Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

In this paper, we study various hyperbolicity properties for a quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite \'etale cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ such that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.


Introduction 1.Background
The classical big Picard theorem says that any holomorphic map from the punctured disk ∆ * into P 1 which omits three points can be extended to a holomorphic map ∆ → P 1 , where ∆ denotes the unit disk.Therefore, we introduce a new notion of hyperbolicity which generalizes the big Picard theorem.We say a complex manifold U is quasi-compact Kähler if it is a Zariski open subset of a compact Kähler manifold Y .Such a Y will be called a smooth Kähler compactification of U .Definition 1.1 (Picard hyperbolicity).A quasi-compact Kähler manifold U is called Picard hyperbolic if there is a smooth Kähler compactification Y of U such that any holomorphic map f : ∆ * → U extends to a holomorphic map f : ∆ → Y .
We will prove in Lemma 5.3 that this definition does not depend on the compactification of U .Picard hyperbolic varieties first attracted the author's interest because of the recent interesting work [JK20] by Javanpeykar-Kucharczyk on the algebraicity of analytic maps.In [JK20, Definition 1.1], they introduce a new notion of hyperbolicity: a reduced quasi-projective variety U is Borel hyperbolic if any holomorphic map from a quasi-projective variety to U is algebraic.In [JK20, Corollary 3.11], they prove that a Picard hyperbolic variety is Borel hyperbolic.We refer the readers to [JK20, Section 1] for their motivation on the Borel hyperbolicity.Picard hyperbolic varieties fascinate us further when we realize in Proposition 5.2 that a more general extension theorem is also valid for them: any holomorphic map from ∆ p × (∆ * ) q to the manifold U in Definition 1.1 extends to a meromorphic map from ∆ p+q to Y .
By Borel [Bor72] and Kobayashi-Ochiai [KO71], it has long been known to us that the quotients of bounded symmetric domains by torsion-free arithmetic lattices are hyperbolically embedded into their Baily-Borel compactification, and thus they are Picard hyperbolic (see [Kob98,Theorem 6.1.3]).Period domains, first introduced by Griffiths [Gri68a] and later systematically studied by him in the seminal work [Gri68b,Gri70a,Gri70b], are classifying spaces of polarized Hodge structures and are a generalization of bounded symmetric domains.In [JK20, Section 1.1], Javanpeykar-Kucharczyk conjectured that an algebraic variety U which admits an integral variation of Hodge structures (Z-VHS for short) with quasi-finite period map is Borel hyperbolic.Their conjecture was recently proved in a remarkable work of Bakker-Brunebarbe-Tsimerman [BBT23].The proof is based on the tame geometry for quotients D Γ of period domains D by arithmetic groups Γ containing the monodromy group of the Z-VHS.However, when one studies Picard (iv) The manifold X is algebraically hyperbolic modulo Z if there is an ε > 0 so that for any compact irreducible curve C ⊂ X not contained in Z, one has where g( C) is the genus of the normalization C of C, and deg ω C := C ω.
Note that Definition 1.3(iv) was first introduced in [JX22].It is easy to show that if X is Kobayashi hyperbolic modulo Z (resp.Picard hyperbolic modulo Z), then X is Brody hyperbolic modulo Z.
The second main result of the paper is the following theorem.

Theorem B.
Let U be a quasi-compact Kähler manifold.Assume that there is a C-PVHS over U so that each fiber of the period map is zero-dimensional.Then there are a quasi-projective manifold Ũ and a finite étale cover Ũ → U such that for any projective compactification X of Ũ , (i) an irreducible Zariski closed subvariety of X not contained in D := X − Ũ is of general type; (ii) the variety X is Picard hyperbolic modulo D; (iii) the variety X is Brody hyperbolic modulo D; (iv) the variety X is algebraically hyperbolic modulo D.
By the work of Deligne (see [Mil13,Theorem 7.10]), the quotient of any bounded symmetric domain by a torsion-free lattice always admits a C-PVHS whose period map is immersive everywhere.Theorem B then yields the following.
Corollary C. Let U be the quotient of a bounded symmetric domain by a torsion-free lattice.Then there is a finite étale cover Ũ → U from a quasi-projective manifold Ũ with any projective compactification X of Ũ Picard and algebraically hyperbolic modulo X − Ũ .
Let us stress here that Nadel [Nad89] and Rousseau [Rou16] proved that the variety X in Corollary C is Brody and Kobayashi hyperbolic modulo X − Ũ , and Brunebarbe [Bru20a] and Cadorel [Cad21,Cad22] proved that any Zariski closed subvariety not contained in X − Ũ is of general type.Theorem B thus incorporates their results, but at the cost of loss of effectivity for the level structures (see Remark 6.4) due to the generality of our result in Theorem B.

Main strategy 1.4.1. Negatively curved Finsler metric.
Let Y be a compact Kähler manifold, and let D be a simple normal crossing divisor on Y .Assume that there is a C-PVHS over U := Y − D. In [Gri70a], Griffiths constructed a metrized line bundle on U whose curvature is semipositive and strictly positive at the points where the period map is immersive.Based on the work by Simpson and Mochizuki, in Proposition 3.5, we can extend this Griffiths line bundle over Y to obtain a more refined positivity result. (1)We then construct a special system of log Hodge bundles (E, θ) = (⊕ p+q=m E p,q , ⊕ p+q=m θ p,q ) on the log pair (Y , D) so that some higher-stage E p 0 ,q 0 contains a big line bundle which admits enough local positivity along D. Inspired by our previous work [Den22a] on the proof of Viehweg-Zuo's conjecture on Brody hyperbolicity of moduli of polarized manifolds, in Theorem 3.9 we show that (E, θ) still enjoys a "partially" infinitesimal Torelli property.These results enable us to construct a negatively curved and generically positive definite Finsler metric on T Y (− log D) in a similar vein as [Den22a].
Let us mention that, though we only construct the (possibly degenerate) Finsler metric over T Y (− log D), it follows from (1.1) that we know exactly the behavior of its curvature near the boundary D since ω is a smooth Kähler form over Y .The proof of Theorem A is then based on Theorem 1.4 and some criteria for the big Picard theorem established in [DLS + 19] (see Theorem 5.4).Let us also mention that the Finsler metric constructed in Theorem 1.4 is also crucially used in the proof of Theorem B.

On the hyperbolicity of the compactification.
The proof of Theorem B is based on Theorem 6.1, whose proof is technically involved.It is worthwhile to mention that our proof is quite different from those in [Nad89,Rou16,Bru20a,Cad22].All these proofs relied heavily on the special property of quotients of bounded symmetric domains by torsion-free lattices.They all applied the toroidal compactifications by Mumford to find the desired finite étale cover Ũ → U when U is a quotient of a bounded symmetric domain by a torsion-free lattice.We construct the cover Ũ → U in Theorem B in a subtle way using the residual finiteness of the global monodromy group.We refer the readers to the beginning of Section 6 for the general strategy.

Some new developments
Shortly after this paper appeared on arXiv, Brunebarbe-Brotbek [BB20, Theorem 1.5] proved the Borel hyperbolicity of U in Theorem A under the additional assumption that the local monodromy of the C-PVHS at infinity is unipotent.Moreover, they also obtained a weaker result than Theorem B(ii) in [BB20, Theorem 1.7], in which they showed that for a quasi-projective manifold U admitting a Z-PVHS with quasi-finite period map, there is a finite étale cover Ũ → U so that the projective compactification X of Ũ is Borel hyperbolic modulo X − Ũ .Our proofs are indeed quite different: Brotbek-Brunebarbe's proof is based on their Second Main Theorem using the Griffiths-Schmid metric, which coincides with the curvature form of the Griffiths line bundle.Let us also mention that result similar to Theorem B(i) is also obtained by Brunebarbe in [Bru20b, Theorem 1.1] when the underlying local system of the C-PVHS is defined over Z.
In [CD21] Cadorel and the author generalized Theorems A and B and Corollary C in this paper to quasi-compact Kähler manifolds admitting nilpotent Higgs bundles.More recently, in [CDY22, Theorem 0.1] Cadorel, Yamanoi and the author proved that for any complex quasi-projective normal variety X, if there is a big representation : π 1 (X) → GL N (C) such that the Zariski closure of (π 1 (X)) is a semisimple algebraic group, then there is a proper Zariski closed subset Z X such that We stress here that Theorem A in this paper is applied in [CDY22].

Notation and Conventions
• A log pair (Y , D) consists of a (possibly non-compact) complex manifold and a simple normal crossing divisor D. It will be called a compact Kähler log pair (resp.projective log pair) if Y is a compact Kähler (resp.projective) manifold.
• For a big line bundle L on a projective manifold, B + (L) denotes its augmented base locus (see [Laz04, Definition 10.3.2]).

Systems of Hodge bundles
Following Simpson [Sim88], a complex polarized variation of Hodge structures (C-PVHS) is equivalent to a system of Hodge bundles.Let us recall the definition in this subsection.
Definition 2.1 (Higgs bundle).A Higgs bundle on a complex manifold Y is a pair (E, θ) consisting of a holomorphic vector bundle E on Y and an O Y -linear map is the adjoint of θ with respect to h.Definition 2.3 (System of Hodge bundles).A system of Hodge bundles of weight m is a harmonic bundle (E, θ, h) satisfying the following: • The vector bundle E = ⊕ p+q=m E p,q is a direct sum of holomorphic vector bundles E p,q .
• The map θ restricts to θ| E p,q : E p,q −→ E p−1,q+1 ⊗ Ω 1 Y .• The splitting E = ⊕ p+q=m E p,q is orthogonal with respect to h.We write h p,q = h| E p,q and θ p,q = θ| E p,q .This harmonic metric h will be called the Hodge metric.
Throughout this paper, we observe the convention that 0 ≤ p, q ≤ m for the decomposition E = ⊕ p+q=m E p,q .This can always be achieved if we make a Tate twist (k, k) to increase the weight by 2k when k ∈ Z >0 is large enough.

Filtered bundles and parabolic Higgs bundles
In this section, we recall the notions of filtered bundles and parabolic Higgs bundles from [Sim88,Moc07].Let (Y , D = c i=1 D i ) be a log pair.Definition 2.4.A filtered bundle (E, P a E) on (Y , D) is a locally free sheaf E on U := Y − D, together with an R c -indexed filtration P a E by locally free sheaves on Y such that (i) a ∈ R c and P a E| U = E; (ii) P a E ⊂ P b E for a ≤ b (i.e., a i ≤ b i for all i); (iii) P a E ⊗ O Y (D i ) = P a+1 i E with 1 i = (0, . . ., 1, . . ., 0) with 1 in the i th component; (iv) P a+ E = P a E for any vector = ( , . . ., ) with 0 < 1; (v) write P <a E = ∪ b<a P b E; the set of weights a such that P a E/P <a E 0 is discrete in R c .
A weight is normalized if it lies in (−1, 0] c .Denote P 0 E by E, where 0 = (0, . . ., 0).Note that the set of weights of (E, P a E) is uniquely determined by the weights lying in (−1, 0] c .Definition 2.5.A parabolic Higgs bundle on (Y , D) is a filtered bundle (E, P a E) together with O Y linear map A natural class of filtered bundles comes from extensions of systems of Hodge bundles, which will be discussed in Section 2.4.• there is a holomorphic isomorphism ϕ : U → ∆ n so that ϕ(D j ) = (z j = 0) for any j = 1, . . ., .We shall write U * := U − D.

Admissible coordinates
Recall that the complete Poincaré metric ω P on (∆ * ) × ∆ n− is described as Note that ω P = dd c ϕ with Remark 2.7 (Global Kähler metric with Poincaré growth).Let (Y , ω) be a compact Kähler manifold, and let D = i=1 D i be a simple normal crossing divisor on Y .Let σ i be the section H 0 (Y , O Y (D i )) defining D i , and pick any smooth metric h i for the line bundle O Y (D i ).One can prove that when ε > 0 is small enough, the closed (1, 1)-current 3) is a Kähler current (i.e., T ≥ δω for some δ > 0), and on any admissible coordinate (U ; z 1 , . . ., z n ), T | U −D is mutually bounded with ω P .

Extension of systems of Hodge bundles
Let (Y , D = i=1 D i ) be log pair.Let (E, h) be a hermitian bundle on Y − D. For any a = (a 1 , . . ., a ) ∈ R , we can prolong E over Y by P h a E as follows: where (U ; z 1 , . . ., z n ) is any admissible coordinate.We still use the notation E in the case a = (0, . . ., 0).In general, P h a E is not coherent.However, by the deep work of Simpson [Sim88, Theorem 3] and Mochizuki [Moc07], this is the case for systems of Hodge bundles.
Theorem 2.8 (Simpson, Mochizuki).If (E = ⊕ p+q=m E p,q , θ, h) is a system of Hodge bundles on Y − D, then (E, P h a E, θ) is a parabolic Higgs bundle on (Y , D).In this case, we write P a E for P h a E to lighten the notation, and denote by θ : P a E −→ P a E ⊗ Ω 1 Y (log D) the prolonged Higgs field by abuse of notation.From Theorem 2.8, one can easily deduce the following.Lemma 2.9.Let (E = ⊕ p+q=m E p,q , θ, h) be as above.
(i) We have P a E = ⊕ p+q=m P a E p,q .Here P a E p,q is the extension of (E p,q , h p,q ).(ii) The map θ restricts to θ| P a E p,q : P a E p,q −→ P a E p−1,q+1 ⊗ Ω 1 Y (log D).Remark 2.10.If (E = ⊕ p+q=m E p,q , θ, h) is a system of Hodge bundles, P a E coincides with the Deligne extension with real part of the eigenvalue in [a, a + 1).See the table in [Sim90,p. 746].
Definition 2.11.Let (Y , D) be a log pair.Let (E = ⊕ p+q=m E p,q , θ, h) be a system of Hodge bundles defined over Y − D. The extension ( E = ⊕ p+q=m E p,q , θ) is called the canonical extension of (E = ⊕ p+q=m E p,q , θ, h).Lemma 2.9 inspires us to introduce the definition of systems of log Hodge bundles.Definition 2.12 (System of log Hodge bundles).Let (Y , D) be a log pair.A system of log Hodge bundles of weight m over (Y , D) consists of a pair (E = ⊕ p+q=m E p,q , θ = ⊕ p+q=m θ p,q ), where • E = ⊕ p+q=m E p,q is a direct sum of holomorphic vector bundles E p,q on Y ; • θ is a direct sum of θ p,q : E p,q −→ E p−1,q+1 ⊗ Ω 1 Y (log D) with θ ∧ θ = 0.

Construction of a special system of log Hodge bundles
In this section, we first study the refined positivity for the Griffiths line bundle associated to a system of Hodge bundles.This positivity is well known when the corresponding C-PVHS has unipotent monodromies near the boundary.We then construct a special system of log Hodge bundles (see Theorem 3.6) over the log pair (Y , D) in Theorem 1.4.Such a system of Hodge bundles will be used to construct a negatively curved Finsler metric in Section 4.

Refined positivity for Griffiths line bundles
For a system of Hodge bundles (E = ⊕ p+q=m E p,q , θ, h) over a complex manifold U , in [Gri70a] Griffiths constructed a line bundle L on U , which can be endowed with a natural metric with semipositive curvature.Precisely, one has Here θ * p,q is the adjoint of θ p,q with respect to h p,q .The Hodge metric h then induces a metric h L on L whose curvature is One can see that √ −1Θ h L (L) > 0 at the point y where θ : T Y ,y → End(E y ) is injective.Note that θ is the differential of the period map (see, e.g., [KKM11,p. 429] for the proof).This means that √ −1Θ h L (L) is strictly positive at the point where the period map is immersive.
Now assume U = Y − D, where (Y , D) is a compact Kähler log pair.Let T be the Kähler current on Y defined in Remark 2.7.Then ω U := T | U is a complete Kähler metric with Poincaré type near D. We recall the following theorem by Simpson [Sim88, Lemma 10.1] and Mochizuki [Moc07].
Theorem 3.1 (Simpson, Mochizuki).Let (E = ⊕ p+q=m E p,q , θ, h) be a system of Hodge bundles on U = Y − D. Then Lemma 3.2.In the notation above, for some constant C > 0. The lemma follows directly from the above inequality.
By Lemma 3.2, the mass of and one can thus apply the Skoda extension theorem (see [Dem97, Theorem 2.3]) so that the trivial extension of over Y is a positive closed (1, 1)-current, which is denoted by S. The current S is therefore less singular than the current T defined in Remark 2.7, which we denote by S T .
Let us consider the extension P 1 L of (L, h L ) defined in (2.4), where 1 = (1, . . ., 1).Then h L can be seen as the singular hermitian metric for P 1 L; this can be seen explicitly from the proof of the next lemma.
Lemma 3.3.The curvature In particular, P 1 L is a pseudo effective line bundle on Y .
Proof.Pick any p ∈ Y .We take an admissible coordinate (W ; z 1 , . . ., z n ) around p as in Definition 2.6.Since S is a closed positive current on Y , over W there is a plurisubharmonic function ψ so that S = dd c ψ.Note that S T .One thus has ϕ ψ, where ϕ is defined in (2.2).For the new metric hL : Let ∇ be the corresponding Chern connection of (L, hL ), which is flat by the relation Θh be the monodromy corresponding to the loop γ i .
Consider the universal covering map where Choose a flat section Φ of the flat line bundle π * (L, ∇).Since (L, hL ) is unitary flat, |Φ|h L is constant, and we may assume that |Φ|h is the monodromy corresponding to the loop γ i ; one has By (3.4), one has Ψ (t 1 , . . ., t , z +1 , . . ., z n ) := Ψ (t 1 , . . ., t i + 1, . . ., t , z +1 , . . ., z n ) for any i = 1, . . ., .It thus descends to a section σ (z) of L| W −D ; i.e., Note that ∇(Φ) = 0; one has Hence Therefore, σ (z) is a holomorphic section trivializing L| W −D .Note that where the second equality follows from the fact that |Φ|h for some N > 0, where the last inequality follows from (2.2).Therefore, The above inequality shows that f extends to a holomorphic function over W . Hence σ is a generator of P 1 L| W .By (3.5), one has where [D i ] is the current of integration associated to D i .This finishes the proof of the theorem.
The following lemma is a consequence of the above proof.Lemma 3.4.For any N ∈ Z >0 , let P 1 (L ⊗N ) be the extension of (L ⊗N , h ⊗N L ) defined in (2.4).Then Proof.We use the same notation as that in the proof of Lemma 3.3.Consider the section σ N , which is a generator of (P 1 L) ⊗N | W .For any section s for any ε > 0. This yields the lemma.
In summary, we have the following positivity result for Griffiths line bundles.Proposition 3.5.Let (Y , D) be a compact Kähler log pair.Let (E = ⊕ p+q=m E p,q , θ, h) be a system of Hodge bundles over Y − D. Assume that its period map is immersive at one point.Then In particular, Y is projective.
Proof.Recall that the closed positive current S is the trivial extension of the semipositive (1, 1)-form Θ h L (L) over Y .By (3.7), one has Lemma 3.4 then yields Note that Therefore, By the discussion at the beginning of this subsection, the semipositive (1, 1)-form Θ h L (L) is strictly positive at the point where the period map is immersive.By Boucksom's criterion [Bou02], the cohomology class {S} is a big (1, 1)-class.Therefore, N {S} − 2D is big for N 1.Note that Since the sum of a big class with an effective class is still big, we conclude that c 1 ( ( This proves the lemma.

Special system of Hodge bundles
Let (Y , D) be a compact Kähler log pair.Let (F = ⊕ p+q=m F p,q , η, h F ) be a system of Hodge bundle over U := Y − D whose period map is immersive at one point.Let us write r p := rank F p,q .Recall that the Griffiths line bundle for (F = ⊕ p+q=m F p,q , η, h F ) is We define a new system of Hodge bundle Precisely, E := F ⊗r , and Then we have θ :

and one can easily check that h = h ⊗r
F is the Hodge metric for (E = ⊕ P +Q=rm E P ,Q , θ).Note that det F p,q = ∧ r p F p,q ⊂ (F p,q ) ⊗r p ⊂ F ⊗r p .Hence where and Q 0 = rm − P 0 .In other words, L ⊗N is a subbundle of E P 0 ,Q 0 .Moreover, their hermitian metrics are compatible in the following sense: h ⊗N L = h| L .By the very definition of the extension (2.4), one has In summary, we construct a special system of log Hodge bundles on (Y , D) as follows (we change the notation for brevity's sake).
Theorem 3.6.Let (Y , D) be a compact Kähler log pair.Let (F = ⊕ p+q=m F p,q , η, h F ) be a system of Hodge bundles over Y − D whose period map is immersive at one point.Then there is a system of log Hodge bundles (E = ⊕ p+q= E p,q , θ = ⊕ p+q= θ p,q ) on (Y , D) satisfying the following properties: (i) The pair (E, θ) is the canonical extension (in the sense of Definition 2.11 ) of some system of Hodge bundles ( Ẽ, θ, h hod ) defined over Y − D. (ii) There is a big line bundle L over Y such that L ⊂ E p 0 ,q 0 for some p 0 + q 0 = , and L ⊗ O Y (−D) is still big.(iii) If the period map moreover has zero-dimensional fibers, then the augmented base locus satisfies B + (L) ⊂ D.
Remark 3.7.The interested readers can compare the Higgs bundle in Theorem 3.6 with the Viehweg-Zuo Higgs bundle in [VZ02,VZ03] (see also [PTW19]).Loosely speaking, a Viehweg-Zuo Higgs bundle for a log pair (Y , D) is a Higgs bundle (E = ⊕ p+q=m E p,q , θ) over (Y , D + S) induced by some (geometric) Z-PVHS defined over a Zariski open subset of Y − (D ∪ S), where S is another divisor on Y so that D + S is simple normal crossing.The extra data is that there is a sub-Higgs sheaf (F = ⊕ p+q=m F p,q , η) ⊂ (E, θ) such that the first stage F n,0 is a big line bundle, and that we have As we explained in Section 1.4.1, the positivity F n,0 comes in a sophisticated way from Kawamata's big line bundle det f * (mK X/Y ), where f : X → Y is some algebraic fiber space between projective manifolds.For our Higgs bundle (E = ⊕ p+q=m E p,q , θ) over the log pair (Y , D) in Theorem 3.6, the global positivity is the Griffiths line bundle which is contained in some intermediate stage E p 0 ,q 0 of (E = ⊕ p+q=m E p,q , θ).

Iterating Higgs fields
Let (E = ⊕ p+q= E p,q , θ) be the system of log Hodge bundles on a compact Kähler log pair (Y , D) satisfying the two conditions in Theorem 3.6.We apply ideas by Viehweg-Zuo [VZ02,VZ03] to iterate Higgs fields.
Since we have θ : The readers might be worried that all τ k might be trivial, so that the above construction will be meaningless.In the next subsection, we will show that this cannot happen.

An infinitesimal Torelli-type theorem
We begin with the following technical lemma.Proposition 3.8.Let (E = ⊕ p+q= E p,q , θ) be a system of log Hodge bundles on a compact Kähler log pair (Y , D) satisfying the two conditions in Theorem 3.6.Then there is a singular hermitian metric h L with analytic singularities for L such that where T is the Kähler current on Y defined in Remark 2.7; , where h hod is the Hodge metric for the system of Hodge bundles Let h D be the canonical singular hermitian metric for D so that We define a singular hermitian metric on L as follows: The first condition is verified.Note that g −1 vanishes on For any point y ∈ D, we pick an admissible coordinate (W ; z 1 , . . ., z n ) and a frame (e 1 , . . ., e r ) for E| W . Since (E, θ) is the canonical extension of a system of Hodge bundles ( Ẽ, θ, h hod ), by (2.4) one has for all ε > 0. Pick a section e ∈ L(W ) which trivializes L| W .By the definition of h L , one has Hence for the frame (e 1 ⊗ e −1 , . . ., e r ⊗ e −1 ) trivializing E ⊗ L −1 | W , one has vanishes on D when ε > 0 is small enough.The proposition is proved.
The proof is almost the same at that of [Den22a,Theorem D].We provide it here for the sake of completeness.
Proof.The inclusion L ⊂ E p 0 ,q 0 induces a global section s ∈ H 0 (Y , L −1 ⊗ E p 0 ,q 0 ) by Theorem 3.6(ii); this section is generically non-vanishing over U = Y − D. Set which is a non-empty Zariski open subset of U .Since the Hodge metric h hod is a direct sum of metrics h p on E p,q , the metric Let D be the (1, 0)-part of its Chern connection over U 1 and Θ to be its curvature form.Then over U 0 , we have where we set and define θ * p,q to be the adjoint of θp,q with respect to the metric h −1 L • h.Hence over U 1 , one has Substituting (3.12) into (3.13),over U 1 , one has . By Proposition 3.8(ii), one has |s| 2 h (y) = 0 for any y ∈ D ∪ B + (L − D).Therefore, there exists a y 0 ∈ U 0 so that |s| 2 h (y 0 ) |s| 2 h (y) for any y ∈ U 0 .Hence |s| 2 h (y 0 ) > 0, and by (3.11), y 0 ∈ U 1 .Since |s| 2 h is smooth over U 0 , dd c log |s| 2 h is seminegative at y 0 by the maximal principle.By Proposition 3.8(i), √ −1Θ L,h L is strictly positive at y 0 .By (3.14) and the relation |s| 2 h (y 0 ) > 0, we conclude that √ −1 θp 0 ,q 0 (s), θp 0 ,q 0 (s) is strictly positive at y 0 .In particular, for any non-zero ξ ∈ T Y ,y 0 , one has θp 0 ,q 0 (s)(ξ) 0. For k = 1, we write τ k in (3.10) as Then over U , it is defined by τ 1 (ξ) := θp 0 ,q 0 (s)(ξ) and is thus injective at y 0 ∈ U 1 .Hence τ 1 is generically injective.The theorem is thus proved.

Construction of a negatively curved Finsler metric
The aim of this technical section is to prove Theorem 1.4 based on Theorem 3.6.We first give the definition of a Finsler metric.Definition 4.1 (Finsler metric).Let E be a holomorphic vector bundle on a complex manifold X.A Finsler metric on E is a real non-negative continuous function h for any a ∈ C and v ∈ E. The metric h is positive definite on a subset U ⊂ X if h(v) > 0 for any non-zero v ∈ E x and any x ∈ U .
We mention that our definition is a bit different from that in [Kob98, Section 2.3], which requires convexity, and the Finsler metric therein can be upper-semicontinuous.
Let (E = ⊕ p+q= E p,q , θ) be a system of log Hodge bundles on a compact Kähler log pair (Y , D) satisfying the two conditions in Theorem 3.6.We adopt the same notation as that in Theorem 3.6 and Section 3.4 throughout this section.Let us denote by n the largest non-negative number for k so that τ k in (3.10) is not trivial.By Theorem 3.9, n > 0. Following [Den22a, Section 2.3], we construct Finsler metrics F 1 , . . ., F n on T Y (− log D) as follows.By (3.10), for each k = 1, . . ., n, there exists a Then it follows from Proposition 3.8(ii) that the (Finsler) metric h on L −1 ⊗ E p 0 −k,q 0 +k induces a Finsler metric h , where τ k is defined in (3.10).By Theorem 3.9, there is a Zariski open subset U • of U such that U • ∩ B + (L) = ∅ and τ 1 is injective at any point of U • .We now fix any holomorphic map γ : C → U with γ(C) ∩ U • ∅.By Proposition 3.8(ii), the metric h for L −1 ⊗ E is smooth and positive definite over where τ k is defined in (3.10).Note that one has τ k+1 (∂ We thus conclude that G k+1 (t) ≡ 0. Hence there exists an m with 1 ≤ m ≤ n so that the set {k | G k (t) > 0 over C • } = {1, . . ., m} and G (t) ≡ 0 for all = m + 1, . . ., n. From now on, all the computations are made over Using the same computations as those in the proof of [Den22a, Proposition 2.10], we have the following curvature formula.Theorem 4.2.For k = 1, . . ., m, over C • , one has Here we make the convention that G m+1 ≡ 0 and 0 0 = 0. We also write ∂ t (resp.∂t ) for dγ(∂ t ) (resp.dγ( ∂t )) abusively, where dγ is defined in (4.2).
Let us mention that in [Den22a, Equation (2.2.11)], we dropped the term Θ L,h L (∂ t , ∂t ) in (4.5), though it can be easily seen from the proof of [Den22a, Lemma 2.7].
We will follow ideas in [Den22a, Section 2.3] (inspired by [TY15, BPW22, Sch18]) to introduce a new Finsler metric F on T Y (− log D) by taking a convex sum of the form where α 1 , . . ., α n ∈ R + are some constants which will be fixed later.
For the above, for γ : C → U with γ(C) ∩ U • ∅, we write Then where G k is defined in (4.3).Recall that for k = 1, . . ., m, G k (t) > 0 for any t ∈ C • .
We first recall a computational lemma by Schumacher.

Lemma 4.3 ( [Sch18, Lemma 17]
).Let α j and G j be positive real numbers for j = 1, . . ., n.Then Now we are ready to compute the curvature of the Finsler metric F based on Theorem 4.2.Proof.By Theorem 3.9 and the assumption that γ(C) ∩ U • ∅, we have G 1 (t) 0.
We first recall a result in [Den22a, Lemma 2.9]; we write its proof here as it is crucial in what follows.

Claim 4.5.
There is a universal constant c 0 > 0 (i.e., it does not depend on γ) so that Θ L,h L (∂ t , ∂t ) ≥ c 0 G 1 (t) for all t ∈ C.
Proof of Claim 4.5.Indeed, by Proposition 3.8(i), it suffices to prove that for some c 0 > 0, where T is a Kähler current on Y , which is a smooth complete metric over Y − D of Poincaré type.It can be seen as a singular hermitian metric for T Y (− log D).Hence for any admissible coordinate (U ; z 1 , . . ., z n ), one has On the other hand, by Proposition 3.8(ii), one has for some constant ε > 0. Hence one has τ * 1 h T .Since Y is compact, there exists a constant c 0 > 0 such that T c 0 τ * 1 h.Therefore, Hence (4.10) holds for any γ : Substituting (4.4) and (4.5) into (4.11), and observing the convention that 0 0 = 0, we obtain One can take α 1 = 1 and choose the further α j > α j−1 inductively so that over C • .By Proposition 3.8(i), this implies that F is continuous and locally bounded from above over C, by the extension theorem of subharmonic function, (4.13) holds over the whole C. Since c 0 > 0 is a constant which does not depend on γ, so are α 1 , . . ., α n by (4.12).The theorem is thus proved.
As a summary of the results in this subsection, we obtain the following theorem.
Theorem 4.6.Let (E = ⊕ p+q= E p,q , θ) be a system of log Hodge bundles on a compact Kähler log pair (Y , D) satisfying the two conditions in Theorem 3.

Algebraic and Picard hyperbolicity
In Definition 1.3, we have seen the definition of algebraic hyperbolicity for a compact complex manifold X, which was introduced by Demailly in [Dem97, Definition 2.2].He proved in [Dem97, Theorem 2.1] that X is algebraically hyperbolic if it is Kobayashi hyperbolic.The notion of algebraic hyperbolicity was generalized to log pairs by Chen [Che04].Note that 2g( C) − 2 + i(C, D) depends only on the complement C − D. Hence the above notion of hyperbolicity also makes sense for quasi-projective manifolds: we say that a quasi-projective manifold U is algebraically hyperbolic if it has a log compactification (X, D) which is algebraically hyperbolic.
However, unlike Demailly's theorem, it is unclear to us that Kobayashi hyperbolicity or Picard hyperbolicity of X − D will imply algebraic hyperbolicity of (X, D).In [PR07], Pacienza-Rousseau proved that if X − D is hyperbolically embedded into X, the log pair (X, D) (and thus X − D) is algebraically hyperbolic.
Before we prove that Definition 1.1 does not depend on the compactification of U , we will need the following proposition, which is a consequence of the deep extension theorem of meromorphic maps by Siu [Siu75].The meromorphic map in this paper is defined in the sense of Remmert, and we refer the reader to [FG02,p. 243] for the precise definition.Y .This proves the first part of the proposition.To prove the second part, we first apply the Hironaka theorem on resolution of singularities to assume that X − X • is a simple normal crossing divisor on X.Then any point x ∈ X − X • has an open neighborhood Ω x which is isomorphic to ∆ p+q so that X • ∆ p × (∆ * ) q under this isomorphism.The above arguments show that g| Ω x ∩X • extends to a meromorphic map from Ω x to Y , and thus g can be extended to a meromorphic map from X to Y .The proposition is proved.
Let us prove that Definition 1.1 does not depend on the compactification of U .This independence also implies the following result.

Proof of Theorem A
This subsection is devoted to the proof of Theorem A. We first recall the following criteria for Picard hyperbolicity established in [DLS + 19], whose proof is Nevanlinna-theoretic.
for some smooth Kähler metric ω on Y .Then f extends to a holomorphic map f : ∆ → Y .
We will combine Theorem 5.4 with Theorem 1.4 to prove Theorem A.
Proof of Theorem A. By Theorem 1.4, there exist finitely many compact Kähler log pairs {(X i , D i )} i=0,...,N so that the following hold: (1) There are morphisms µ i : X i → Y with µ −1 i (D) = D i so that each µ i : X i → µ i (X i ) is a birational morphism and X 0 = Y with µ 0 = 1.
(2) There are smooth Finsler metrics h i for ) There are smooth Kähler metrics ω i on X i such that for any holomorphic map γ : Let us explain how to construct these log pairs.By the assumption, there is a C-PVHS on Y − D so that each fiber of the period map is zero-dimensional.In particular, the period map is generically immersive.We then apply Theorem 1.4 to construct a Finsler metric on T Y (− log D) which is positive definite over some Zariski open subset U • of U = Y − D with the desired curvature property (4.14).Set X 0 = Y , µ 0 = 1 and For each i, we take a desingularization µ i : X i → Z i so that D i := µ −1 i (D) is a simple normal crossing divisor in X i .We pull back the C-PVHS to U i = X i − D i via µ i .Then its period map is still generically immersive.We then apply Theorem 1.4 to construct the desired Finsler metrics in item (4) for T X i (− log D i ).We iterate this construction, and since at each step the dimension of X i is strictly decreasing, this algorithm stops after finitely many steps.
(i) We will first prove that U is Picard hyperbolic.Fix any holomorphic map f : . By item (4), there is a smooth Kähler metric ω 0 on X 0 such that We now apply Theorem 5.4 to conclude that f extends to a holomorphic map f : By item (5), there exists an I 0 ⊂ {0, . . ., N } so that f (∆ * ) ⊂ µ 0 (U 0 ) − µ 0 (U • 0 ) ⊂ ∪ j∈I 0 µ j (X j ).Since the µ j (X j ) are all irreducible, there exists a k ∈ I 0 so that f By Theorem 5.4 and item (4) again, we conclude that f k extends to a holomorphic map f we apply item (5) to iterate the above arguments, and after finitely many steps, there exists an Denote by ν i : Ci → C i ⊂ X i the normalization of C i , and set ) induces a (non-trivial) hermitian pseudo metric hi := ν * i h i over T Ci (− log P i ).By (5.2), the curvature current Then we conclude that for any reduced and irreducible curve C ⊂ Y with C D, one has where C → C is its normalization.This shows the algebraic hyperbolicity of U .The proof of the theorem is accomplished.
Let us mention that the idea of using Finsler metrics to prove the hyperbolicity in the above theorem was inspired by the work of To-Yeung in [TY15].
Remark 5.5.Let U be a quasi-projective manifold admitting an integral variation of Hodge structures whose period map is quasi-finite.In [JL19, Theorem 4.2], Javanpeykar-Litt proved that U is weakly bounded in the sense of Kovács-Lieblich [KL10, Definition 2.4] (which is weaker than algebraic hyperbolicity).Though not mentioned explicitly, their proof of [JL19, Theorem 4.2] implicitly shows that such a U is also algebraically hyperbolic when the local monodromies of the C-PVHS at infinity are unipotent.Their proof is based on the work [BBT23] as well as the Arakelov-type inequality for Hodge bundles by Peters [Pet00].
We end this section with the following remark.
Remark 5.6.Let (E, θ) be the system of log Hodge bundles on a log pair (Y , D) as that in Theorem 4.6.One can also use the idea by Viehweg-Zuo [VZ02] in constructing their Viehweg-Zuo sheaf (based on the negativity of kernels of Higgs fields by Zuo It follows from our proof of Theorem A that one can also combine Theorem 5.4 with this result, which is only weaker in appearance, to prove Theorem A. The more general result Theorem 1.4 will be used to prove Theorem 6.1(ii) in the next section.

Hyperbolicity for the compactification after a finite étale cover
In this section, we will prove Theorem B and Corollary C. We first prove the following theorem.Theorem 6.1.Let U be a quasi-compact Kähler manifold.Assume that there is a C-PVHS over U whose period map is immersive at one point.Then there are a finite étale cover Ũ → U together with a compact Kähler compactification X of Ũ and a proper Zariski closed subvariety Z X so that (i) the variety X is of general type; (ii) the variety X is Kobayashi hyperbolic modulo Z; (iii) the variety X is Picard hyperbolic modulo Z; (iv) the variety X is algebraically hyperbolic modulo Z.
Let us briefly explain the idea of the proof of Theorem 6.1 and some related results.Let Y be a compact Kähler manifold compactifying U with D := Y − U a simple normal crossing divisor.By Theorem 3.6, there is a special system of log Hodge bundles (E, θ) := (⊕ p+q= E p,q , ⊕ p+q= θ p,q ) on (Y , D) satisfying the properties therein.We divide the proof into four steps.
(1) The first step is devoted to constructing a compact Kähler log pair (X, D) and a generically finite surjective log morphism µ : (X, D) → (Y , D) which is étale over U so that for each irreducible component Di of D, • either ord Di (µ * D) 1, • or the local monodromy of the pull-back C-PVHS over Ũ around Di is trivial.To find this µ, we apply the well-known result that that monodromy group of a C-PVHS is residually finite and use the Cauchy argument principle to show the high ramification over irreducible components of D around which the local monodromies are not trivial.Let us mention that this step is quite different from those in [Nad89,Rou16,Bru20a,Cad22] for the hyperbolicity of compactifications of quotients of bounded symmetric domains by a torsion-free lattice, as they all applied Mumford's work on toroidal compactifications of quotients of bounded symmetric domains [Mum77] so that ord Di (µ * D) 1 for all Di .In general, we are not sure that such a covering can be found in our case.
(2) The second step is to construct a new system of log Hodge bundles (G = ⊕ p+q= G p,q , η) over (X, D) which is the canonical extension of the pull-back of the C-PVHS via µ.This system of log Hodge bundles (G = ⊕ p+q= G p,q , η) on (X, D) satisfies the two conditions in Theorem 3.6.Moreover, some G p 0 , −p 0 contains L ⊗ O X ( D X ) with L a big line bundle.Here D X is the sum of irreducible components of D around which the local monodromies of the pull-back C-PVHS are not trivial (hence µ is highly ramified over D X ).Note that (G, η) has singularities along D X instead of D since the pull-back C-PVHS extends across the components where the local monodromies are trivial (see (6.4).) (3) The third step is to prove Theorem 6.1(i).We start with G p 0 , −p 0 and iterate the Higgs field η, ending at finitely many steps.By the negativity of the kernel of θ, L ⊗ O( D X ) ⊂ G p 0 , −p 0 , and (6.4), we can construct an ample sheaf contained in some symmetric differential Sym β Ω 1 X (rather than on Sym β Ω 1 X (log D)!).It follows from a celebrated theorem of Campana-Pȃun [CP19] that X is of general type.Let us mention that this idea of iterating Higgs fields to their kernels, originally due to Viehweg-Zuo [VZ02], has been used by Brunebarbe in [Bru20a], in which he proved similar results for quotients of bounded symmetric domains by arithmetic groups.There are also some similar results for quotients of bounded domains by Boucksom-Diverio [BD21] and Cadorel-Diverio-Guenancia [CDG19].(4) The last step is to prove Theorem 6.1(ii)-Theorem 6.1(iv).We use the above system of log Hodge bundles (G, η) and ideas in Section 4 to construct a Finsler metric F on T X (rather than T X (− log D)!) due to the extra positivity L ⊗ O( D X ) ⊂ G p 0 , −p 0 .Such a metric F is generically positive and has holomorphic sectional curvature bounded from above by a negative constant by Theorem 4.4.By the Ahlfors-Schwarz lemma, we conclude that X is Kobayashi hyperbolic modulo a proper closed subvariety, and by Theorem 5.4, the Picard hyperbolicity modulo a proper subset of X follows.Let us mention that Rousseau [Rou16] has proved a similar result for hermitian symmetric spaces, which was later refined by Cadorel [Cad22].Their methods use Bergman metrics for bounded symmetric domains instead of Hodge theory.Now we start the detailed proof of Theorem 6.1.
Proof of Theorem 6.1.By Theorem 3.6, there is a system of log Hodge bundles (E, θ)=(⊕ p+q= E p,q , ⊕ p+q= θ p,q ) over (Y , D) satisfying the two conditions therein.In particular, there are a big line bundle L on Y and an inclusion L ⊂ E p 0 , −p 0 for some 0 ≤ p 0 ≤ .Pick m 1 so that L − +1 m D is a big Q-line bundle.
Step 1a.Fix a base point y ∈ U := Y − D. Let us denote by ρ : π 1 (U , y) → GL(r, C) the monodromy representation of the corresponding C-PVHS and denote by Γ := ρ(π 1 (U , y)) its monodromy group, which is a finitely generated linear group, hence residually finite by a theorem of Malcev [Mal40].Let us cover Y by finitely many admissible coordinate systems where S is a finite set, so that Let S ⊂ Γ be the finite subset defined by where m is the integer chosen at the beginning.It follows from the definition of a residually finite group that there is a normal subgroup Γ of Γ with finite index so that S ∩ Γ = {0}.(6.2) Then ρ −1 ( Γ ) is a normal subgroup of π 1 (U , y) with finite index.Let ν : Ũ → U be the finite étale cover of U so that for the induced map of the fundamental group ν * : π 1 ( Ũ , x) → π 1 (U , y), its image is ρ −1 ( Γ ).Here x ∈ Ũ with µ(x) = y.We consider π 1 ( Ũ , x) as a subgroup of π 1 (U , y) of finite index.Since the monodromy representation of the pull-back of the C-PVHS on Ũ is the restriction ρ| π 1 ( Ũ ,x) : π 1 ( Ũ , x) −→ GL(r, C), its monodromy group is thus Γ .
Step 1b.Note that U is quasi-projective.Hence Ũ is also quasi-projective.Let us take a smooth projective compactification X of Ũ with D := X − Ũ simple normal crossing so that ν : Ũ → U extends to a log morphism µ : (X, D) → (Y , D). Write D = n j=1 Dj , where the Dj are irreducible components of D. Claim 6.2.For each j = 1, . . ., n, one has • or the local monodromy group of the pull-back C-PVHS around Dj is trivial.
Pick a path h : [0, 1] → Ũ connecting x and x 0 , which lifts the above path h By (6.1) and (6.2), either ρ (ν * ( γ0 )) = 0, or there is some i ∈ {1, . . ., k} so that n i ≥ m.The first case means that the local monodromy of the pull-back C-PVHS around Dj is trivial.In the latter case, one has The claim is proved.
Step 2. Set D X ⊂ D to be the sum of all Dj so that the local monodromy group of the pull-back C-PVHS around Dj is not trivial.Then by the dichotomy in Claim 6.2, µ * D − mD X is an effective divisor, and the pull-back C-PVHS on Ũ around Di with Di D X is trivial.Note that the pull-back C-PVHS extends to a C-PVHS defined over X − D X .By the second condition in Theorem 3.6(i), (E, θ) is the canonical extension (in the sense of Definition 2.11) of some system of Hodge bundles ( Ẽ = ⊕ p+q= Ẽp,q , θ, h hod ) defined over Y − D. Hence for any admissible coordinate (U ; z 1 , . . ., z d ) and any holomorphic frame (e 1 , . . ., e r )| U for E p,q , one has Here n i := ord (x i =0) (µ * (z 1 • • • z k )).It then follows from the definition of the extension (2.4) that µ * E p,q ⊂ (µ * Ẽp,q ).(6.3)Note that µ * ( Ẽ, θ, h hod ) is still a system of Hodge bundles over Ũ , which corresponds to the pull-back of the given C-PVHS on U .Recall that the pull-back C-PVHS extends to a C-PVHS defined over X − D X .Hence µ * ( Ẽ, θ, h hod ) extends to a system of Hodge bundles over X − D X .
We denote by (G = ⊕ p+q= G p,q , η = ⊕ p+q= η p,q ) the canonical extension (in the sense of Definition 2.11) of µ * ( Ẽ, θ, h hod ) (which is defined over X − D X ) over the log pair (X, D X ), which is thus a system of log Hodge bundles on (X, D X ).In particular, one has η p,q : G p,q −→ G p−1,q+1 ⊗ Ω 1 X (log D X ).(6.4)By Lemma 2.9(i), one has G p,q = (µ * Ẽp,q ).(6.5) Since L is a subsheaf of E p 0 , −p 0 , by (6.3) and (6.5), one has µ * L ⊂ µ * E p 0 ,q 0 ⊂ G p 0 ,q 0 .Recall that µ * D − mD X is an effective divisor and L − +1 m D is a big Q-line bundle.Write L := µ * L − D X .Then L and L − D X are both big line bundles.The above inclusion yields L ⊗ O X ( D X ) ⊂ G p 0 ,q 0 .(6.6) Step 3. Now we iterate η k times as in Section 3.3 to obtain a morphism G p 0 , −p 0 −→ G p 0 −k, −p 0 +k ⊗ Sym k Ω 1 X (log D X ).(6.7) The inclusion (6.6) then induces a morphism κ k : L ⊗ O X ( D X ) −→ G p 0 −k, −p 0 +k ⊗ Sym k Ω 1 X (log D X ).(6.8) Write k 0 for the largest k so that κ k 0 is non-trivial.Then 0 ≤ k 0 ≤ p 0 ≤ .Let us denote by N p the kernel of θ p, −p .Hence κ k 0 admits a factorization κ k 0 : L ⊗ O X ( D X ) −→ N p 0 −k 0 ⊗ Sym k 0 Ω 1 X (log D X ).We first note that k 0 > 0; or else, there is a morphism from the big line bundle L ⊗ O X ( D X ) to N p 0 , whose dual N * p 0 is weakly positive in the sense of Viehweg by [Bru17] (see also [Den22b,Theorem 4.6]).Hence Step 4. Let us prove that X is both pseudo Picard and pseudo Kobayashi hyperbolic.Note that κ k in (6.8) induces a morphism By Theorem 3.9, we know that τ 1 is injective on a Zariski open set Ũ ⊂ Ũ .The morphism τ k induces a morphism τk : Sym k T X −→ Sym k T X (− log D X ) ⊗ O X ( D X ) −→ G p 0 −k, −p 0 +k ⊗ L−1 which coincides with τ k over Ũ .Hence τ1 is also injective over Ũ .By Proposition 3.8, we can take a singular hermitian metric hL for L so that h := h −1 L ⊗ hhod on G ⊗ L−1 is locally bounded on Y and smooth outside D X ∪ B + ( L − D X ), where hhod is the Hodge metric for the system of Hodge bundles (G, η)| X−D X .Moreover, h vanishes on D X ∪ B + ( L − D X ).This metric h on G ⊗ L−1 induces a Finsler metric F k on T X defined as follows: for any e ∈ T X,x , F k (e) := h τk e ⊗k 1 k We apply the same method as in Section 4 to construct a new Finsler metric F on T X by taking a convex sum in the form where α 1 , . . ., α k 0 ∈ R + are certain constants.This Finsler metric F on T X is positive definite over Ũ • := Ũ − B + ( L − D X ) as τ1 is injective over Ũ and h is smooth on Ũ − B + ( L − D X ).Set Z := X \ Ũ • , which is a proper Zariski closed subvariety of X.By Theorem 4.4 one can choose α 1 , . . ., α k 0 ∈ R + properly so that for any γ : C → X with C an open subset of C and γ(C) ∩ Ũ • ∅, one has dd c log |γ (t)| 2 F ≥ γ * ω (6.9) for some fixed smooth Kähler form ω on X.Indeed, it follows from the proof of Theorem 4.4 that there is an open subset C • of C whose complement is a discrete set such that (6.9) holds over C • .By Definition 4.1, |γ (t)| 2 F is continuous and locally bounded from above over C, and by the extension theorem of subharmonic functions, (6.9) holds over the whole unit disk C. Applying Theorem 5.4 to (6.9), we conclude that X is Picard hyperbolic modulo Z. Hence Theorem 6.1(iii) follows.
Let C be an irreducible compact curve in X not contained in Z. Write h C the induced singular hermitian metric for T C by F, where C is the normalization of C. Then by (6.9), one has This proves Theorem 6.1(iv).By Definition 4.1 again, there is an ε > 0 so that ω ≥ εF 2 .Hence (6.9) implies that The proof of Theorem B(iv) is exactly the same as that of Theorem A. We will not repeat the arguments and leave the proof to the interested readers.
Theorem 1.4 ( = Theorem 3.6 + Theorem 4.6).Let Y be a compact Kähler manifold, and let D be a simple normal crossing divisor on Y .Assume that there is a C-PVHS over U := Y − D whose period map is immersive at one point.Then there are a Finsler metric h (see Definition 4.1 ) on T Y (− log D) which is positive definite on a (1) If the local monodromy around D is unipotent, this is well known.dense Zariski open subset U • of Y − D and a smooth Kähler form ω on Y such that for any holomorphic map γ : C → U from an open set C ⊂ C to U , one has and h i is a smooth metric for the line bundle O Y (D i ).Proof.By Theorem 3.6(ii), the line bundle L ⊗ O Y (−D) is big, and thus by [Bou04, Theorem 3.17], we can put a singular hermitian metric g 0 on it with analytic singularities for L ⊗ O Y (−D) such that g 0 is smooth on Y \ B + (L ⊗ O Y (−D)), where B + (L ⊗ O Y (−D)) is the augmented base locus of L ⊗ O Y (−D), and the curvature current satisfies √ −1Θ g 0 (L−D) ω for some smooth Kähler form ω on Y .Take g F k on T Y (− log D) defined as follows: for any e ∈ T Y (− log D) y , F k (e) := h τ k e ⊗k 1 k .(4.1) Let C ⊂ C be any open subset of C. For any holomorphic map γ : C → U := Y − D, one has dγ : T C −→ γ * T U = γ * T Y (− log D). (4.2) We denote by ∂ t := ∂ ∂t the canonical vector field in C ⊂ C, and by ∂t := ∂ ∂ t its conjugate.The Finsler metric F k induces a continuous hermitian pseudo metric on C, defined by

Theorem 4. 4 .
Fix a smooth Kähler metric ω on Y .There exist universal constants 0 < α 1 < . . .< α n and δ > 0 such that for any holomorphic map γ : C → U = Y − D with C an open subset of C and γ(C) ∩ U • ∅, one has dd c log |γ (t)| 2 F ≥ δγ * ω. (4.9) 6. Then there are a Finsler metric h on T Y (− log D) which is positive definite on a dense Zariski open subset U • of U := Y − D and a smooth Kähler form ω on Y such that for any holomorphic map γ : C → U from any open subset C of C with γ(C) ∩ U • ∅, one has dd c log |γ | 2 h ≥ γ * ω. (4.14) Proof of Theorem 1.4.Theorem 3.6 together with Theorem 4.6 imply Theorem 1.4.

Definition 5. 1 (
Algebraic hyperbolicity).Let (X, D) be a compact Kähler log pair.For any reduced irreducible curve C ⊂ X such that C D, we denote by i X (C, D) the number of distinct points in the set ν −1 (D), where ν : C → C is the normalization of C. The log pair (X, D) is algebraically hyperbolic if there is a smooth Kähler metric ω on X such that 2g( C) − 2 + i(C, D) ≥ deg ω C := C ω for all curves C ⊂ X as above.

Proposition 5. 2 .
Let Y • be a Zariski open subset of a compact Kähler manifold Y .Assume that Y • is Picard hyperbolic.Then any holomorphic map f : ∆ p × (∆ * ) q → Y • extends to a meromorphic map f : ∆ p+q Y .In particular, any holomorphic map g from a Zariski open subset X • of a compact complex manifold X to Y • extends to a meromorphic map from X to Y .Proof.By [Siu75, Theorem 1], any meromorphic map from a Zariski open subset Z • of a complex manifold Z to a compact Kähler manifold Y extends to a meromorphic map from Z to Y provided that the codimension of Z − Z • is at least 2. The complement ∆ p × (∆ * ) q in ∆ p+q is a simple normal crossing divisor D. We remove a subvariety Z ⊂ ∆ p+q of codimension at least 2 with D − Z smooth.Then any point x ∈ D − Z has an open neighborhood Ω x ⊂ ∆ p+q − Z which is isomorphic to ∆ p+q−1 × ∆ * .It then suffices to prove the extension theorem for any holomorphic map f : ∆ r × ∆ * → Y • .By the assumption that Y • is Picard hyperbolic, for any z ∈ ∆ r , the holomorphic map f | {z}×∆ * : {z} × ∆ * → Y • can be extended to a holomorphic map from {z} × ∆ to Y .It then follows from [Siu75, p.442, ( * )] that f extends to a meromorphic map f : ∆ r+1 Let U be a Zariski open subset of a compact Kähler manifold Y .If any holomorphic map f : ∆ * → U extends to f : ∆ → Y , then Y is bimeromorphic to any other compact Kähler manifold Y which contains U as a Zariski open set.In particular, f : ∆ * → U also extends to a holomorphic map ∆ → Y .Proof.By blowing up Y − U and Y − U , we can assume that both Y − U and Y − U are simple normal crossing divisors.By the same arguments as those in the proof of Proposition 5.2, the identity map of U extends to meromorphic maps a : Y Y and b : Y Y .Note that a • b| U and b • a| U are identity maps.Hence Y and Y are bimeromorphic.Composing b with f , one obtains the desired extension ∆ → Y of f : ∆ * → U in Y .By Chow's theorem, Proposition 5.2 in particular gives an alternative proof of the fact that a Picard hyperbolic variety is moreover Borel hyperbolic, proven in [JK20, Corollary 3.11].
Theorem 5.4 ( [DLS + 19, Theorem A]).Let Y be a projective manifold, and let D be a simple normal crossing divisor on Y .Let f : ∆ * → Y − D be a holomorphic map.Assume that there is a (possibly degenerate) Finsler metric h of T Y (− log D) such that |f (t)| 2 h 0 and By Theorem 5.4 and item (4) again, f i extends to the origin, and so does f .This proves the Picard hyperbolicity of U = Y − D. (ii) Let us prove the algebraic hyperbolicity of U in as similar vein as [DLS + 19, Proof of Theorem D].Fix any reduced and irreducible curve C ⊂ Y with C D. By the above arguments, there exists an [Zuo00]) to prove a weaker result than Theorem 4.6: for any holomorphic map γ : C → U from any open subset C of C with γ(C) ∩ U • ∅, there exist a Finsler metric h C of T Y (− log D) (depending on C) and a Kähler metric ω C for Y (also depending on C) so that |γ (t)| 2 h 0 and α) i is a clockwise loop around the origin in the i th factor ∆ * .Pick a path h α : [0, 1] → Y − D connecting y α with y, and denote by γ , x d ) | {|x 1 | ≤ ε, x 2 = ζ 2 , . . ., x d = ζ d } ⊂ W so that ν i (x) 0 for any x ∈ S and any i = 1, . . ., k.Let us define a loop e(θ) : [0, 1] → W * := W − D by e(θ) := (εe 2πiθ , ζ 2 , . . ., ζ d ) which is the generator of π 1 (W * , x 0 ), where x 0 ∈ W * is a point with µ(x 0 ) = y α ∈ U * .By Cauchy's argument principle, the winding number of µ p • e(θ) around 0 is n p for p = 1, . . ., k. Hence by the diagram

Definition 2.6 (Admissible
coordinate).Let (Y , D = c i=1 D i ) be log pair.Let p be a point of Y , and let {D j } j=1,..., be the components of D containing p.An admissible coordinate around p is a tuple (U ; z 1 , . . ., z n ; ϕ) (or simply (U ; z 1 , . . ., z n ) if no confusion arises) where • U is an open subset of Y containing p;