive hull, and the morphisms τ n−q,q and ρ n−q,q extend naturally.
To conclude that n q=0 L −1 ⊗ E n−q,q , n q=0 1 ⊗ θ n−q,q is a VZ Higgs bundle as in Definition 1.1, we have to introduce a sub-Higgs sheaf with log poles supported on D. Writeθ n−q,q := 1 ⊗ θ n−q,q for short. Following [VZ02, Corollary 4.5] (cf. also [PTW19]), for each q = 0, . . . , n, we define a coherent torsion-free sheaf F q := ρ n−q,q (F n−q,q ) ⊂ E n−q,q . By , and let us by η q the restriction ofθ n−q,q to F q . Then (F , η) := n q=0 F q , n q=0 η q is a sub-Higgs bundle of (Ẽ ,θ ) := n q=0 L −1 ⊗ E n−q,q , n q=0θ n−q,q . 1.3. Proper metrics for logarithmic Higgs bundles. We adopt the same notations as Definition 1.1 in the rest of § 1. As is well-known, E can be endowed with the Hodge metric h induced by the 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 unipotent monodromies) and [PTW19, §3] (for quasi-unipotent monodromies), one can take a proper singular metric α on L such that the induced singular hermitian metric −1 α ⊗ h onẼ := L −1 ⊗ E is locally bounded from above. Before we summarize the above-mentioned results in [PTW19, §3], we introduce some notations in loc. cit.
Write the simple normal crossing divisor D = D 1 + · · · + D k and Let be a singular hermitian metric with analytic singularities of the big and nef line bundle L such that is smooth on The following proposition is a slight variant of [PTW19, Lemma 3.1, Corollary 3.4].   . Let H = F 0 ⊃ F 1 ⊃ · · · ⊃ F N ⊃ 0 be a variation of Hodge structures defined over (∆ * ) p × ∆ q , where ∆ (resp. ∆ * ) is the (resp. punctured) unit disk. Consider the lower canonical extension l F • over ∆ p+q ⊃ (∆ * ) p × ∆ q , and denote by (E , θ ) the associated Higgs bundle. Then for any holomorphic section s ∈ Γ(U , E ), where U ∆ p+q is a relatively compact open set containing the origin, one has the following norm estimate where α is some positive constant independent of s, and t = (t 1 , . . . , t p+q ) denotes to be the coordinates of ∆ p+q .
Let us mention that the estimates of Hodge metric for upper canonical extension were obtained by Peters [Pet84] in one variable, and by Catanese-Kawamata [CK17] in several variables, based on the work [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 1 (∆ * ) p × ∆ q is generated by elements γ 1 , . . . , γ p , where γ j may be identified with the counter-clockwise generator of the fundamental group of the jth copy of ∆ * in (∆ * ) p . Set T j to be the monodromy transformation with respect to γ j , which pairwise commute and are known to be quasi-unipotent; that is, for any multivalued section (t 1 , . . . , t p+q ) of H , one has and [T j ,T k ] = 0 for any j, k = 1, . . . , p. Set T j = D j ·U j to be the (unique) Jordan-Chevally decomposition, so that D j diagonalizable and U j is unipotent with [D j , U j ] = 0. SinceT j is quasi-unipotent by the theorem of Borel, all the eigenvalues of D j are thus the roots of unity. Set Fix a point t 0 ∈ (∆ * ) p × ∆ q , and take a basis 1 , . . . , r ∈ V t 0 so that S 1 , . . . , S p are simultaneously diagonal, that is, one has Let us define 1 (t), . . . , r (t) to be the induced multivalued flat sections. Then is single-valued and forms a basis of holomorphic sections for the lower canonical extension l H .
Recall that d jℓ are all roots of unity. One thus can take a positive integer m so that m jℓ := −mλ jℓ /2πi are all non-negative integers. Equivalently, each T m j is unipotent. Define a ramified cover 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 ′ * j which are multivalued sections for the local system π ′ * H . Then which forms a basis of c π ′ * H . Based on the work of [Sch73,CKS86], it was shown in [VZ03, Claim 7.8] that one has the upper bound of norms for some positive constants C 0 and α. One the other hand, we have By the definition of lower canonical extension, m ij are all non-negative integers, and thus Hence This shows the estimate (1.3.1). Remark 1.6. For the Hodge metric of upper canonical extension, one makes the choice that λ jℓ ∈ [0, 2πi) instead of λ jℓ ∈ (−2πi, 0] in the proof of Theorem 1.5. Then the same computation as above can easily show that which were obtained in [CK17].
1.4. A generic local Torelli for VZ Higgs bundle. In this section we prove that the generic local Torelli holds for any VZ Higgs bundle, which is a crucial step in the proofs of Theorems A and B.
Remark 1.7. Viehweg-Zuo [VZ02] showed that τ 1 : Proof of Theorem A. By the stratified arguments of Viehweg-Zuo [VZ03], it suffices to prove that there 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 modelṼ of V is known to us by Theorem 1.2. Letγ : C →Ṽ is the lift of γ which is also Zariski dense. In [VZ03, PTW19], the authors proved that the restriction of τ 1 defined in (1.1.1) on C, say τ 1 | C : T C →γ * (L −1 ⊗ E n−1,1 ), has to vanish identically, or else, they can construct a pseudo hermitian metric on C with strictly negative Gaussian curvature, which violates the Ahlfors-Schwarz lemma. By Theorem D, this cannot happen sinceγ : C →Ṽ is Zariski dense. The theorem is proved.

P K
In this section we first establish an algorithm to construct Finsler metrics whose holomorphic sectional curvatures are bounded above by a negative constant via VZ Higgs bundles. By our construction and generic local Torelli Theorem D, those Finsler metrics are positively definite over a Zariski open set, and by the Ahlfors-Schwarz lemma, we prove that a quasi-projective manifold is pseudo 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 will denote 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 3 on E is a real non-negative continuous function F : for any a ∈ C and ∈ E . The Finsler metricF is positively definite at some subset S ⊂ X if for any x ∈ S and any non-zero vector ∈ E x , F ( ) > 0.
When 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 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 : for every pair of points p, q ∈ X , where the infimum is taken over all differentiable curves ℓ : [0, 1] → 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) > 0 for every pair of distinct points p, q ∈ X not both contained in ∆. When ∆ is an empty set, the manifold X is Kobayashi hyperbolic; when ∆ is proper and Zariski closed, the manifold X is pseudo Kobayashi hyperbolic.
By definition it is easy to show that if X is Kobayashi hyperbolic (resp. pseudo Kobayashi hyperbolic), 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 Kobayashi hyperbolicity, no criteria is known for pseudo Kobayashi hyperbolicity of a compact complex space 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 Hermitian pseudo-metric on D where λ(t) is a non-negative continuous function on D. The Gaussian curvature K γ * F 2 of the pseudometric γ * F 2 is defined to be Definition 2.3. Let X be a complex manifold endowed with a Finsler metric F .
(i) For any x ∈ X , and ∈ T X ,x , let [ ] denote the complex line spanned by . We define the holomorphic sectional curvature where the supremum is taken over all γ : D → X such that γ (0) = x and [ ] is tangent to γ ′ (0).
for some nonzero ∈ T X ,x , and the set of such points is denoted by ∆ F .
As mentioned in § 0, our negatively curved Finsler metrics are only constructed on birational models of the base spaces in Theorems B and C, we thus have to establish bimeromorphic criteria for (pseudo) Kobayashi hyperbolicity to prove the main theorems.
Lemma 2.4 (Bimeromorphic criteria for pseudo Kobayashi hyperbolicity). Let µ : X → Y be a bimeromorphic morphism between complex manifolds. If there exists a Finsler metric F on X which is negatively curved in the sense of Definition 2.3.(ii), then X is Kobayashi hyperbolic modulo ∆ F , and Y is Kobayashi hyperbolic modulo µ Ex(µ) ∪ ∆ F , 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 may assume that K F −1. By the Ahlfors-Schwarz lemma, one has F κ X . Let δ F : X × X → [0, +∞[ be the distance function on X defined by F in a similar way as d X : for every pair of points p, q ∈ X , where the infimum is taken over all differentiable curves ℓ : [0, 1] → X joining p to q. Since F is continuous and positively definite over X \ ∆ F , for any p ∈ X \ ∆ F , one has d X (p, q) δ F (p, q) > 0 for any q p, which proves the first statement.
Let us denote by Hol(Y , ) to be the set of holomorphic maps γ : D → Y with γ (0) = . Pick any point ∈ U := Y \ µ Ex(µ) , then there is a unique point x ∈ X with µ(x) = . Hence µ induces a bijection between the sets X is a meromorphic map, so is µ −1 • γ for any γ ∈ Hol(Y , ). Since dim D = 1, the map µ −1 • γ is moreover holomorphic. It follows from (2.1.2) that for any ξ ∈ T X ,x . Hence one has Let G : T U → [0, +∞[ be the Finsler metric on U so that µ * G = F | µ −1 (U ) . Then G is continuous and positively definite over U \ µ(∆ F ), and one has κ Y | U G.
Therefore, for any ∈ Y \ µ ∆ F ∪Ex(µ) , one has d Y ( , z) > 0 for any z , which proves the second statement.
The above criteria can be refined further to show the Kobayashi hyperbolicity of the complex manifold.
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 withX equipped with a negatively curved Finsler metric F such that p µ ∆ F ∪ Ex(µ) . Then X is Kobayashi hyperbolic.
Proof. It suffices to show that d X (p, q) > 0 for every pair of distinct points p, q ∈ X . We take the bimeromorphic morphism µ :X → X in the lemma with respect to p. By Lemma 2.4, X is Kobayashi hyperbolic modulo µ ∆ F ∪ Ex(µ) , which shows that d X (p, q) > 0 for any q p. The lemma follows.
2.2. Curvature formula. Let (Ẽ ,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 metric on V via (Ẽ ,θ ). Our main result is the following. Let us first construct the desired Finsler metric F , and we then proved the curvature property. By  (1.1.1), for each k = 1, . . . , n, there exists and thus the Finsler metric F k induces a continuous Hermitian pseudo-metric on D, denoted by In general, G k (t) may be identically equal to zero for all k. However, if we further assume that γ (D) ∩ V 1 ∅, by the assumption in Theorem 2.6 that the restriction of τ 1 to V 1 is injective, one has G 1 (t) 0. Denote by ∂ t := ∂ ∂t the canonical vector fields in D, and∂ t := ∂ ∂t its conjugate. Set C := γ −1 (V 1 ), and note that D \ C is a discrete set in D.
Lemma 2.7. Assume that G k (t) 0 for some k > 1. Then the Gaussian curvature K k of the continuous pseudo-hermitian metric γ * F 2 k on C satisfies that Proof. For i = 1, . . . , n, let us write e i := τ i dγ (∂ t ) ⊗i , which can be seen as a section of γ * (L −1 ⊗ E n−i,i ). Then by (2.2.2) one observes that Let R k = Θ h α (L −1 ⊗ E n−k,k ) be the curvature form of L −1 ⊗ E n−k,k on V 0 := Y \ D ∪ S induced by the metric h α = −1 α · h defined in Proposition 1.3.(ii), and let D ′ be the (1, 0)-part of the Chern connection D of (L −1 ⊗ E n−k,k , h α ). Then for k = 1, . . . , n, one has Recall that for the logarithmic Higgs bundle ( n k=0 E n−k,k , n k=0 θ n−k,k ), the curvature Θ k on E n−k,k ↾V 0 induced by the Hodge metric h is given by where we recall that θ n−k,k : E n−k,k → E n−k−1,k+1 ⊗ Ω Y log(D + S) . Setθ n−k,k := 1 ⊗ θ n−k,k : L −1 ⊗ E n−k,k → L −1 ⊗ E n−k−1,k+1 ⊗ Ω Y log(D + S) , and one has , , l l whereθ * n−k,k is the adjoint ofθ n−k,k with respect to the metric h α over Y \ D ∪ S. Here we also write ∂ t (resp.∂ t ) for dγ (∂ t ) (resp. dγ (∂ t ) ) abusively. Then over V 0 , we have By the definition of τ k in (1.1.1), for any k = 2, . . . , n one has e k =θ n−k+1,k−1 (∂ t )(e k−1 ), (2.2.8) and we can derive the following curvature formula Putting this into (2.2.6), we obtain (2.2.4).
Remark 2.8. For the final stage E 0,n of the Higgs bundle ( n q=0 E n−q,q , n q=0 θ n−q,q ). We make the convention that G n+1 ≡ 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 We need the following lemma to control the negative term in the above inequality.
Lemma 2.9. When α ≫ 0, there exists a universal constant c > 0, such that for any γ : In particular, Proof. By Proposition 1.3.(ii), it suffices to prove that where τ * 1 (r 2 D · h α ) 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 uniform constant c > 0 such that ω α cτ * 1 (r 2 D · h α ). We thus obtained the desired inequality (2.2.9).
In summary, we have the following curvature estimate for the Finsler metrics F 1 , . . . , F n defined in (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 1 ∅. Assume that G k 0 for k = 1, . . . , q, and G q+1 ≡ 0 (thus G j ≡ 0 for all j > q + 1). Then q 1, and over C := γ −1 (V 1 ), which is a complement of a discrete set in D, one has Here the constant c > 0 does not depend on the choice of γ .

2.3.
Construction of the Finsler metric. By Proposition 2.10, we observe that none of the Finsler metrics F 1 , . . . , 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 of all F 1 , . . . , 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 mainly follow 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.

The lemma follows from Definition 2.3.(i).
For any γ : D → V with C := γ −1 (V 1 ) ∅, we define a Hermitian pseudo-metric σ := √ −1H(t)dt ∧ dt on D by taking convex sum in the following form where G k is defined in (2.2.3), and α 1 , . . . , α n ∈ R + are some universal constants which will be fixed later. Following the similar estimate in [Sch18, Proposition 11], one can choose those constants properly such that the Gaussian curvature K σ of σ is uniformly bounded.
Proposition 2.14. There exists universal constants 0 < α 1 . . . α n and K > 0 (independent of γ : D → V ) such that the Gaussian curvature Proof. It follows from (2.3.4) that By Proposition 2.10, one has One can take α 1 = 1, and choose the further α j > α j−1 inductively such that min j β j > 0. Set Note that α 1 , . . . , α n and K is universal. The lemma is thus proved.
It follows from Proposition 2.14 and (2.1.3) that one has the following estimate Since H(t) ∈[0, +∞[ is continuous (in particular locally bounded from above) over D, log H(t) is a subharmonic 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 where F k is defined in (2.2.2), such that γ * F 2 = √ −1H(t)dt ∧ dt for any γ : D → V . Since we assume that τ 1 is injective over V 0 , the Finsler metric F 1 is positively definite on V 0 , and a fortiori F . Therefore, we finish the proof of Theorem 2.6.

Proof of Theorem B.
Proof of Theorem B. By Theorem 1.2, there is a VZ Higgs bundle over some birational modelṼ of V . By Theorem D and Theorem 2.6, we can associate this VZ Higgs bundle with a negatively curved Finsler metric which is positively definite over some Zariski dense open set ofṼ . The theorem follows directly from the bimeromorphic criteria for pseudo Kobayashi hyperbolicity in Lemma 2.4.
Remark 2.15. Let me mention that Sun and Zuo also have the similar idea in constructing Finsler metric over the base using Viehweg-Zuo Higgs bundles combining with To-Yeung's method [TY15].

K
In this section we will prove Theorem C. We first refine Viehweg-Zuo's result on the positivity of direct images. We then apply this result to take different branch covering in the construction of VZ Higgs bundles to prove the Kobayashi hyperbolicity of the base in Theorem C.
Theorem 3.1 (Pluricanonical L 2 -extension). Let f : X → Y be an algebraic fiber space so that the Kodaira dimension of the general fiber is non-negative. Assume that f is smooth over a dense Zariski open set of Y 0 ⊂ Y so that both B := Y \ Y 0 and f * B are normal crossing. Let L be any pseudo-effective line bundle L on X equipped with a positively curved singular metric h L with algebraic singularities satisfying the following property (i) There exists some regular value z ∈ Y of f , such that for some m ∈ N, all the sections H 0 X z , (mK X + L) ↾X z extends locally near z.
Then for any regular value of f satisfying that (i) all sections H 0 X , mK X + L ↾X extends locally near , (ii) the metric h L↾X is not identically equal to +∞, the following restriction map Here A Y is a universal ample line bundle on Y which does not depend on L, f and m, and where the sum is taken over all prime divisors V j of f * B with multiplicity a j and its image f (V j ) a divisor in Y .
We will apply a technical lemma in [CaoP17,Claim 3.5] to prove Theorem 3.7.(i). Let us first recall some 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 0 (X , L ) ∅. One defines Viehweg showed that one can control the lower bound of e(L ). Lemma 3.4 (Cao-Păun). Let f : X → Y be an algebraic fiber space so that the Kodaira dimension of the general fiber is non-negative. Assume that f is smooth over a dense Zariski open set of Y 0 ⊂ Y so that both B := Y \ Y 0 and f * B are normal crossing. Then there exists some positive integer C 2 so that for any m m 0 and a ∈ N, any ∈ Y 0 and any section σ ∈ H 0 (X , amCK X ), there exists a section 1.4) whose restriction to the fiber X is equal to σ ⊗r m . Here F m and P m are effective divisors on X (independent of a) such that F m is f -exceptional with f (F m ) ⊂ Supp(B), Supp(P m ) ⊂ Supp(∆ f ), r m := rankf * (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).
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 : Z → W be a surjective morphism between quasi-projective manifolds with connected fibers, which is birational to an slc family ′ : Z ′ → W whose generic fiber has at most Gorenstein canonical 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 is birational to (Z × W W ′ )~→ W ′ . (iii) Denote by Z ′r the r -fold fiber product Z ′ × W · · · × W Z ′ . Then ′r : Z ′r → W is also an slc family whose generic fiber has at most Gorenstein canonical singularities. Moreover, Z ′r is birational to the main component (Z r )~of Z r dominating W . (iv) Let Z (r ) be a desingularization of (Z r )~. Then ( (r ) ) * (ℓK Z (r ) /W ) ≃ ( ′r ) * (ℓK Z ′r /W ) is reflexive for every sufficiently divisible ℓ > 0.
3.2. Positivity of direct images. This section is devoted to prove Theorem 3.7 on positivity of direct images, which refines results by Viehweg-Zuo [VZ02, Proposition 3.4] and [VZ03, Proposition 4.3]. It will be crucially used to proved Theorem C.
Theorem 3.7. Let f 0 : X 0 → Y 0 be a smooth family of projective manifolds of general type. Assume that for any ∈ Y 0 , the set of z ∈ Y 0 with X z bir ∼ X is finite.
(i) For any smooth projective compactification f : X → Y of f 0 : X 0 → Y 0 and any sufficiently ample line bundle A Y over Y , f * (ℓK X /Y ) ⋆⋆ ⊗ A −1 Y is globally generated over Y 0 for any ℓ ≫ 0. In particular, f * (ℓK X /Y ) is ample with respect to Y 0 . (ii) In the same setting as (i), det f * (ℓK X /Y ) ⊗ A −r ℓ Y is also globally generated over Y 0 for any ℓ ≫ 0, where r ℓ = rankf * (ℓK X /Y ). In particular, (iii) For some r ≫ 0, there exists an algebraic fiber space f : X → Y compactifying X r 0 → Y 0 , so that f * (ℓK X /Y ) ⊗ A −ℓ Y is globally generated over Y 0 for ℓ large and divisible enough. Here X r 0 denotes to be the r -fold fiber product of X 0 → Y 0 , and A Y is some sufficiently ample line bundle over Y .
Proof. Let us first show that, to prove Claims (i) and (ii), one can assume that both B := Y \ Y 0 and f * B are normal crossing.
For the arbitrary smooth projective compactification f ′ : X ′ → Y ′ of f 0 : X 0 → Y 0 , we take a log resolution ν : Y → Y ′ with centers supported on Y ′ \ Y 0 so that B := ν −1 (Y ′ \ Y 0 ) is a simple normal crossing divisor. Define X to be strong desingularization of the main component ( so that f * B is normal crossing. By [Vie90, Lemma 2.5.a], there is the inclusion which is an isomorphism over Y 0 for each m ∈ N. Hence for any ample line bundle A over Y ′ , once f * (mK X /Y ) ⋆⋆ ⊗ (ν * A) −1 is globally generated over ν −1 (Y 0 ) ≃ Y 0 for some m 0, f ′ * (mK X ′ /Y ′ ) ⋆⋆ ⊗ A −1 will be also globally generated over Y 0 . 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 0 and f * B are normal crossing. It follows from [Vie90, Theorem 5.2] that one can take some b ≫ a ≫ 0, µ ≫ m ≫ 0 and s ≫ 0 such that L := det f * (µmK X /Y ) ⊗a ⊗det f * (mK X /Y ) ⊗b is ample over Y 0 . In other words, B + (L ) ⊂ Supp(B). By the definition of augmented base locus, one can even arrange a, b ≫ 0 such that there exists a singular hermitian metric h 1 of L − 4A Y which is smooth over Y 0 , and the curvature current √ −1Θ h L (L ) ω for some Kähler form ω in Y . Denote by r 1 := rankf * (µmK X /Y ) and r 2 := rankf * (mK X /Y ). It follows from Lemma 3.4 that for any sections there exists effective divisors Σ 1 and Σ 2 such that Write N := amµr 1 C +bmr 2 C, P := P 1 +P 2 and F := F 1 +F 2 . Fix any ∈ Y 0 . Then the effective divisor Σ 1 +Σ 2 induces a singular hermitian metric h 2 for the line bundle L 2 := N K X /Y − f * L +2f * A Y +P +F such that h| X is not identically equal to +∞, and so is the singular hermitian metric h : In particular, when ℓ sufficiently large, the multiplier ideal sheaf J (h 1 ℓ ↾X ) = O X . By Siu's invariance of plurigenera, all the global sections H 0 X , (ℓK X + L 0 ) ↾X ≃ H 0 X , (ℓ + N )K X extends locally, and we thus can apply Theorem 3.1 to obtain the desired surjectivity Recall that Supp(P) ⊂ Supp(∆ f ). Then ℓ f * B P for ℓ ≫ 0, and one has the inclusion of sheaves which is an isomorphism over X 0 . By (3.2.3) this implies that the direct image sheaves f * (ℓK X /Y − f * A Y + F ) are globally generated over some Zariski open set U ⊂ Y 0 containing for ℓ ≫ 0. Since is an arbitrary point in Y 0 , the direct image f * (ℓK X /Y + F ) ⊗ A −1 Y is globally generated over Y 0 for ℓ ≫ 0 by noetherianity. Recall that F is f -exceptional with f (F ) ⊂ Supp(B). Then there is an injection which is an isomorphism over Y 0 . Hence f * (ℓK X /Y ) ⋆⋆ ⊗A −1 Y is also globally generated over Y 0 . Hence f * (ℓK X /Y ) is ample with respect to Y 0 for ℓ ≫ 0. The first claim follows. (ii) The trick to prove the second claim has already appeared in [Den17] in proving a conjecture by Demailly-Peternell-Schneider. We first recall that f * (ℓK X /Y ) is locally free outside a codimension 2 analytic subset of Y . By the proof of Theorem 3.7.(i), for ℓ sufficiently large and divisible, f * (ℓK X /Y + F ) ⊗ A −1 Y is locally free and generated by global sections over Y 0 , where F is some f -exceptional effective divisor. Therefore, its determinant det f * (ℓK X /Y + F ) ⊗ A −r ℓ Y is also globally generated over Y 0 , where r ℓ := rankf * (ℓK X /Y ). Since F is f -exceptional and effective, one has Y is also globally generated over Y 0 . By the very definition of the augmented base locus B + (•) we conclude that The second claim is proved. (iii) We combine the ideas in [VZ03, Proposition 4.1] as well as the pluricanonical extension techniques in Theorem 3.1 to prove the result. By Corollary A.2, there exists a smooth projective compactification Y of Y 0 with B := Y \Y 0 simple normal crossing, a non-singular finite covering ψ : W → Y , and an slc family ′ : Z ′ → W , which extends the family X 0 × Y 0 W . By Lemma 3.6.(iii) for any r ∈ Z >0 , the r -fold fiber product ′r : Z ′r → W is still an slc family, which compactifies the smooth family . Note that Z ′r has canonical singularities. Take a smooth projective compactification f : X → Y of X r 0 → Y 0 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 (X × Y W )~. Then : Z → W is smooth over LetZ be a strong desingularization of Z ′ , which is thus smooth over W 0 := ψ −1 (Y 0 ). For the new family˜ :Z → W , we denote byZ 0 :=˜ −1 (W 0 ). ThenZ 0 → W 0 is also a smooth family, and any fiber of Z w with w ∈ W 0 is a projective manifold of general type. By our assumption in the theorem, for any w ∈ W 0 , the set of w ′ ∈ W 0 withZ w ′ bir ∼Z w is finite as ψ : W → Y is a finite morphism. We thus can apply Theorems 3.7.(i) and 3.7.(ii) to our new family˜ :Z → W .
From now on, we will always assume that ℓ ≫ 0 is sufficiently divisible so that ℓK 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,˜ * (ℓKZ /W ) = ′ * (ℓK Z ′ /W ). It follows from Theorem 3.7.(ii) that, for any ℓ ≫ 0, the line bundle is globally generated over W 0 , where r := rank ′ * (ℓK Z ′ /W ) depending on ℓ. where pr i : Z ′r → Z ′ is the i-th directional projection map. Hence ℓK Z ′r is Cartier as well, and we have By Lemma 3.6.(iv), * (ℓK Z /W ) is reflexive, and we thus have det ′ * (ℓK Z ′ /W ) → r ′ * (ℓK Z ′ /W ) ≃ * (ℓK Z /W ), which induces a natural effective divisor Γ ∈ |ℓK Z /W − * det ′ * (ℓK Z ′ /W )| such that Γ ↾Z w 0 for any (smooth) fiber Z w with w ∈ W 0 . By Lemma 3.3 for any w ∈ W 0 the log canonical threshold For any n ∈ N * , applying Theorem 3.1 to nF we obtain the surjectivity for all w ∈ W 0 . In other words, * CℓnK Z /W ) ⊗ A −(nr −1) W is globally generated over W 0 for any ℓ ≫ 0 and any n 1.
Since K X is big, one thus has r = r ℓ ∼ ℓ d as ℓ → +∞ where d := dim Z w 2 (if the fibers of f are curves, one can take a fiber product to replace the original family). Recall that C is a constant which does not depend on ℓ. One thus can take an a priori ℓ ≫ 0 so that r ≫ Cℓ. In conclusion, for sufficiently large and divisible m, * mK Z /W ) ⊗ A −2m is globally generated over W 0 . Therefore, we have a morphism which is surjective over W 0 . On the other hand, by [Vie90, Lemma 2.5.b], one has the inclusion * mK Z /W ֒→ ψ * f * (mK X /Y ), which is an isomorphism over W 0 . (3.2.7) thus induces a morphism which is surjective over Y 0 . Note that that even if f * (mK X /Y ) is merely a coherent sheaf, the projection formula ψ * ψ * f * (mK X /Y ) = f * (mK X /Y ) ⊗ ψ * O W still holds for ψ is finite (see [Ara04,Lemma 5.7]). The trace map and is thus surjective. Hence (3.2.8) gives rise to a morphism which is surjective over Y 0 . By taking m sufficiently large, we may assume that ψ * O W ⊗ A m Y is generated by its global sections. Then f * (mK X /Y ) ⊗ A −m Y is globally generated over Y 0 . We complete the proof.
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). Then for any given smooth fiber Proof. By Siu's invariance of plurigenera and Grauert-Grothedieck's "cohomology and base change", we know that f * (ℓK X /Y ) ⊗ A −ℓ Y is locally free on Y 0 , and the natural map is an isomorphism for any ∈ Y 0 . Since K X is assumed to be semi-ample, one can take ℓ ≫ 0 so that |ℓK X | is base point free. By the Bertini theorem, one can take a section s ∈ H 0 (X , ℓK X ) whose zero locus is a smooth hypersurface on X . By Theorem 3.7.(iii), one has the surjection Hence there is which extends the section s. In other words, for the zero divisor H = (σ = 0), its restriction to X is smooth. Hence there is a Zariski open neighborhood V 0 ⊃ so that H is smooth over V 0 . The lemma is proved.
Remark 3.9. Note that we do not know how to find a hypersurface H ∈ |ℓK X /Y − ℓ f * A Y | so that its discriminant 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 . 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 |ℓK X ′ /Y ′ − ℓ f ′ * ν * A Y |. We will overcome this problem in Theorem 3.11 at the cost of the appearance of some f ′ -exceptional divisors.
Since the Q-mild reduction in Corollary A.2 holds for any smooth surjective projective morphism with connected fibers and smooth base, it follows from our proof in Theorem 3.7.(iii) and Kawamata's theorem [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 varieties with 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 ≫ 0 and a smooth projective compactification f : X → Y of U r → V so that f * (mK X /Y ) ⊗ A −m is globally generated over some Zariski open subset of V . Here U r → V is the r -fold fiber product of U → V , and A is some ample line bundle on Y .
3.3. Sufficiently many "moving" hypersurfaces. As we have seen in § 1.2 on the construction of VZ Higgs bundles, one has to apply branch cover trick to construct a negatively twisted Hodge bundle on the compactification of the base, which is well-defined outside a simple normal crossing divisor. This means that the hypersurface H ∈ |ℓK X /Y − ℓ f * A Y | in constructing the cyclic cover is smooth over the complement of an SNC divisor of the base. As we discussed in Lemma 3.8, in general we cannot perform a simple blow-up of the base to achieve this. In this subsection we will overcome this difficulty in applying the methods in [PTW19,Proposition 4.4]. It will be our basic setup in constructing refined VZ Higgs bundles in § 3.4.
Theorem 3.11. Let X 0 → Y 0 be a smooth family of minimal projective manifolds of general type over a quasi-projective manifold Y 0 . Suppose that for any ∈ Y 0 , the set of z ∈ Y 0 with X z bir ∼ X is finite. Let Y ⊃ Y 0 be the smooth compactification in Corollary A.2. Fix any 0 ∈ Y 0 and some sufficiently ample line bundle A Y on Y . Then there exist a birational morphism ν : Y ′ → Y and a new algebraic fiber space f ′ : X ′ → Y ′ which is smooth over ν −1 (Y 0 ), so that for any sufficiently large and divisible ℓ, one can find a hypersurface • There exists a reduced divisor S in Y ′ , so that D +S is simple normal crossing, and H → Y ′ is smooth over Y ′ \ D ∪ S. • The exceptional locus Ex(ν) ⊂ Supp(D + S), and 0 ν(D ∪ S).
• The divisor E is effective and f ′ -exceptional with f ′ (E) ⊂ Supp(D + S). Moreover, when X 0 → Y 0 is effectively parametrized over some open set containing 0 , so is the new family X ′ → Y ′ .
Z ′′ := Z ′r × W W ′ → W ′ is still an slc family, which compactifies the smooth family X ′ 0 × Y ′ 0 W ′ → W ′ 0 . Let M ′ be a desingularization of Z ′′ so that it resolves the rational maps to X ′ as well as Z .
By the properties of slc families, µ ′ * ω Z ′′ /W ′ , which induces a natural map Since both Z ′r and Z ′′ have canonical singularities, one has the following natural morphisms * (ℓK Z /W ) ≃ ( ′r ) * (ℓK Z ′r /W ), h ′ * (ℓK M ′ /W ′ ) = ′′ * (ℓK Z ′′ /W ′ ). We can leave out a subvariety of codimension at least two in Y ′ supported on D + S (which thus avoids 0 by our construction) so that ψ ′ : W ′ → Y ′ becomes a flat finite morphism. As discussed at the beginning of the proof, there is also a natural map is also an isomorphism, and thus the restriction of X → Y to V is isomorphic to that of X ′ → Y ′ to ν −1 (V ). Hence by our construction,the restriction of Z ′r → W to ψ −1 (V ) is isomorphic to that of Z ′′ → W ′ to (ν • ψ ′ ) −1 (V ) = (ν ′ • ψ ) −1 (V ). In particular, under the above isomorphism, for the section σ ∈ H 0 Z ′r , ℓK Z ′r /W − ℓ( ′r ) * A W in (3.3.3) with ϒ(σ ) defining H 1 , one has where µ * and ϒ ′ are defined in (3.3.4) and (3.3.5). Denote byH the zero divisor defined by Recall that H 1 is smooth over V , thenH is also smooth over ν −1 (V ).
Note that ϒ ′ (µ * σ ) ∈ H 0 Y ′ , f ′ * (ℓK X ′ /Y ′ )⊗ν * A −ℓ Y is only defined over a big open set of Y ′ containing ν −1 (V ). Hence it extends to a global section where E is an f ′ -exceptional effective divisor with f ′ (E) ⊂ Supp(D + S). Denote by H the hypersurface in X ′ defined by s.
Note that the property of effective parametrization is invariant under fiber product. The theorem follows.
3.4. Kobayashi hyperbolicity of the moduli spaces. In this subsection, for effectively parametrized smooth family of minimal projective manifolds of general type, we refine the Viehweg-Zuo Higgs bundles in Theorem 1.2 so that we can apply Theorem 2.6 and the bimeromorphic criteria for Kobayashi hyperbolicity in Lemma 2.5 to prove Theorem C.
Theorem 3.12. Let U → V be an effectively parametrized smooth family of minimal projective manifolds of general type over the quasi-projective manifold V . Then for any given point ∈ V , there exists a smooth projective compactification Y for a birational model ν :Ṽ → V , and a VZ Higgs bundle (Ẽ ,θ ) ⊃ (F , η) over Y satisfying the following properties: (i) there is a Zariski open set V 0 of V containing so that ν : ν −1 (V 0 ) → V 0 is an isomorphism. (ii) Both D := Y \Ṽ and D + S := Y \ ν −1 (V 0 ) are simple normal crossing divisors in Y . (iii) The Higgs bundle (Ẽ,θ ) has log poles supported on D ∪ S, that is,θ :Ẽ →Ẽ ⊗ log(D + S) . (iv) The morphism induced by the sub-Higgs sheaf (F , η) is injective over V 0 .
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 ∈ V , the set of z ∈ V with X z bir ∼ X 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 to isomorphisms and up to fiberwise numerical equivalence for H . By [Vie95, Section 7.6], there exists a 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 is finite. Note that a projective manifold of general type has finitely many minimal models. Hence the set of z ∈ V ′ with X z bir ∼ X is finite as well. Now we will choose the hypersurface in (1.2.1) carefully so that the cyclic cover construction in Theorem 1.2 can provide the desired refined VZ Higgs bundle. Let Y ′ ⊃ V be the smooth compactification in Corollary A.2. By Theorem 3.11, for any given point ∈ V and any sufficiently ample line 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 satisfying that • the inverse image D := ν −1 (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 smooth over V 0 := Y \ (D ∪ S). • The restriction ν : ν −1 (V 0 ) → V 0 is an isomorphism.
• The given point is contained in V 0 .
• The divisor E is effective and f -exceptional with f (E) ⊂ Supp(D + S).
• The restricted family f −1 (V 0 ) → V 0 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 (Ẽ ,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.
The main result in this appendix is the following: Theorem A.1. Let f 0 : S 0 → T 0 be a projective family of smooth varieties with T 0 quasi-projective.
(i) There are compactifications S 0 ⊂ S and T 0 ⊂ T , with S and T Deligne-Mumford stacks with projective coarse moduli spaces, and a projective morphism f : S → T extending f 0 which is a Kollár family of slc varieties. (ii) Given a finite subset Z ⊂ T 0 there is a projective variety W and finite surjective lci morphism ρ : W → T , unramified over Z , such that ρ −1 T sm = W sm .
Here the notion of Kollár family refers to the condition that the sheaf ω [m] S/T is flat and its formation commutes 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 the pullback S 0 × T 0 W 0 → W 0 of the original family to W 0 := W × T T 0 . This is applied in the present paper, where some mild regularity assumption on T 0 and W is required: Corollary A.2 (Q-mild reduction). Assume further T 0 is smooth. For any given finite subset Z ⊂ T 0 , there exist (i) a compactification T 0 ⊂ T with T a regular projective scheme, (ii) a simple normal crossings divisor D ⊂ T containing T T 0 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 0 × 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 wheneverT 1 → T is a dominant morphism with T 1 having at most Gorenstein canonical singularities, then the total space S 1 = T 1 × S T has canonical singularities. It was shown by Kollár-Shepherd-Barron [KSB88, Theorem 5.1] and Karu [Kar00, Theorem 2.5] that Kollár 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 of stable maps parametrized by T if π is a Kollár family of slc varieties and K U /T is ϕ × π -ample. Note that this condition is very flexible and does not require the fibers to be of general type, although key applications in Theorems 3.7.(iii) and 3.11 require some positivity of the fibers. The existence of an algebraic stack satisfying the valuative criterion for properness was known to Alexeev, and can also be deduced directly from the results of [AH11], which presents it as a global quotient 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 provided in [Kar00, Corollary 1.2].
Proof of Theorem A.1. (i) Let T 0 ⊂ T and S 0 ⊂ S be projective compactifications with π : S → T extending f 0 . The family S 0 → T 0 with the injective morphism ϕ : S 0 → S is a family of stable maps into S, providing a morphism T 0 → M(S) which is in fact injective. Let T be the closure of T 0 . 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 0 is a compactification as needed.
(ii) The existence of W follows from the main result of [KV04].
Proof of Corollary A.2. Consider the coarse moduli space T of the stack T provided by the first part of the main result. This might be singular, but by Hironaka's theorem we may replace it by a resolution of singularities such that D ∞ := T T 0 is a simple normal crossings divisor. Thus condition (i) is satisfied.
For each component D i ⊂ D ∞ denote by m i the ramification index of T → T . In particular any covering W → T whose ramification indices over D i are divisible by m i lifts along the generic point of D i to T .
Choosing a Kawamata covering package [AK00] disjoint from Z we obtain a simple normal crossings 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 .