Benoît Cadorel

Y A Deng

PICARD HYPERBOLICITY OF MANIFOLDS ADMITTING NILPOTENT HARMONIC BUNDLES

Keywords: 2010 Mathematics Subject Classification. 32H25, 14D07, 32H30, 32Q45 Picard hyperbolicity, nilpotent harmonic bundles, holomorphic sectional curvature, criterion for Picard hyperbolicity, Nevanlinna characteristic function, non-Kähler locus

For a quasi-compact Kähler manifold endowed with a nilpotent harmonic bundle whose Higgs field is injective at one point, we prove that is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic, and is of log general type. Moreover, we prove that there is a finite unramified cover ˜ of from a quasi-projective manifold ˜ so that any projective compactification of ˜ is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic and is of general type. As a byproduct, we establish some criterion of pseudo-Picard hyperbolicity and pseudo-algebraic hyperbolicity for quasi-compact Kähler manifolds.

The notion of Picard hyperbolicity for quasi-compact Kähler manifolds, which was introduced in [START_REF]Algebraicity of analytic maps to a hyperbolic variety[END_REF][START_REF]Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures[END_REF], is motivated by the classical big Picard theorem, which states that a holomorphic map Δ * → P 1 \{0, 1∞} extends as holomorphic map to the whole disk Δ. Complex manifolds sharing this property with P 1 \ {0, 1, ∞} are then said to be Picard hyperbolic. This notion turns out to be an important hyperbolicity property since it implies the algebraicity of analytic maps from quasi-projective manifolds to Picard hyperbolic ones; this was first proven in [START_REF]Algebraicity of analytic maps to a hyperbolic variety[END_REF]. The study of Picard hyperbolicity continues to have interesting developments: see e.g. the work of He-Ru [HR21] where a quantitative version is introduced, or Etesse [Ete20], who introduces a notion of intermediate Picard hyperbolicity, and gives applications to finiteness properties of automorphism groups.

In [START_REF]Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures[END_REF], the second named author proved the Picard hyperbolicity for quasicompact Kähler manifolds admitting a complex polarized variation of Hodge structures (C-PVHS for short) whose period map has zero dimensional fibers. C-PVHS is a subcategory of nilpotent harmonic bundles. Our goal of this paper is to extend the results in [START_REF]Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures[END_REF] to manifolds admitting nilpotent harmonic bundles. The first result is the following.

Theorem A. Let be a quasi-compact Kähler manifold. Assume that there is a nilpotent harmonic bundle ( , , ℎ) over so that :

→ End( ) is injective at one point. Then is pseudo-Picard hyperbolic, pseudo-algebraically hyperbolic, and is of log general type. Moreover, can be equipped with a unique algebraic structure that makes it quasiprojective, and any dominant meromorphic map from another complex quasi-projective manifold to is algebraic.

See Definitions 1.3, 1.4 and 1.6 for definitions of nilpotent harmonic bundles, pseudoalgebraic hyperbolicity and pseudo-Picard hyperbolicity. Using ideas of [START_REF]Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures[END_REF], we can prove a stronger result on the hyperbolicity of compactifications after taking finite unramified cover of , which is the main result of this paper.

Theorem B (⊂Theorem 6.1). Let be a quasi-compact Kähler manifold. Assume that there is a nilpotent harmonic bundle ( , , ℎ) on so that :

→ End( ) is injective at one point. Then there is a finite unramified cover ˜ → from a quasi-projective manifold ˜ so that any smooth projective compactification of ˜ is of general type, pseudoalgebraically hyperbolic and pseudo-Picard hyperbolic.

The proofs of Theorems A and B both rely on some new criterion of pseudo-Picard hyperbolicity and pseudo-algebraic hyperbolicity for quasi-compact Kähler manifolds, which is a novelty of this paper.

Theorem C (⊂Theorem 3.1+Theorem 4.1). Let be a compact Kähler manifold, and let be a simple normal crossing divisor on . Assume that := -is equipped with a pseudo-Kähler metric whose holomorphic sectional curvature is bounded from above by a negative constant -2 , then

(i)
is pseudo-algebraically hyperbolic and pseudo-Picard hyperbolic. (ii) If the (1, 1) cohomology class { } -{ } is big, then is pseudo-Picard hyperbolic and pseudo-algebraically hyperbolic. Here is the closed positive (1, 1)-current on which is the trivial extension of . 0.2. Related works. After the work [START_REF]Algebraicity of analytic maps to a hyperbolic variety[END_REF], Picard hyperbolicity drew a lot of attention over the last years. In [BBT18], the authors proved the algebraicity of analytic maps from a quasi-projective manifold to another one admitting a quasi-finite period map.

In [START_REF]Picard theorems for moduli spaces of polarized varieties[END_REF] the second named author with Lu, Sun and Zuo proved the Picard hyperbolicity for moduli of polarized manifolds with semi-ample canonical sheaf. In [START_REF]Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures[END_REF], the second named author proved Theorems A and B when the nilpotent harmonic bundle ( , , ℎ) is moreover a complex polarized variation of Hodge structures. Similar results were later also obtained by Brotbek-Brunebarbe [START_REF] Deng | Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications[END_REF] and Brunebarbe [START_REF]Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures[END_REF]. Indeed, this paper is strongly inspired by the works [Den20b, [START_REF] Deng | Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications[END_REF][START_REF]Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures[END_REF]: the proof of Theorem 3.1 is inspired by the Second Main Theorem of Brotbek-Brunebarbe [START_REF] Deng | Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications[END_REF], especially by their study of Nevanlinna characteristic function relative to positive currents; the proof of Theorem B follows the same line as that of [Den20b, Theorem B]. Though the methods in [START_REF]Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures[END_REF] and [START_REF] Deng | Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications[END_REF][START_REF]Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures[END_REF] are quite different, a common ingredient is the use of Griffiths line bundle for systems of Hodge bundles, which has a nice global positivity property. For nilpotent harmonic bundles, we do not have such analogous algebraic objects, and thus the methods and results in [Den20b, BB20] cannot be applied directly. A novelty in this paper is the use of a transcendental cohomology big class { } (see Theorem 6.1 for the precise definition) which plays a similar role in the proof as the Griffiths line bundle for complex variation of Hodge structures. This also enables us to simplify previous work [START_REF]Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures[END_REF][START_REF] Deng | Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications[END_REF][START_REF]Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures[END_REF] since we do not use deep results in Hodge theory such as Schmid's nilpotent orbit theorems and Hodge norm estimates etc.

In [START_REF]Picard theorems for moduli spaces of polarized varieties[END_REF] the second named author with Lu, Sun and Zuo obtained the criterion for Picard hyperbolicity in terms of a Finsler metric for (-log ) with a stronger curvature property than the negativity of holomorphic sectional curvature. Indeed, we are not sure that our new criterion for Picard hyperbolicity Theorem 3.1 still holds if is only assumed to a (1, 1)-hermitian form; its global positivity is crucial in the proof.

Theorem 6.1 is a generalization of several earlier work, stemming from the seminal paper of Mumford [Mum77], who proved that given an arithmetic lattice on a bounded symmetric domain, then all compactifications of quotients by sublattices of sufficiently high index are of general type. It was then shown later by the work of Brunebarbe [Bru20a], Rousseau [Rou16], the first named author [Cad18] that these compactification satisfy very strong algebraic or hyperbolicity properties. These results were later extended to varieties supporting variations of Hodge structures in [Den20b, Theorem B] and [START_REF] Deng | Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications[END_REF][START_REF]Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures[END_REF]; Theorem 6.1 can be seen as a generalization of these last results for varieties supporting nilpotent harmonic bundles.

After the completion of this paper, Yohan Brunebarbe informed us that they were able to prove that the quasi-projective manifold in Theorem A is of log general type in an ongoing work with Jeremy Daniel towards the Shafarevich conjecture for open varieties. 0.3. Acknowledgment. The second named author would like to thank Professors Takuro Mochizuki and Carlos Simpson, and Jeremy Daniel for very helpful discussions on the proof of Proposition 6.4. Both the authors would like to thank Yohan Brunebarbe and Damian Brotbek for their remarks and interest on this work.

N

• A complex manifold is called quasi-compact Kähler if it is the Zariski dense open set of a compact Kähler manifold. • For two real functions and on a complex manifold, we write or if ≥ for some constant > 0.

• A compact Kähler log pair (resp. projective log pair) ( , ) consists of a compact Kähler (resp. projective) manifold and a simple normal crossing divisor on . • A map : ( , ˜ ) → ( , ) between compact Kähler log pairs is called a log morphism if : → is a holomorphic map with ˜ ⊂ -1 ( ). • The unit disk is denoted by Δ and Δ * denotes the punctured unit disk.

• For any closed positive (1, 1) current on a compact Kähler manifold, we write { } for its cohomology class. For two cohomology (1, 1) class and , we write ≥ if is pseudo effective. • For a line bundle with a singular hermitian metric ℎ, its curvature current is denoted by Θ ℎ ( ) := -dd c log ℎ, where dd c := 2 ¯ .

T

In this section we first recall the definitions of nilpotent harmonic bundles and algebraic hyperbolicity. We then state and prove some results on Picard hyperbolicity and closed positive (1, 1) currents, which will be used throughout this paper.

Harmonic bundles.

Definition 1.1 (Higgs bundle). A Higgs bundle on a complex manifold is a pair ( , ) consisting of a holomorphic vector bundle on and an O -linear map : → ⊗ Ω 1 so that ∧ = 0. The map is called the Higgs field.

Definition 1.2 (Harmonic bundle). A harmonic bundle ( , , ℎ) consists of a Higgs bundle ( , ) and a hermitian metric ℎ for so that the connection := ℎ + + * ℎ is flat. Here ℎ is the Chern connection of ( , ℎ), and * ℎ ∈ ∞ ( , End( ) ⊗ Ω 0,1 ) is the adjoint of with respect to ℎ. Definition 1.3 (Nilpotent harmonic bundle). A harmonic bundle ( , , ℎ) is called nilpotent if the characteristic polynomial det( -) = rank .

Note that a complex polarized variation of Hodge structures induces a nilpotent harmonic bundle. Definition 1.4 (Pseudo-algebraic hyperbolicity). Let ( , ) be a compact Kähler manifold and let be a simple normal crossing divisor on . For any reduced irreducible curve ⊂ such that ⊄ , we denote by ( , ) the number of distinct points in the set -1 ( ), where : ˜ → is the normalization of . Assume that is Zariski closed proper subset of . If there is > 0 so that

2 ( ˜ ) -2 + ( , ) ≥ deg := ∫
for all reduced irreducible curve ⊂ not contained in ∪ , ( , ) is called algebraically hyperbolic modulo , and pseudo-algebraically hyperbolic. If = , ( , ) is called algebraically hyperbolic.

Note that the number 2 ( ˜ ) -2 + ( , ) depends only on the intersection of with the complement -. Hence the above notion of hyperbolicity also makes sense for quasi-projective manifolds: we say that a quasi-projective manifold is algebraically hyperbolic if it has a log compactification ( , ) which is algebraically hyperbolic.

However, it is unclear to us if Demailly's theorem extends to the non-compact case, i.e. if Kobayashi hyperbolicity, or Picard hyperbolicity, ofwill imply the algebraic hyperbolicity of ( , ). Note that Pacienza-Rousseau [START_REF]On the logarithmic Kobayashi conjecture[END_REF] have proved that ifis hyperbolically embedded into , the log pair ( , ) (and thus -) is algebraically hyperbolic.

Picard hyperbolicity.

Let us first recall the definition of Picard hyperbolicity introduced in [START_REF]Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures[END_REF]. We start with the following definition of admissible coordinate systems which will be used frequently.

Definition 1.5. (Admissible coordinates) Let be an -dimensional complex manifold, and let be a simple normal crossing divisor. Let be a point of , and assume that { } =1,...,ℓ are the components of containing . An admissible coordinate system around is a tuple (Ω; 1 , . . . , ; ) (or simply (Ω; 1 , . . . , ) if no confusion arises) where

• Ω is an open subset of containing .

• is a holomorphic isomorphism : Ω → Δ so that ( ) = ( = 0) for any = 1, . . . , ℓ.

Definition 1.6 (pseudo-Picard hyperbolicity). Let be a quasi-compact Kähler manifold, and let be a smooth Kähler compactification. is called pseudo-Picard hyperbolic if there is a Zariski closed proper subset so that any holomorphic map : Δ * → with (Δ * ) ⊄ extends to a holomorphic map ¯ : Δ → . We also say that is Picard hyperbolic modulo . If = , is simply called Picard hyperbolic.

In [Den20b, Lemma 4.3] we proved that Definition 1.6 does not depend on the compactification of when = . The proof of this statement is based on the deep extension theorem of meromorphic maps by Siu [Siu75], and is also valid when is not empty. Let us now give some interesting properties of pseudo-Picard hyperbolic manifolds, which generalizes [Den20b, Lemma 4.3].

Proposition 1.7. Let , be as in Definition 1.6, and assume that is pseudo-Picard hyperbolic. Let be a compact complex manifold and let be a simple normal crossing divisor on . If there is a meromorphic map : -which is dominant, that is, its image contains a non-empty open set of , then extends to a meromorphic map . In particular, (i) any compact complex manifold containing as a Zariski dense open set is bimeromorphic to . (ii) the pseudo-Picard hyperbolicity of in Definition 1.6 does not depend on the compactification .

Proof. Write := -. To prove that extends to a meromorphic map , it suffices to check that locally around . By [Siu75, Theorem 1], any meromorphic map from a Zariski open set • of a complex manifold to a compact Kähler manifold extends to a meromorphic map from to provided that the codimension of -• is at least 2. It then suffices to consider the extensibility of around smooth points on . Pick any such point ∈ and choose admissible coordinates (Ω; 1 , . . . , ) around so that Ω ∩ = ( 1 = 0). The theorem follows if we can prove that : Δ * × Δ -1 extends to a meromorphic map Δ +1 . Let be the Zariski closed proper subset of as in Definition 1.6. Denote by the subvariety of Δ * ×Δ -1 of codimension at least two so that is not a holomorphic map. Since is assumed to be dominant, there is thus a dense open set ⊂ Δ -1 so that for any ∈ , Δ * × { } ⊄ and (Δ * × { } -) ⊄ . Then the restriction | :Δ * ×{ } : Δ * × { } is well-defined, which is moreover holomorphic. Since is Picard hyperbolic modulo , : Δ * × { } → then extends to Δ × { } → for ∈ . It then follows from [Siu75, p.442, ( * )] that extends to a meromorphic map : Δ . We thus can conclude that : -extends to a meromorphic map . Let ′ be another compact complex manifold containing as a Zariski dense open set. We can apply the Hironaka theorem on resolution of singularities to assume that ′is a simple normal crossing divisor. By the above result, the identity map of extends to a meromorphic map ′ which is thus bimeromorphic. The second statement follows, which also implies the last claim.

1.4. Closed positive (1, 1)-currents. In this subsection we first recall some results concerning closed positive (1, 1)-currents (see [Dem12a]). We then prove Lemma 1.15 which will be crucial in the proofs of our main results.

Definition 1.8 (Pseudo-Kähler metric). Let be a complex manifold. A (1, 1)-form on is called a pseudo-Kähler metric (or pseudo-Kähler form) if = 0, is semipositive, and strictly positive on a Zariski open set of . Definition 1.9. Let ( , ) be a compact Kähler manifold. Let ∈ 1,1 ( , R) be a cohomology (1, 1)-class of . The class is nef if for any > 0 there is a smooth closed (1, 1)-form ∈ so that ≥ -. The class is pseudo-effective if there is a closed positive (1, 1)-current ∈ . is called big if there is a closed positive (1, 1)-current ∈ so that ≥ for some > 0. Such a current will be called a Kähler current.

For two cohomology (1, 1) classes and , we write ≥ ifis a pseudo-effective class.

Boucksom's criterion [Bou02] asserts that a class is big if there is a closed positive current ∈ so that ∫ ( ac ) dimX > 0, where ac denotes the absolutely continuous part of with respect to any smooth measure on .

The non-Kähler locus ( ) of a big class introduced by Boucksom [Bou04] measures how far is from being Kähler. It is the transcendental generalization of the augmented base locus for big line bundles.

Definition 1.10 (non-Kähler locus). Let be a compact Kähler manifold and let be a big class on . The non-Kähler locus ( ) of is

( ) := ∈ Sing( ),
where the intersection ranges over all Kähler currents ∈ , and Sing( ) is the complement of the set of points ∈ such that is smooth around .

Let us quote the following result by Boucksom [Bou04, Theorem 3.17].

Theorem 1.11 (Boucksom). Let be a big class on a compact Kähler manifold. Then its non-Kähler locus ( ) is a proper analytic subvariety. Moreover, there is a Kähler current ∈ with analytic singularities which is smooth outside ( ).

If the class is big and nef, in [CT15] Collins-Tosatti proved the following theorem on the characterization of its non-Kähler locus ( ). It is a transcendental generalization of the Nakamaye theorem.

Theorem 1.12 (Collins-Tosatti). Let ( , ) be a compact Kähler manifold. Let be a big and nef (1, 1) class on . Then

( ) = Null( ) := ∫ dim =0 (1.1)
where the union is taken over all positive dimensional irreducible analytic subvarieties in .

Let us recall the following extension theorem of Skoda (see e.g. [Dem12b, (2.4) Theorem]) which will be used frequently.

Theorem 1.13 (Skoda). Let be a (not necessarily compact) complex manifold and let be a closed analytic subset of . Assume that is a closed positive ( , )-current defined onso that has locally finite mass in a neighborhood of any point of . Then the trivial extension of , denoted by , is also a closed positive ( , )-current on .

Recall that is defined as follows. For any smooth test ( -1, -1)-form , we let

( ) := ∫ - ∧ . (1.2)
In particular, Skoda's theorem implies the following result due to Bishop [Bis64, Theorem 3], which will be used to prove Lemma 3.3.

Theorem 1.14 (Bishop). Let be a (not necessarily compact) complex manifold and let be a closed analytic subset of . Let be an analytic subset of pure dimension of -. Assume that has locally finite volume near . Then the topological closure of is an analytic subset of .

The following result, which is a variant of in [Den20a, Lemma 5.4], will be crucial throughout this paper.

Lemma 1. 15. Let ( , ) be a compact Kähler log pair. Let be a pseudo-Kähler form on := -with holomorphic sectional curvature bounded from above by a negative constant. Then (i) the trivial extension of , denoted by , is a closed positive current; (ii) the cohomology class { } is big and nef; (iii) for any admissible coordinates (Ω; 1 , . . . , ), the local potential of = dd c satisfies

-log ℓ =1 (-log | | 2 ) . (1.3)
Proof. Pick any point ∈ , and choose admissible coordinates (Ω; 1 , . . . , ) centered at so that ∩ Ω = ( 1 • • • ℓ = 0). Since the holomorphic sectional curvature of is bounded from above by a negative constant, by Ahlfors-Schwarz lemma, we can use [Cad16, Proposition 3.1.2], which implies that there is a constant > 0 so that

1 ≤ := ℓ =1 √ -1 ∧ ¯ | | 2 (log | | 2 ) 2 + =ℓ+1 √ -1 ∧ ¯ (1 -| | 2 ) 2 . (1.4)
Consequently, the local mass of is bounded. By Skoda's theorem, its trivial extension is a closed positive current. Since ∫ -> 0, by Boucksom's criterion, { } is big. Since is a closed positive (1, 1)-current, there is a psh function on Ω so that dd c = . Now, by the very definition of trivial extension (1.2), ≤

. Note that

= -dd c log ℓ =1 (-log | | 2 ) • =ℓ+1 (1 -| | 2 ) .
Since -≥ 0, the function

-dd c log ℓ =1 (-log | | 2 ) • =ℓ+1 (1 -| | 2 ) -
is thus a psh function, and as such, it is locally bounded from above. The inequality (1.3) then follows. Therefore, has zero Lelong numbers everywhere. By the regularization theorem for closed positive currents of Demailly (see [Dem92, Corollary 6.4]), the class { } is nef. The lemma is proved.

The following result due to Brunebarbe [Bru20b, Proposition 3.3] will be used to prove Theorem 6.1.(i). For completeness sake, we provide a proof here.

Lemma 1. 16. Let ( , ) be a compact Kähler log-pair, and let be a pseudo-Kähler form on := -so that it has non-positive holomorphic bisectional curvature and holomorphic sectional curvature bounded from above by a negative constant -2 . Then the class + -{ } is pseudo-effective.

Proof. Let 0 be the Zariski open set of so that is strictly positive definite. Since has non-positive holomorphic bisectional curvature and holomorphic sectional curvature bounded from above by a negative constant -2 , one has -Ric( ) ≥ 2 .

Let ℎ + be the singular hermitian metric on + induced by . We will prove that its curvature current Θ ℎ + ( + ) is positive.

Pick any point ∈ , and choose admissible coordinates (Ω; 1 , . . . , ) centered at so that

∩ Ω = ( 1 • • • ℓ = 0). Then for the local frame := log 1 ∧ • • • log ℓ ∧ ℓ+1 ∧ • • • ∧ of + | Ω , one has -:= | | 2 ℎ + = 1 ∧ ¯ 1 ∧ • • • ∧ ∧ ¯ | 1 • • • ℓ | 2 ℓ =1 (-log | | 2 )
where the last inequality follows from (1.4). Hence the local potential of ℎ + = - is always locally bounded. One the other hand, dd c = -1 2 Ric( ) ≥ 0 over the Zariski dense open set 0 . This implies that the curvature current Θ ℎ + ( + ) = dd c is positive everywhere. On the other hand, since is the trivial extension of , one has thus

Θ ℎ + ( + ) ≥ . (1.5)
The lemma follows.

P K

In this section we prove that the nilpotent harmonic bundle on a complex manifold induces a pseudo-Kähler metric with nice curvature properties similar to the case of period domains.

Proposition 2.1. Assume that is a complex manifold that supports a harmonic bundle ( , , ℎ) so that :

→ End( ) is injective at one point. Then admits a pseudo-Kähler metric with non-positive holomorphic bisectional curvature. If ( , , ℎ) is moreover nilpotent, then the holomorphic sectional curvature of is bounded from above by -1 4 rank -1 .

Proof. We define a metric ℎ as the pullback metric of ℎ by the map → End( ). This gives ℎ ( 1 , 2 ) := ( 1 ), ( 2 ) ℎ for any 1 , 2 ∈ . The fundamental (1, 1)-form relative to ℎ can thus be written as = -tr( * ℎ ∧ ), (2.1) which shows that ≥ 0. Since : → End( ) is immersive at one point, is therefore strictly positive at a general point. Moreover,

= -tr( * ℎ ∧ ) = -tr( ℎ * ℎ ∧ ) + tr( * ℎ ∧ ℎ )
where ℎ is the Chern connection of ( , ℎ). Note that ℎ = 0 = ℎ * ℎ , hence = 0. Thus is a pseudo-Kähler form.

Let ∈ so that → End( ) is injective. Pick local coordinates ( 1 , . . . , ) centered at , and set := ( ). Denote by * the adjoint of with respect to ℎ. Write to be the curvature tensor of , and denote by End( ) the curvature tensor of End( ) induced by the harmonic metric ℎ. By the curvature decreasing properties of subbundles, the holomorphic bisectional curvature in the direction and is

¯ ¯ : = ¯ ( ), ℎ ≤ End( ) ¯ ( ), ℎ .
By the flatness of ℎ + + * ℎ , we have

= -[ , * ], so End( ) ¯ ( ) = -[[ , * ], ]. This gives ¯ ¯ ≤ -[[ , * ], ], ℎ = -tr([[ , * ], ] * ) = -tr([ , * ] [ , * ]) = -| [ , * ]| 2 ≤ 0.
We conclude that has non-positive holomorphic bisectional curvature at any point ∈ where is injective. Assume now ( , , ℎ) is moreover nilpotent. Then : → is a nilpotent endomorphism for each and ∈ . Recall that the holomorphic sectional curvature in the direction is defined by

( ) := ¯ ¯ | | 4 ≤ -| [ , * ]| 2 | | 4
Since is nilpotent, by Lemma 2.2 below, one has

| [ , * ]| ≥ | | 2 2 rank -1 . This proves that ( ) ≤ - 1 4 rank -1 .
Since the local coordinate is arbitrary, this proves that the holomorphic sectional curvature of is bounded from above by -1 4 rank -1 . The following lemma of linear algebra was outlined in [Sim92, p. 27]. = 0 ( ≤ ) and such that ⊂ +1 . Applying the standard orthonormalization process, we may assume that the flag ( ) is adapted to a -unitary base of C . Changing the standard base to this new base, we may now assume that is strictly upper triangular, and is the identity.

Write := ( ) 1≤ , ≤ . Denote by [ , * ] := ( ) 1≤ , ≤ . Since is strictly upper triangular, then

= = +1 | | 2 - -1 =1 | 1 | 2 . Set := = +1 | | 2 . Then -1 =1 = | | 2 .
There exists an integer with 1 ≤ ≤ -1 so that

< 1 2 -| | 2 for < and ≥ 1 2 -| | 2 . Note that ≥ --1 =1 . This implies ≥ 1 2 -1 | | 2 . The lemma follows from the fact that | [ , * ]| 2 ≥ | | 2 .

C P

In this section we will establish our criterion for pseudo-Picard hyperbolicity of quasicompact Kähler manifolds. Theorems 3.1.(i) and 3.1.(ii) will be used to prove Theorems A and B respectively. Their proofs are inspired by the Second Main theorem of Brotbek-Brunebarbe [START_REF] Deng | Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications[END_REF] and by [Yam19, Lemma 5.1]. Since we work on Kähler manifolds rather than projective ones, we have to establish the criterion on removable singularities of holomorphic maps from punctured disks into compact Kähler manifold in term of the growth of Nevanlinna characteristic functions (see Lemma 3.3).

Theorem 3.1. be a compact Kähler manifold, and let be a simple normal crossing divisor on . Assume that := -is equipped with a pseudo-Kähler metric whose holomorphic sectional curvature is bounded from above by a negative constant -2 , then

(i)
is Picard hyperbolic modulo the non-Kähler locus ({ }). Here is the closed positive (1, 1)-current on which is the trivial extension of , and its cohomology class { } is big. Moreover,

({ }) ⊂ -{ ∈ | is strictly positive at }. (3.1) (ii) If { } -{ } is a big class, then is Picard hyperbolic modulo ( { } -{ }) ∪ .
Proof. Our first step will be to prove an inequality similar to the Arakelov-Nevanlinna inequality of [BB20, Theorem 4.1] (see (3.5)). The method of using a current with Poincaré singularities to define a first Nevanlinna characteristic function is essentially the same; the arguments can be explained quite shortly in our context so we will recall them for completeness.

For any : Δ * → with (Δ * ) ⊄ , write * = ( ) ∧ ¯ . Since has negative holomorphic sectional curvature, ( ) ∈ 1 loc (Δ * ), and dd c log | ′ | 2 ≥ * (3.2) outside -1 ( ). Indeed, let 0 ∈ Δ * be so that ( 0 ) ∈ . By the Ahlfors-Schwarz lemma, around 0 we have

( ) 1 | -0 | 2 (log | -0 | 2 ) 2 .
If is a local potential for , this shows that log ( ) - * + log | -0 | 2 is locally bounded from above near 0 , and thus extends as a psh function on the whole disk. Applying the -operator, one gets the inequality of (1, 1)-currents

dd c log ( ) ≥ ( ) ∧ ¯ -[ -1 ].
Here -1 is the reduced divisor on Δ * , and [ -1 ] is the associated current. In other words, where

dd c log | ′ | 2 ≥ * -[ -1 ( )]. (3.
Δ 2, := { ∈ Δ * | 2 < | | < }.
By Jensen formula, one has

∫ 2 ∫ Δ 2, dd c log | ′ | 2 = ∫ 2 0 log | ′ ( )| 2 - ∫ 2 0 log | ′ (2 )| 2 (3.4) -2 log 2 ∫ 2 0 log | ′ | (2 ) 2 .
Using concavity of log, we have

∫ 2 0 log | ′ ( )| 2 ≤ 1 2 log ∫ 2 0 | ′ ( )| 2 2 . Note that 1 2 ( , ( )) = ∫ 2 0 | ′ ( )| 2 2 .
Since , ( ) and , ( ) are both monotone increasing functions, we apply Borel's lemma [NW14, Lemma 1.2.1] twice so that, for any > 0 one has log ( , ( )) ≤ (1 + ) log , ( )

= (1 + ) log + (1 + ) log , ( ) ≤ (1 + ) log + (1 + ) 2 log , ( )
Here means that the inequality holds outside a Borel set ⊂ (2, ∞) of finite Lebesgue measure. The above inequalities yield

1 2 log ∫ 2 0 | ′ ( )| 2 2 ≤ (1 + ) 2 2 log , ( ) + 2 log - 1 2 log(2 ) .
Putting this into (3.4), we get

∫ 2 ∫ Δ 2, dd c log | ′ | 2 ≤ (1 + ) 2 2 log , ( ) + 2 log - 1 2 log(2 ) - ∫ 2 0 log | ′ (2 )| 2 -2 log 2 ∫ 2 0 log | ′ | (2 ) 2 .
.

By (3.3), this implies the requested inequality

1 log , ( ) + 2 log + 3 ≥ , ( ) - [1] , ( ) (3.5) for some positive constants 1 , 2 , 3 . Here [1] , ( ) is the truncated counting function defined by

[1] , ( ) := ∫ 2 ∫ Δ 2, [ -1 ( )].
Obviously, it is zero if avoids . Note that in this case, we would have , ( ) ≤ log( ), and the requested extension of follows directly from Lemma 3.3 below if were the restriction of a Kähler form in on . Since is merely pseudo-Kähler and is only defined on , one needs some additional work, which makes use of the condition that is closed.

By Lemma 1.15, the trivial extension of over , denoted by , is a closed positive current. Moreover, the cohomology class { } is big. By Theorem 1.11 there is a Kähler current 2 ∈ { } which is a smooth Kähler form on -({ }). We also choose a smooth closed (1, 1)-form ∈ { }. . Set to be a section 0 ( , O ( )) defining , and pick a smooth metric ℎ for O ( ). Since ∈ { }, there is a quasi-psh function ≤ 0 defined on so that = -dd c . By (1.3), one has

≥ -1 log( ℓ =1 log 2 | • | 2 ℎ )
for some 1 > 0 and > 0. By Jensen's formula, one has

, ( ) -, ( ) = ∫ 2 0 • ( ) 2 - ∫ 2 0 • (2 ) 2 -log 2 ∫ 2 0 • (2 ) 2 ≥ -1 ∫ 2 0 log( ℓ =1 log 2 | • | 2 ℎ ) • ( ) 2 -4 log -5
By the concavity of log, one has

- ∫ 2 0 log( ℓ =1 log 2 | • | 2 •ℎ ) • ( ) 2 ≥ -2 ℓ =1 log ∫ 2 0 (-log | • | 2 ℎ ) • ( ) 2
Using Jensen formula again, one obtains

∫ 2 0 (-log | • | 2 ℎ ) • ( ) 2
≤ ,Θ ℎ ( ) ( ) + (log ).

(3.6) follows from the fact that ,Θ ℎ ( ) ( ) ≤ 2 , ( ) for some positive constants 2 .

Since 2 and are both in { }, there is a quasi-psh function ≤ 0 defined on so that = 2 -dd c . Since 2 is smooth over 0 := -( ), and (Δ * ) ∩ 0 ≠ , * 2 is thus well defined on Δ * . By Jensen formula again, so one has

, ( ) -, 2 ( ) = - ∫ 2 0 • ( ) 2 + ∫ 2 0 • (2 ) 2 -log 2 ∫ 2 0 • (2 ) 2 (3.9) ≥ -7 log -8 .
On the other hand, 2 ≥ for some constant > 0 since 2 is a Kähler current, one has , 2 ( ) ≥ , ( ). This proves (3.7).

In a similar vein as in (3.9), one can prove that , ( ) -, ( ) ≥ -3 log -4 . Since , ( ) ≥ 5 , ( ), (3.8) follows.

Let us prove Theorem 3.1.(i). For any : Δ * → with (Δ * ) ⊄ ({ }), one has

[1]

, ( ) = 0. Putting (3.6) and (3.7) into (3.5), we immediately conclude that , ( ) ∼ log when → ∞. This proves that extends across the point ∞ by Lemma 3.3 below, hence is Picard hyperbolic modulo ({ }). Let us prove (3.1). By Lemma 1.15 { } is big and nef. By Theorem 1.12, one has

({ }) = Null({ }) := ∫ { } dim =0
where the union is taken over all positive dimensional irreducible analytic subvarieties in

. If ⊄ -{ ∈ | is strictly positive at }. by (1.3) one has ∫ { } dim = ∫ reg ∩ dim > 0.
This yields (3.1) by (1.12). (3.1) is proved.

Let us now prove Theorem 3.1.(ii). Since { } -{ } is big, by Theorem 1.11 one can take a Kähler current 3 ∈ { } -{ } which is smooth outside the non-Kähler locus ({ } -{ }). Let : Δ * → be a curve which is not contained in

({ } - { }) ∪ . Since { 3 + [ ]} = { } = { }, similar arguments as (3.9) show that , ( ) -, 3 +[ ] ( ) ≥ -9 log -10 . Moreover, , 3 +[ ] ( ) = , 3 ( ) + ,[ ] ( ) ≥ 11 , ( ) + [1] , ( ).
Combining these inequalities with (3.6), (3.5) and (3.8), we conclude that , ( ) ∼ log . This proves that extends across the point ∞ by Lemma 3.3 below.

We state and prove the following criterion on the extendibility across the origin of the holomorphic map from the punctured disk to a compact Kähler manifold. We now apply Theorem 1.14 to conclude that the closure of in Δ × , denoted by , is an one dimensional closed analytic subset. Hence the map 1 | : → Δ is a proper holomorphic map, which is an isomorphism over Δ * . Therefore, 1 | is moreover an isomorphism. The composition 2 • ( 1 | ) -1 : Δ → is a holomorphic map which extends . The proposition is proved.

Remark 3.4. Note that Lemma 3.3 is a well-known result when is a projective manifold; see e.g. [Dem97b, 2.11. Cas «local »] or [Siu15, Lemma 6.5]. For their strategy of the proof, they use sufficiently many global rational functions on to reduce the theorem to holomorphic maps Δ * → P 1 and then apply Nevanlinna's logarithmic derivative lemma to conclude. Our proof of Lemma 3.3 thus also provides an alternative and simplified proof in the projective setting.

C

In this section we will establish an algebraic analogue to Theorem 3.1.

Theorem 4.1. Let ( , ) be a compact Kähler log pair. Assume that := -is equipped with a pseudo-Kähler metric whose holomorphic sectional curvature is bounded above by a negative constant -2 , then (i) is algebraic hyperbolic modulo ({ }), where is the closed positive (1, 1)current on which is the trivial extension of . (ii) If { } -{ } is a big class, then is algebraically hyperbolic modulo ( { } -{ }) ∪ .

Proof. By Lemma 1.15, we know that { } is big. Let be any irreducible reduced curve not contained in ∪ ({ }). Set : ˜ → to be the normalization. Write ˜ • := -1 ( ), and denote by := -1 ( ) the reduced divisor on ˜ . By (3.1), * is also a pseudo-Kähler metric on ˜ • . Since the holomorphic sectional curvature of is bounded from above by a negative constant -, by the curvature decreasing property, the holomorphic sectional curvature of * is also bounded above by -2 . As in the proof of Lemma 1.16, * induces a singular hermitian metric ℎ ˜ + whose curvature current is positive. Moreover, by (1.5), one has

Θ ℎ ˜ + ( ˜ + ) ≥ *
where * is the closed positive current on ˜ which is the trivial extension of * . By

  Main results.

1. 2 .

 2 Algebraic hyperbolicity. Algebraic hyperbolicity for a compact complex manifold was introduced by Demailly in [Dem97a, Definition 2.2]. He proved in [Dem97a, Theorem 2.1] that a compact complex manifold is algebraically hyperbolic if it is Kobayashi hyperbolic. The notion of algebraic hyperbolicity was generalized to log pairs by Chen [Che04].

Lemma 2. 2 .

 2 Let be a nilpotent × -matrix with values in the complex numbers, and let be an hermitian definite positive matrix of size . Let * := -1 be the adjoint of with respect to . Then | [ , * ]| ≥ 1 2 -1 | | 2 , where | | 2 = 1 2 -1 tr( * ). Proof. Since is nilpotent, there is a strictly decreasing flag C = 0 1 . . .

2 ∫

 2 3) holds over the whole Δ * . We now change our model of the disk into Δ * := { ∈ C | 1 < | | < ∞} by taking ↦ → 1 , and define a Nevanlinna characteristic function , ( ) := ∫ Δ 2, *

Claim 3. 2 .

 2 Fix any smooth Kähler metric over . For any : Δ * → with (Δ * ) ⊄ ({ }), there are positive constants so that , ( ) ≥ , ( ) -4 log , ( ) -4 log -5 (3.6) , ( ) ≥ 6 , ( ) -7 log -8 (3.7) , ≥ 9 , ( ) -10 log -11 (3.8) Proof of Claim 3.2. Write = ℓ =1

2 ∫ 1 + * 2 ,

 212 Lemma 3.3. Let ( , ) be a compact Kähler manifold, and let : Δ * → be a holomorphic map from the punctured disk to . If, ( ) := ∫ Δ 2, *is bounded from above by log when → ∞ for some constant > 0. Here we consider our model of the punctured disk asΔ * := { ∈ C | 1 < | | < ∞} by taking ↦ → 1 . Then extends to a holomorphic map Δ * ∪ {∞} → . Proof. We claim that ∫ Δ 2, *< 3 for any > 0. Or else, there is 0 > 0 so that ∫ Δ 2, * ≥ 3 when ≥ 0 . Then , ( ) ≥ 3 (log -log 0 ) ≥ 2 log if ≫ 0. This contradicts with our assumption. For simplicity, let us now change our model of the punctured disk to Δ * := { ∈ C | 0 < | | < 1} by taking ↦ → 1 . Then one has ∫ { ∈C|0<| |< 1 2 } * < 3 . Consider the graph of , which is an one dimensional closed analytic subvariety of Δ * × . Let us equip Δ× with the Kähler metric ′ := * where 1 : Δ× → Δ and 2 : Δ × → is the projection map, and := ∧ ¯ . Then the volume of the analytic set ∩ { | 0 < | | < 1 2 } × with respect to the Kähler metric ′ is ∫

  (1.3), the Lelong numbers of the local potentials of are 0, so using * loc = ( • ), one can easily check that * = * . Hence2 ( ˜ ) -2 + ( , ) = ∫ ˜ Θ ℎ ˜ + ( ˜ + ) ≥ ∫ ˜ * = { } • { }, (4.1)where we use the notation in Definition 1.4.Fix a Kähler form on . By Theorem 1.11 one can choose a Kähler current 1 ∈ { } which is smooth outside ({ }). Hence, there is a constant > 0 so that 1 ≥ . Since is not contained in ({ }, one has

  Email address: benoit.cadorel@univ-lorraine.fr URL: http://www.normalesup.org/~bcadorel/ ya.deng@univ-lorraine.fr ya.deng@math.cnrs.fr URL: https://ydeng.perso.math.cnrs.fr

Putting this inequality into (4.1), we obtain 2 ( ˜ ) -2 + ( , ) ≥ deg .

The first claim follows since > 0 and > 0 does not depend on .

If { } -{ } is big, by Theorem 1.11 again there is a Kähler current 2 ∈ { } -{ } which is smooth outside ( { } -{ }). Hence there is a constant 2 > 0 so that 2 ≥ 2 . If is not contained in ∪ ( { } -{ }), by 2 + [ ] ∈ { } one has

Putting this to (4.1), we obtain 2 ( ˜ ) -2 ≥ 2 deg .

This proves the second claim.

P T A

We are now ready to prove Theorem A.

Proof of Theorem A. Take a compact Kähler manifold compactifying so that := -is simple normal crossing. By Proposition 2.1, the nilpotent harmonic bundle induces a pseudo-Kähler metric on whose holomorphic bisectional curvature is nonpositive and holomorphic sectional curvature is bounded from above by -1 2 rank -1 . One can then apply the criterion in [START_REF]Symmetric differentials on complex hyperbolic manifolds with cusps[END_REF]Theorem 2] or [BC20, Theorem 1.6] to conclude that is of log general type. Alternatively, by Lemma 1.16, + ≥ 1 4 rank -1 •2 { } where is the closed positive current on which is the trivial extension of . Since { } is big, + is also big. This also proves that is of log general type. Hence is both a Kähler and Moishezon manifold, hence projective. By Proposition 1.7.(i), any compact complex manifold compactifying is bimeromorphic to . This proves the uniqueness of algebraic structure of by Chow's theorem.

It follows from Theorems 3.1.(i) and 4.1.(i) that is pseudo-Picard hyperbolic and pseudo-algebraically hyperbolic.

A direct consequence of Theorems 3.1.(i) and 4.1.(i) is the following result.

Corollary 5.1. Let be a quasi-projective manifold. If is equipped with a Kähler metric with holomorphic sectional curvature bounded from above by a negative constant, then is Picard hyperbolic and algebraically hyperbolic. Theorem 5.2. Let be a quasi-projective quotient of bounded symmetric domain by a torsion free lattice. Then is Picard hyperbolic and algebraically hyperbolic.

Proof. Since the Bergman metric on is Kähler with holomorphic sectional curvature bounded from above by a negative constant, the Picard hyperbolicity and algebraic hyperbolicity of follows from the above corollary immediately.

H

In this section we will prove Theorem B using ideas similar to [Den20b, Proof of Theorem 5.1].

Theorem 6.1. Let ( , ) be a compact Kähler log pair. Assume that there is a nilpotent harmonic bundle ( , , ℎ) on := -so that :

→ End( ) is injective at one point. Then there is a log morphism : ( , ˜ ) → ( , ) from a projective log pair ( , ˜ ) which is a finite unramified cover over such that (i) any irreducible subvariety of non contained in the analytic subvariety -1 (

Here is the trivial extension of the pseudo-Kähler form = -tr( * ℎ ∧ ) on defined in (2.1). Moreover, we have

We will need the following crucial result proved in [Den20b, Claim 5.2] to find the desired covering : → in Theorem 6.1. The proof is based on residual finiteness of the global monodromy group and Cauchy's argument theorem. Lemma 6.2. Let be a projective manifold and let = ℓ =1 be a simple normal crossing divisor on . Assume that there is a complex local system L over := -. Then for any > 0, there is a smooth projective log pair ( , ˜ ) and a log morphism : ( , ˜ = =1 ˜ ) → ( , ) which is unramified over so that for each = 1, . . . , ℓ, one has

• or the local monodromy group of * L around ˜ is trivial.

Let us now prove Theorem 6.1.

Proof of Theorem 6.1. By the proof of Theorem A, is a projective manifold. For the (1, 1)-form on defined by = -tr( * ℎ ∧ ), (6.2) by Lemma 1.16, we know that is a pseudo-Kähler form whose holomorphic sectional curvature is bounded from above by -1 4 rank -1 . Let be the positive closed (1, 1)-current on which is the trivial extension of . By Lemma 1.15, the class { } is big and nef. Choose > 0 so that { } -is still big and

Let L be the local system relative to the tame harmonic bundle. By Lemma 6.2, we find a log morphism : ( , ˜ = =1 ˜ ) → ( , ) from a smooth projective log pair ( , ˜ ) which is unramified over satisfying the properties therein.

Set ⊂ ˜ to be the sum of all ˜ 's so that the local monodromy group of * L around ˜ is not trivial. Then by the dichotomy in Lemma 6.2, * -is an effective divisor, and the monodromy of * L around ˜ with ˜ ⊄ is trivial. By Proposition 6.4 below, the pull-back harmonic bundle extends to a nilpotent harmonic bundle over -. Such a nilpotent harmonic bundle induces a pseudo-Kähler metric 2 on -. One has 2 = * over ˜ . 2 thus has non-positive holomorphic bisectional curvature and holomorphic sectional curvature bounded from above by -1 4 rank -1 . Denote by 2 the closed positive current which is the trivial extension of 2 .

Proof of Claim 6. 3. By the very definition of trivial extension, we have (6.4) * ≥ * -2 ≥ 0.

Pick any point ∈ ˜ , and choose admissible coordinates (Ω; 1 , . . . , ) and (Ω 2 ; 1 , . . . , ) around and = ( ) with (Ω 1 ) ⊂ Ω 2 so that

3), the local potential of = dd c satisfies log

Hence the local potential

Therefore, the Lelong numbers of * are zero everywhere, and by (6.4), the same holds for the positive current * -2 . On the other hand, since 2 = * , * -2 is thus supported on ˜ . By the support theorem [Dem12b, (2.14

Recall that * -≥ 0, and ≥ 2 2rank -1 . Hence

where ′ is some effective R-divisor supported in ˜ . Therefore, { 2 } -2 2rank -1 { } is big with its non-Kähler locus

Recall that the holomorphic sectional curvature of 2 is bounded from above by -1 4 rank -1 . By Theorems 3.1.(ii) and 4.1.(ii), we conclude that is both Picard hyperbolic and algebraically hyperbolic modulo -1 ( ({ })) ∪ ˜ . Theorems 6.1.(ii) and 6.1.(iii) follows. Let ˜ ⊂ be any irreducible closed subvariety which is not contained in -1 ( ({ }))∪ ˜ . Let : → ˜ be a desingularization so that := -1 ( ) is a simple normal crossing divisor. Applying Theorem 1.11 we can pick a Kähler current

which implies that ˜ ∈ ({ 2 }) by Theorem 1.12. Since

this contradicts with the assumption that ˜ ⊂ is not contained in -1 ( ({ })) ∪ ˜ . Therefore, 3 is a pseudo Kähler form.

By the curvature decreasing property of submanifolds, we conclude that the holomorphic bisectional curvature of 3 is non-positive, and the holomorphic sectional curvature of 3 is bounded from above by -1 4 rank -1 . Let 3 be the closed positive (1, 1) current on which is the trivial extension of 3 . One can employ a similar method as for Claim 6.3 to show that

Hence is big. Theorem 6.1.(i) follows. Lastly, by (6.2), is strictly positive at any point where is injective. (6.1) then follows from (3.1).

We state and prove the following crucial extension result for nilpotent tame harmonic bundles across the boundary components around which the local monodromies of the corresponding local system are trivial. Its proof was communicated to us by C. Simpson, and it uses the deep theorem by Mochizuki on the correspondence between tame pure imaginary harmonic bundles and semisimple local systems over quasi-projective manifolds.

Proposition 6. 4. Let be a projective manifold and let = =1 be a simple normal crossing divisor on . Let ( , , ℎ) be a nilpotent harmonic bundle on := -, whose corresponding complex local system is denoted by L. Assume that for = 1, . . . , the local monodromy of L around the component is trivial. Then ( , , ℎ) extends to a nilpotent harmonic bundle on ′ := -= +1 .

Proof. Since is assumed to be nilpotent, the eigenvalue of the residue Res( ) at each component is thus zero. Hence ( , , ℎ) is a tame pure imaginary harmonic bundle in the sense of [Moc07, Definition 22.3]. By [Moc07, Proposition 22.15], L is semisimple. Hence it is a direct sum L := ⊕ L ⊗C , where L is a simple local system and > 0. Since the local monodromy of L around the component is trivial for = 1, . . . , , so is L for each . Hence L extends to a local system L ′ on ′ . Since the map between fundamental groups 1 ( ) → 1 ( ′ ) is surjective, L ′ is thus also simple.

By [Moc07, Theorem 25.21], there is a tame pure imaginary harmonic bundle ( , , ℎ ) on whose corresponding local system is L . Moreover, by the uniqueness property of the correspondence between semisimple local systems and tame pure imaginary harmonic bundles proved in [Moc07, Theorem 25.28], one has

where denotes a hermitian metric of C . Since rank = det( -) = det( -) , one has det( -) = rank . Hence each ( , , ℎ ) is a nilpotent harmonic bundle.

Again by [Moc07, Theorem 25.21], there is a tame pure imaginary harmonic bundle on ( ′ , ′ , ℎ ′ ) on ′ whose corresponding local system is L ′ . The restriction ( ′ , ′ , ℎ ′ )| is thus a tame pure imaginary harmonic bundle with the corresponding local system L ′ | = L . By the uniqueness result in [Moc07, Theorem 25.28], ( ′ , ′ )| = ( , ) and • ℎ ′ | = ℎ for some constant > 0. Since characteristic polynomial det( -′ )| = det( -) = rank , by continuity det( -′ ) = rank on ′ . Hence ( ′ , ′ , ℎ ′ ) is also nilpotent. Therefore, the nilpotent harmonic bundle

defined on ′ extends ( , , ℎ). The proposition is proved.

The following result allows us to control non-Kähler locus on ramified covers. Lemma 6.5. Let : ( , ˜ ) → ( , ) be a log morphism between compact Kähler log pairs, which is unramified over -˜ . Let be a big class on . Then ( * ) ⊂ -1 ( ( )) ∪ ˜ (6.6) Proof. By Theorem 1.11, one can take a Kähler current ∈ with analytic singularities which is smooth outside ( ). Choose a Kähler form on so that ≥ . Then { * } is a big and nef class, and by Theorem 1.12, one has

Applying Theorem 1.11 again, there is a global quasi-psh function on with analytic singularities which is smooth outside ˜ so that * + dd c is a Kähler current on . It follows from ≥ that * + dd c ≥ * + dd c . Hence * + dd c is a Kähler current with analytic singularities, which is smooth outside -1 ( ( )) ∪ ˜ . Since * + dd c ∈ * , by the very definition of non-Kähler locus Definition 1.10, one has

The lemma is proved. R