Y A Deng

A CHARACTERIZATION OF COMPLEX QUASI-PROJECTIVE MANIFOLDS UNIFORMIZED BY UNIT BALLS

Keywords: Mathematics Subject Classification. 14D07, 14C30, 32M15 Uniformization, non-compact ball quotient, Simpson-Mochizuki correspondence, log Higgs bundles, principal variation of Hodge structures, toroidal compactification

A

. In 1988 Simpson extended the Donaldson-Uhlenbeck-Yau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds and quasi-projective curves whose universal coverings are complex unit balls. In this paper we give a necessary and sufficient condition for quasi-projective manifolds to be uniformized by complex unit balls. This generalizes the uniformization theorem by Simpson. Several byproducts are also obtained in this paper. 0. I 0.1. Main result. The main goal of this paper is to characterize complex quasi-projective manifolds whose universal coverings are complex unit balls.

Theorem A (=Theorem 4.7.(i)). Let be an -dimensional complex projective manifold and let be a smooth divisor on (which might contain several disjoint components). Let be an ample polarization on . For the log Higgs bundle (Ω 1 (log ) ⊕ O , ) on ( , ) with the Higgs field defined by : Ω 1 (log ) ⊕ O → (Ω 1 (log ) ⊕ O ) ⊗ Ω 1 (log ) (0.1.1) ( , ) ↦ → (0, ), if it is -polystable (see § 1.3 for the definition), then one has the following inequality

2 2 (Ω 1 (log )) -+ 1 1 (Ω 1 (log )) 2 • 1 ( ) -2 ≥ 0. (0.1.2)
When the equality holds, then -≃ B Γ for some torsion free lattice Γ ⊂ ( , 1) acting on B . Moreover, is the (unique) toroidal compactification of B Γ , and each connected component of is the smooth quotient of an Abelian variety by a finite group acting freely on .

Let us stress here that the smoothness of in Theorem A is indeed necessary if one would like to characterize non-compact ball quotients: in Theorem 4.7.(ii) we prove that the universal cover ofis not the complex unit ball B if is assumed to be simple normal crossing but not smooth, leaving other conditions in Theorem A unchanged. Thus, it might be more appropriate to say that in this paper we give a characterization of smooth toroidal compactification of non-compact ball quotients.

Note that when is empty or when dim = 1, Theorem A has already been proved by Simpson [START_REF]Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF]Proposition 9.8]. As we will see later, we follow his strategy closely to prove the above theorem. Let us also mention that the inequality (0.1.2) is a direct consequence of Mochizuki's deep work on the Bogomolov-Gieseker inequality for parabolic Higgs bundles [Moc06, Theorem 6.5]. Our main contribution is the uniformization result when the equality in (0.1.2) is achieved. The proof builds on Simpson's ingenious ideas [START_REF]Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF] on characterizations of complete varieties uniformized by Hermitian symmetric spaces, as well as Mochizuki's celebrated work on Simpson correspondence for tame harmonic bundles [START_REF] Kobayashi | Hitchin correspondence for tame harmonic bundles and an application[END_REF]. Since the Kobayashi-Hitchin correspondence for general slope polystable parabolic Higgs bundles is still unproven, we need some additional methods to prove the above uniformization result (see § 0.3 for rough ideas).

We will show that the conditions in Theorem A is indeed necessary, by proving the following slope stability (with respect to a more general polarization) result for the natural log Higgs bundles associated to toroidal compactification of non-compact ball quotient by torsion free lattice.

Theorem B (= § 5.4). Let Γ ⊂ ( , 1) be a torsion free lattice with only unipotent parabolic elements. Let be the (smooth) toroidal compactification of the ball quotient B Γ . Write := -B Γ for the boundary divisor, which is a disjoint union of Abelian varieties. Let ∈ 1,1 ( , R) be a big and nef cohomology (1, 1)-class on containing a positive closed (1, 1)-current ∈ so that | -is a smooth Kähler form and has at most Poincaré growth near (for example, = 1 ( + ) or contains a Kähler form ). Then one has the following equality for Chern classes

2 2 (Ω 1 (log )) -+ 1 1 (Ω 1 (log )) 2 = 0. (0.1.3)
The log Higgs bundle (Ω 1 (log ) ⊕ O , ) defined in (0.1.1) is -polystable for the above big and nef polarization . In particular, it is slope polystable with respect to any Kähler polarization and the polarization by the big and nef class 1 ( + ).

Since both stability of log Higgs bundles and Chern equality (0.1.3) are invariant under taking conjugates with respect to the Galois action, a direct consequence of Theorems A and B is the following rigidity result of ball quotient under the automorphism of complex number field C to its coefficients of defining equations.

Corollary C. Let Γ ⊂ ( , 1) be a torsion free lattice, and let := B Γ be the ball quotient, which carries a unique algebraic structure, denoted by alg . For any automorphism ∈ Aut(C), let alg := alg × Spec(C) be the conjugate variety of alg under the automorphism , and denote by the analytification of alg . Then is also a ball quotient, namely there is another torsion free lattice Γ ⊂ ( , 1) so that = B Γ .

When Γ is arithmetic, Corollary C has been proved by Kazhdan [Kaz83]. When Γ is non-arithmetic, it was proved by Mok-Yeung [MY93, Theorem 1] and by Baldi-Ullmo [BU20, Theorem 8. 4.2].

In this paper we obtain some byproducts, and let us mention a few. We prove the Simpson-Mochizuki correspondence for principal system of log Hodge bundles over projective log pairs (see Theorem 3.1). We give a characterization of slope stability with respect to big and nef classes for log Higgs bundles on Kähler log pairs (see Theorem 5.7). We also give a very simple proof of the negativity of kernels of Higgs fields of tame harmonic bundles by Brunebarbe [START_REF]Semi-positivity from Higgs bundles[END_REF] (originally by Zuo [START_REF]On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applications[END_REF] for system of log Hodge bundles), using some extension theorems of plurisubharmonic functions in complex analysis (see Theorem 4.6). In the appendix written jointly with Benoît Cadorel, we prove a metric rigidity result for toroidal compactification of non-compact ball quotients (see Theorem A.7). 0.2. A few histories. Since the main purpose of this paper is to prove the uniformization result rather than the Miyaoka-Yau type inequality (0.1.2), we shall only recall some earlier work related to the characterization of ball quotient, and we refer the readers to [GKT16, GT16] for more references on the Miyaoka-Yau type inequalities.

Based on his proof of the Calabi conjecture [Yau78], Yau established the inequality (0.1.2) when is a projective manifold and = with ample. He proved that is uniformized by the complex unit ball in case of equality. Miyaoka-Yau inequality and uniformization result were extended to the context of compact Kähler varieties with quotient singularities by Cheng-Yau [CY86] using orbifold Kahler-Einstein metrics. A partial uniformization result for smooth minimal models of general type have been obtained by Zhang [Zha09]. More recently, uniformization result has been extended to projective varieties with klt singularities in the series of work [GKPT19b, GKPT19a] by Greb-Kebekus-Peternell-Taji.

All the above works dealt with compact varieties. A strong uniformization result was established by Kobayashi [Kob84,[START_REF]Einstein-Kähler V-metrics on open Satake V-surfaces with isolated quotient singularities[END_REF] in the case of open orbifold surfaces (see also [CY86]). In [CY86] Cheng-Yau also gave a differential geometric characterization of quasi-projective ball quotients of any dimensions using the method of bounded geometry in [CY80]. At almost the same time, based on [CY86], Tian-Yau [TY87] and Tsuji [Tsu88] independently established similar algebraic geometric characterizations of non-compact ball quotient of any dimension. See also [KNS89, Kob90, Yau93] for more related works on uniformization results.

All these aforementioned uniformization results are built on the positivity of the (log) canonical sheaf of the varieties together with existence of Kähler-Einstein metrics. In [START_REF]Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF], Simpson established a remarkable uniformization result in terms of stability of Higgs bundles. We essentially follow his approaches in this paper. In next subsection, we shall recall his ideas and discuss main difficulties in generalizing his methods to the context of non-compact varieties. 0.3. Main strategy. We mainly follow Simpson's strategy [START_REF]Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF] to prove Theorem A. Let us explain our rough ideas in the proof of Theorem A when the equality in (0.1.2) holds.

Step 1. Following Simpson in the compact setting, we first define systems of log Hodge bundles over log pairs. We prove that, a system of log Hodge bundles on a projective log pair with vanishing first and second Chern classes admits an adapted Hodge metric. The proof is based on Mochizuki's celebrated theorem [START_REF] Kobayashi | Hitchin correspondence for tame harmonic bundles and an application[END_REF]Theorem 9.4] on the existence of harmonic metric, and C * -action invariant property of log Hodge bundles.

Step 2. We generalize the result in Step 1 to the context of principal bundles. Fix a Hodge group 0 . Following Simpson again, we define a principal system of log Hodge bundles ( , ) on log pairs ( , ) with the structure group ⊂ , where is the complexification of 0 . Based on the result in Step 1 together with some similar Tannakian arguments in [START_REF]Harmonic bundles on noncompact curves[END_REF], in Theorem 3.1 we prove that if there is a faithful Hodge representation : → ( ) for some polarized Hodge structure ( = ⊕ + = , , ℎ ) so that the system of log Hodge bundles ( × , ( )) is -polystable with ∫ ℎ 2 ( × ) • 1 ( ) dim -2 = 0, then there is a metric reduction for | -so that the triple ( | -, | -, ) gives rise to a principal variation of Hodge structures on -.

Step 3. For the system of log Hodge bundles ( := Ω 1 (log ) ⊕ O , ) in Theorem A, we first associate it a principal system of log Hodge bundles ( , ) in Proposition 2.11, whose Hodge group 0 = ( , 1) is of Hermitian type (see Definition 2.5). One can easily show that 2 ( × ) = 2 (End( ) ⊥ ) = 0 when the equality in (0.1.2) holds, where End( ) ⊥ denotes the trace free part of End( ). By Theorem 1.9, the system of log Hodge bundles ( × , ( )( )) = (End( ) ⊥ , ( ) ⊥ ) is also slope polystable if ( , ) is slope polystable. Since the adjoint representation : → ( ) is a faithful Hodge representation, by the result in Step 2, there is a metric reduction for | -so that the triple ( | -, | -, ) gives rise to a principal variation of Hodge structures on -. Since : (-log ) → × -1,1 is an isomorphism, this implies that the period map : -→ ( , 1) ( ) associated to ( | -, | -, ) from the universal cover ofto the period domain 0 0 = ( , 1) ( ) is locally biholomorphic.

For more details, see Step one of the proof of Theorem 4.7.

Step 4. We have to prove that the period map in Step 3 is moreover a biholomorphism.

Note that when = , this step is quite easy. In Remark 2.7 we show that it suffices to prove that the hermitian metric * ℎ onis complete, where ℎ is the hermitian metric on × -1,1 | -induced by the metric reduction together with the Killing form of . This step is slightly involved and the readers can find it in Step two of the proof of Theorem 4.7. To be brief, we establish a precise model metric (ansatz) for ( , ) ⊗ ( * , * ) locally around with at most log growth, and we prove that this local metric is indeed mutually bounded with ℎ using similar ideas in [START_REF]Harmonic bundles on noncompact curves[END_REF]§4]. Based on this model metric, we obtain a precise norm estimates for ℎ near , so that we can prove that * ℎ is a complete metric on -. This concludes that the universal cover ofis the unit ball ( , 1) ( ) . 0.4. Acknowledgments. This work owes a lot to the deep work [START_REF]Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF][START_REF]Harmonic bundles on noncompact curves[END_REF][START_REF]Higgs bundles and local systems[END_REF][START_REF] Kobayashi | Hitchin correspondence for tame harmonic bundles and an application[END_REF], to which I express my deepest gratitude. I sincerely thank Professor Carlos Simpson for answering my questions, as well as his suggestions and encouragements. I thank Professor Takuro Mochizuki for sending me his personal notes on the proof of Theorem 1.9. I also thank Professors Jean-Pierre Demailly, Henri Guenancia, Emmanuel Ullmo, Shing-Tung Yau, and Gregorio Baldi, Jiaming Chen, Chen Jiang, Jie Liu, Mingchen Xia for very helpful discussions and their remarks on this paper. My special thanks go to Benoît Cadorel for his very fruitful discussions on the toroidal compactification, which lead to a joint appendix with him in this paper. Last but not least, I am grateful to the referee for his/her careful readings and very helpful comments to improve this manuscript.

L H H

1.1. Higgs bundles and tame harmonic bundles. In this section we recall the definition of Higgs bundles and tame harmonic bundles. We refer the readers to [Sim88, Sim90, Sim92, Moc02, Moc07] for further details.

Definition 1.1. Let be a complex manifold. A Higgs bundle on is a pair ( , ) where is a holomorphic vector bundle with ¯ its complex structure, and : → ⊗ Ω 1 is a holomorphic one form with value in End( ), say Higgs field, satisfying ∧ = 0.

Let ( , ) be a Higgs bundle over a complex manifold . A smooth hermitian metric ℎ of is called harmonic if ℎ := ℎ + + ℎ is flat. Here ℎ is the Chern connection of ( , ℎ), and ℎ is the adjoint of with respect to ℎ. Definition 1.2 (Harmonic bundle). A harmonic bundle on a complex manifold is triple ( , , ℎ) where ( , ) is a Higgs bundle and ℎ is a harmonic metric for ( , ).

A log pair consists of an -dimensional complex manifold , and a simple normal crossing divisor on .

Definition 1.3. (Admissible coordinate) Let be a point of , and assume that { } =1,...,ℓ be components of containing . An admissible coordinate around is the tuple ( ; 1 , . . . , ; ) (or simply ( ; 1 , . . . , ) if no confusion arises) where

• is an open subset of containing .

• there is a holomorphic isomorphism :

→ Δ so that ( ) = ( = 0) for any = 1, . . . , ℓ. We shall write * := -, ( )

:= { ∈ | | | < , ∀ = 1, . . . , } and * ( ) := ( ) ∩ * .
Recall that the Poincaré metric on (Δ * ) ℓ × Δ -ℓ is described as

= ℓ =1 √ -1 ∧ ¯ | | 2 (log | | 2 ) 2 + =ℓ+1 √ -1 ∧ ¯ (1 -| | 2 ) 2 .
Definition 1.4 (Poincaré growth). Let ( , ) be a log pair. A hermitian metric on has at most (resp. the same) Poincaré growth near if for any point ∈ , there is an admissible coordinate ( ; 1 , . . . , ) centered at and a constant > 0 so that ≤ (resp. ∼ ) holds over * ( ) for some 0 < < 1.

Remark 1.5 (Global Kähler metric with Poincaré growth). Let ( , ) be a compact Kähler manifold and = ℓ =1 is a simple normal crossing divisor on . By Cornalba-Griffiths [CG75], one can construct a Kähler current over , whose restriction onis a complete Kähler form, which has the same Poincaré growth near as follows.

Let be the section 0 ( , O ( )) defining , and we pick any smooth metric ℎ for the line bundle O ( ). One can prove that the closed (1, , 1)-current

:= - √ -1 log(- ℓ =1 log | • | 2 •ℎ ), (1.1.1)
the desired Kähler current when 0 < ≪ 1. 1.2. Log Higgs bundle and adapted harmonic metrics. Throughout this paper, we mainly consider log Higgs bundles ( , ) over log pairs. Definition 1.6 (Log Higgs bundles). Let ( , ) be a log pair. A log Higgs bundle consists of a pair ( , ) with a holomorphic vector bundle on and : → ⊗ Ω 1 (log ) with ∧ = 0. Definition 1.7 (Adapted harmonic metric). Let ( , ) be a log pair, and let ( , ) be a log Higgs bundle on ( , ). Suppose that ℎ is a harmonic metric for the Higgs bundle ( , )| -. It is called adapted to the log Higgs bundle if for any admissible coordinate ( ; 1 , . . . , ) and any ( 1 , . . . , ℓ ) ∈ [0, 1) ℓ , one has

( ) = { ∈ ( -) | | | ℎ = O( ℓ =1 | | --) for any > 0}
In the terminology of [START_REF] Kobayashi | Hitchin correspondence for tame harmonic bundles and an application[END_REF][START_REF]Asymptotic behaviour of tame harmonic bundles and an application to pure twistor -modules[END_REF], the above definitions are equivalent that ( , ) is a parabolic Higgs bundle with trivial parabolic structures over ( , ) of weight (0, . . . , 0), and the harmonic bundle ℎ for ( , )| -is adapted to its parabolic structures. 1.3. Slope stability. Let ( , ) be a compact Kähler manifold of dimension and let be a simple normal crossing divisor on . Let ( , ) be a log Higgs bundle on ( , ). Let be a big and nef cohomology (1, 1)-class on . For any torsion free coherent sheaf , its degree with respect to is defined by deg ( ) := 1 ( ) • -1 , and its slope with respect to is defined by ( ) := deg ( ) rank . Consider a log Higgs bundle ( , ) on ( , ). A Higgs sub-sheaf is a saturated coherent torsion free subsheaf ′ ⊂ so that ( ′ ) ⊂ ′ ⊗ Ω 1 (log ). We say ( , ) is -stable if for Higgs sub-sheaf ′ of , with 0 < rank ′ < rank , the condition ( ′ ) < ( ) is satisfied. ( , ) is -polystable if it is a direct sum of -stable log Higgs bundles with the same slope. When = { } where is a Kähler form on , we write instead of . When = 1 ( ) for some ample line bundle on , we use the notation instead of . By Simpson [START_REF]Harmonic bundles on noncompact curves[END_REF], there is a C * -action on log Higgs bundles ( , ) defined by ( , ) for any ∈ C * . It follows from the definition that, if ( , ) is -stable, then ( , ) is also -stable for any ∈ C * .

The following celebrated Simpson correspondence for tame harmonic bundles proved by Mochizuki [START_REF] Kobayashi | Hitchin correspondence for tame harmonic bundles and an application[END_REF] is a crucial ingredient in this paper.

Theorem 1.8 (Mochizuki). Let ( , ) be a projective log pair endowed with an ample polarization . A log Higgs bundle ( , ) on ( , ) is -polystable with

∫ 1 ( ) • 1 ( ) dim -1 = ∫ ℎ 2 ( ) • 1 ( ) dim -2 = 0 if
and only if there is a harmonic metric ℎ for ( | -, | -) adapted to ( , ). When ( , ) is moreover stable, such a harmonic metric ℎ is unique up to some positive constant multiplication.

Let us mention that in [Biq97] Biquard has proved a stronger theorem when the divisor in Theorem 1.8 is smooth. The poly-stability is also preserved under tensor product and dual by Mochizuki [Moc19, Proposition 4.10].

Theorem 1.9 (Mochizuki). Let ( , ) be a projective log pair endowed with an ample polarization . Let ( , ) be a -polystable log Higgs bundle on ( , ). Then the tensor product , ( , ) is still a -polystable log Higgs bundle for , ∈ Z ≥0 . Here , ( , ) := Hom( ⊗ , ⊗ ), , is the induced log Higgs bundle by taking the tensor product.

Since [Moc19, Proposition 4.10] worked with the much more general case than what we need, we shall provide a quick proof for Theorem 1.9 for completeness sake. The idea essentially follows [START_REF]Higgs bundles and local systems[END_REF]Corollary 3.8] in the compact setting.

Proof of Theorem 1.9. By the Mehta-Ramanathan type theorem proved by Mochizuki [Moc06, Proposition 3.29], , ( , ) is -polystable if and only if , ( , )| is -polystable, where denotes a complete intersection of sufficiently ample general hypersurfaces in . This enables us to reduce the desired statement to the case of curves. Assume now that dim = 1. By [START_REF]Harmonic bundles on noncompact curves[END_REF] or [Biq97, Théorème 8.1], ( , )| -admits a Hermitian-Yang-Mills metric ℎ: Λ ℎ ( ) = ⊗ , where is some Kähler form in 1 ( ), and is some topological constant. Moreover, ℎ is adapted to ( , ), and is adapted to log order in the sense of Definition 4.1. Hence (ℎ * ) ⊗ ⊗ ℎ ⊗ is the Hermitian-Yang-Mills metric for , ( , )| -, which is also adapted to log order. It follows from Theorem 5.7 below that , ( , ) is also -polystable.

1.4. Simpson-Mochizuki correspondence for systems of log Hodge bundles. A typical and important class of log Higgs bundle is the system of log Hodge bundles. In this subsection, we shall apply Theorem 1.8 to prove the Simpson-Mochizuki correspondence for systems of log Hodge bundles.

Definition 1.10 (System of log Hodge bundles). Let ( , ) be a log Higgs bundle on a log pair ( , ). We say that ( , ) is a system of log Hodge bundles if there is a decomposition of into holomorphic vector bundles := ⊕ + = , such that : , → -1, +1 ⊗ Ω 1 (log ).

When = , such ( , ) is called a system of Hodge bundles. A system of log Hodge bundles is -(poly)stable if it is -(poly)stable in the sense of log Higgs bundles.

Definition 1.11 (Hodge metric). Let ( := ⊕ + = , , ) be a system of Hodge bundles on a complex manifold . A hermitian metric ℎ for is called a Hodge metric if ℎ is harmonic, and it is a direct sum of metrics on the bundles , .

By Simpson [START_REF]Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF], a system of Hodge bundles equipped with a Hodge metric is equivalent to a complex variation of Hodge structures. He then established his correspondence for Hodge bundles over compact Kähler manifolds in [Sim88, Proposition 8.1]. In the rest of this subsection, we will extend his result to the log setting.

Let us state and prove the main result in this subsection.

Proposition 1.12. Let ( , ) be a projective log pair. Let ( , ) = (⊕ + = , , ) be a system of log Hodge bundles on ( , ) which is -polystable with

∫ 1 ( ) • 1 ( ) dim -1 = ∫ ℎ 2 ( ) • 1 ( ) dim -2 = 0.
Then there is a decomposition ( , ) = ⊕ ∈ ( , ) where each ( , ) is -stable system of log Hodge bundles so that there is a Hodge metric ℎ (unique up to a positive multiplication) for ( | -, | -) which is adapted to ( , ).

Proof. Let us first prove the proposition when ( , ) is stable. By [Moc06, Theorem 9.1 & Propositions 5.1-5.3], there is a harmonic metrics ℎ for ( | -, | -) which is adapted to ( , ), and such a harmonic metric is unique up to a positive constant multiplication. We introduce automorphism : → of parametrized by ∈ (1), defined by

( + = , ) = + = , . (1.4.1) for every , ∈ , . Then : ( , ) → ( , ) is an isomorphism since • = • .
Hence by the uniqueness of harmonic metrics, there is a function

( ) : (1) → R + such that * ℎ = ( ) • ℎ.
For every , ∈ , , one has

( ) • ℎ( , , , ) = * ℎ( , , , ) = ℎ( ( , ), ( , )) = | | 2 ℎ( , , , ) = ℎ( , , , )
Hence ( ) ≡ 1 for ∈ (1), namely * ℎ = ℎ. On the other hand, ℎ( , , , ) = * ℎ( , , , ) = ℎ( ( , ), ( , )) = -ℎ( , , , )

for any ∈ (1). Therefore, ℎ( , , , ) = 0 if ≠ . Hence ℎ is a direct sum of hermitian metrics for , , namely ℎ is a Hodge metric. The proposition is proved if ( , ) is stable.

Let us prove the general cases. By [Moc06, Corollary 3.11 & Theorem 9.1 & Propositions 5.1-5.3], there is a canonical and unique decomposition ( , ) = ⊕ ∈ ( , ) ⊗ C where is a finite set and harmonic metrics ℎ for ( | -, | -) which is adapted to ( , ) so that ( , ) is a -stable log Higgs bundle. By the above arguments, it suffices to prove that each ( , ) is system of log Hodge bundles. Since ( , ) is a system of log Hodge bundles, ( , ) is isomorphic to ( , ) for any ∈ (1). We have the following decomposition ( ,

) = ⊕ ( , ) ⊗ C . Note that ( , ) is still -stable.
By the uniqueness of the decomposition, ( , ) ≃ ( , ) for some ∈ . Since is a finite set, there exists 1 , 2 so that 1 / 2 is not a root of unity and 1 = 2 . In other words, ( , 1 ) ≃ ( , 2 ). By [Sim90, Lemma 4.1] or [Sim92, Theorem 8], ( , 1 ) is a system of log Hodge bundles, and so is ( , ). Hence ( , ) is a direct sum of -stable system of log Hodge bundles ( , ), and each ( | -, | -) admits a Hodge metric ℎ adapted to ( , ). The proposition is proved.

P H

In this section, we will extend Simpson's principal system of log Hodge bundles in [Sim88, §8] to the log setting. We will provide all necessary proofs for the claims for completeness sake. Let us mention that most results in this section follows from [Sim88, §8 & §9] with minor changes.

Let 0 be a real algebraic group which is semi-simple with its Lie algebra denoted by 0 . Let be the complexification of 0 with its Lie algebra denoted by . Then = 0 + √ -1 0 . 0 is called a Hodge group if the following conditions hold.

• The Lie algebra of admits a Hodge structure of weight 0, namely, one has a decomposition

= ⊕ ,- so that [ ,-, ,-] ⊂ + ,--. • If • denotes the complex conjugation with respect to 0 , then ,-= -, . • The form ℎ ( , ) := (-1) +1 ( ¯ ) for , ∈ ,- (2.0.1)
is a positively definite hermitian metric for . let 0 ⊂ 0 be the Lie subgroup of 0 so that its Lie algebra 0 is 0 ∩ 0,0 . Let ⊂ (resp.

) be the complexification of 0 (resp. 0 ), and thus the Lie algebra of is . Then the restriction of the Killing form of 0 on 0 is positively definite, and thus 0 is a compact real Lie group.

In the rest of the paper, we shall use the above notations without recalling their meanings.

The following concrete example of the Hodge group will be used in this paper, especially in the proof of Theorem A.

Example 2.1. Consider the a direct sum of C-vector spaces

= ⊕ + = ,
Denote by := rank , , and := rank . Fix a hermitian metric ℎ = ⊕ + = ℎ for where ℎ is a hermitian metric for , . We take a sesquilinear form ( , ) := ( √ -1) -ℎ( , ) for , ∈ , . Define 0 := ( , ) ≃ ( 0 , 0 ), where 0 := and 0 := . We shall show that 0 is a Hodge group. First we note that the complexification of 0 is := ( ) ≃ ( , C). Then the Lie algebra of is = ( ) ≃ ( , C), and the Lie algebra of 0 is 0 = ( 0 , 0 ). Let us define the Hodge decomposition as follows:

,-= ⊕ Hom( , , + , -) ∩ ( ).

Then = ⊕ ,-. One can check that ,-= -, , where the conjugate is taken with respect to the real form 0 of . Let be the subgroup of which fix each , . Then = ( + = ( , )), and its Lie algebra is = 0,0 . Define 0 := ∩ 0 = ( + = ( , , ℎ )), whose Lie algebra is

0 = 0,0 ∩ 0 .
More precisely, if we fix a unitary frame 1 , . . . , 0 for (⊕ , , ⊕ ℎ ) and a unitary frame 1 , . . . , 0 for (⊕ , , ⊕ ℎ ), elements in 0 can be expressed as the ones in ( × , C) with the form * where ∈ ( 0 ) and ∈ ( 0 ) so that ( ) + ( ) = 0. Note that the Killing form

( ) = 2 ( ),
if we consider , as elements in ( , C). Moreover, for ∈ ,-, one can show that

= - * if is even * if is odd.
where * denotes the conjugate transpose of . Hence the hermitian metric ℎ defined in (2.0.1) can be simply expressed as

ℎ ( , ) = 2 ( * )
once we consider , as elements in ( , C). In other words, for the natural inclusion : ↩→ ( ), one has ℎ = 2 • * ℎ ( ) , where ℎ ( ) is the hermitian metric on End( ) induced by ℎ . This fact is an important ingredient in the proof of Theorem A.

Let us generalize Simpson's definition of principal system of Hodge bundles in [START_REF]Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF]§8] to the log setting as follows.

Definition 2.2 (Principal system of log Hodge bundles). A principal system of log Hodge bundles on a log pair ( , ) is a pair ( , ), where is a holomorphic -fiber bundle endowed with a holomorphic map

: (-log ) → × -1,1 such that [ ( ), ( )] = 0. A metric for | -is a reduction
⊂ | -whose structure group is 0 . Let be the Chern connection for . Define to be the complex conjugate of | -with respect to the reduction . Then

∈ C ∞ ( -, ( × 0 1,-1 ) ⊗ Ω 0,1 -). Set := + | -+ , (2.0.2)
which is a connection on the smooth 0 -bundle × 0 0 . Such triple ( | -, | -, ) is called a principal variation of Hodge structures overof Hodge group 0 , if the induced connection in (2.0.2) is flat, namely the curvature of is zero.

Remark 2.3. Note that the metric reduction for a principal system of Hodge bundles ( , ) on a complex manifold induces a hermitian metric ℎ on × ≃ × 0 defined by ℎ ( , ), ( ,

) := ℎ ( , ) (2.0.3)
for any ∈ and , ∈ . Here ℎ is the hermitian metric defined in (2.0.1). Note that 0 preserves the decomposition = ⊕ + = -, . It thus also preserves ℎ . Indeed, for , ∈ -, and ∈ 0 , one has

(-1) +1 ℎ ( , ) = (-1) +1 ℎ ( , ).
By the equivalence relation ( , ) ∼ ( -1 , ), the metric ℎ is thus well-defined.

Remark 2.4 (Period map of principal variation of Hodge structures). By Simpson [Sim88, p. 900], for a principal variation of Hodge structures ( , , ) on a complex manifold , one can also define its period map as follows. Denote by : ˜ → the universal cover of . Set ( ˜ := * , ˜ := * , ˜ := * ), which is a principal variation of Hodge structures on the simply connected complex manifold ˜ . The flat connection thus induces a flat trivialization ˜ × 0 0 ≃ ˜ × 0 . Denote by : ˜ → 0 the composition of the inclusion ˜ ⊂ ˜ × 0 0 ≃ ˜ × 0 and the projection ˜ × 0 → 0 . It induces a map

: ˜ → 0 0 =: D (2.0.4) ˜ ↦ → ( ) • 0 ∀ ∈ ˜ , ˜ .
Alternatively, we view 0 → D as a principal 0 -fiber bundle over D, and its pull-back on ˜ via is nothing but the principal 0 -fiber bundle ˜ by our definition of . Hence the complexified differential of is

C : C ˜ → * C D ≃ * ( 0 × 0 ⊕ ≠0 ,-) = ˜ × 0 ⊕ ≠0 ,-
One can prove that C = ˜ + ˜ , where ˜ is the conjugate of ˜ with respect to ˜ .

Hence the restriction of C to the holomorphic tangent bundle ˜ is ˜ , which is a holomorphic map since the holomorphic tangent bundle of

D is D ≃ 0 × 0 ⊕ <0 ,-.
In conclusion, is a holomorphic map, which is called the period map associated to the principal variation of Hodge structures ( , , ), whose differential is given by = ˜ .

The uniformization is related by Hodge group of Hermitian type.

Definition 2.5 ( [Sim88, §9]). A Hodge group 0 is called Hermitian type if the Hodge decomposition of the Lie algebra of is

= -1,1 ⊕ 0,0 ⊕ 1,-1
and that 0 has no compact factor. In this case, 0 ⊂ 0 is the maximal compact subgroup and D := 0 0 is a Hermitian symmetric space of non-compact type. Let us generalize the definition of uniformizing bundle by Simpson [Sim88, §9] to the log setting.

Definition 2.6 (Uniformizing bundle). Let 0 be a Hodge group of Hermitian type. A uniformizing bundle on a log pair ( , ) is a principal system of log Hodge bundles ( , ) such that : (-log ) ≃ → × -1,1 is an isomorphism. A uniformizing variation of Hodge structures is a uniformizing bundle on a complex manifold together with a flat metric ⊂ .

Remark 2.7 (Uniformization via uniformizing bundles). It follows from Definition 2.6 that, for a uniformizing variation of Hodge structures ( , , ) over a complex manifold , the period map : ˜ → D defined in (2.0.4) is locally biholomorphic. This follows from the fact that = ˜ , which is isomorphic at any point of ˜ by the definition. Recall that in Remark 2.3 the metric reduction together with the positively definite form ℎ for in (2.0.1) induce a metric ℎ for × -1,1 . For the period domain D which is a hermitian symmetric space, one can also define the hermitian metric ℎ D for D ≃ 0 × 0 -1,1 in a similar way. By Remark 2.4, ˜ = * 0 when we consider 0 → D as a principal 0 -fiber bundle over D. One thus has * * ℎ = * ℎ D .

(2.0.5) In other words, : ( ˜ , ℎ ˜ := * * ℎ ) → (D, ℎ D ) is a local isometry. Hence for the action of 1 ( ) on ˜ , the metric ℎ ˜ is invariant under this 1 ( )-action. If * ℎ is a complete metric, so is * * ℎ , and by [Cha06, Theorem IV.1.2],

: ˜ → D is a Riemannian covering map, which is thus a biholomorphism since ˜ and D are both simply connected. In other words, is uniformized by the hermitian symmetric space D when the metric * ℎ on is complete.

One can construct systems of log Hodge bundles from principal ones via Hodge representations.

Definition 2.8 ([Sim88, p. 900]). Let ( = ⊕ + = , , ℎ ) be a polarized Hodge structure. A Hodge representation of 0 is a complex representation : → ( ) satisfying the following conditions.

• The action of is compatible with Hodge type, and such that 0 preserves Hodge type.

In other words, ( ,-)( , ) ⊂ + , - and ( 0 )( , ) ⊂ , . 1 • The sesquilinear form defined by

( , ) := ( √ -1) -ℎ ( , ) for , ∈ , (2.0.6) is 0 invariant. Namely, one has ( 0 ) ⊂ ( , ).
Example 2.9. For the Hodge group 0 , ( = ⊕ ,-, ℎ ) is a polarized Hodge structure of weight 0, where ℎ is the polarization defined in (2.0.1) via the Killing form. One can easily check that the adjoint representation : → ( ) is a Hodge representation for this polarized Hodge structure.

A principal system of log Hodge bundles together with a Hodge representation induces a system of log Hodge bundles as follows.

Lemma 2.10. If : → ( ) is a Hodge representation of the Hodge group 0 and ( , ) is a principal system of log Hodge bundles on the log pair ( , ), then ( := × , := ( )) is a system of log Hodge bundles. A polarization ℎ for together with a metric for | -give a metric ℎ on the system of Hodge bundles ( , )| -over -. When ( | -, | -, ) is a principal variation of Hodge structures over -, ( | -, | -, ℎ ) gives rise to a complex variation of Hodge structures. Proof. By Definition 2.8, one has ( )( , ) ⊂ , . Hence := × admits a decomposition of holomorphic vector bundles = ⊕ + = , with , := × , . Let us define := ( ). Since : (-log ) → × -1,1 satisfies [ ( ), ( )] = 0, and ( -1,1 )( , ) ⊂ -1, +1 , one thus has : , → -1, +1 ⊗ Ω 1 (log ), with ∧ = 0. Hence ( , ) is a system of log Hodge bundles.

Let us now prove that | 0 : 0 → ( ) has image on ( , ℎ ). Since ( )( , ) ⊂ , , one thus has

( ) ⊂ + = ( , ).
Since the sesquilinear form in (2.0.6) is 0 invariant, one thus has

( 0 ) = ( , ).
Hence

( 0 ) ⊂ ( 0 ∩ ) ⊂ + = ( , , ℎ , ) ⊂ ( , ℎ ). (2.0.7) Note that = × ≃ × 0 .
We define the hermitian metric ℎ for by setting ℎ (( , ), ( , )) := ℎ ( , ) (2.0.8) for any ∈ and for any , ∈ . Since ( 0 ) ⊂ ( , ℎ ), one can check as Remark 2.3 that ℎ is well-defined. 1 As remarked by Simpson [START_REF]Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF], this is not automatic if 0 is not connected. However, in Example 2.1, 0 is always connected, and thus such condition will be superfluous in that case.

If ( | -, | -, ) is a principal variation of Hodge structures on -, the connection := + + is flat. By construction, the connection ℎ := ℎ + + ℎ for | -is also flat, where ℎ is the Chern connection for the metrized vector bundle ( , ℎ ), and ℎ is the conjugate of with respect to ℎ . Indeed, it can be seen from that ℎ is naturally induced by , := ( ), and ℎ = ( ) by (2.0.8). By [Sim88, p. 898], the triple ( | -, | -, ℎ ) gives rise to a complex variation of Hodge structures on -.

such together with the polarization ℎ for gives rise to a Hodge metric ℎ for ( , )| - (defined in Lemma 2.10) which is adapted to ( , ).

Proof. We first prove that ( , )| -admits a Hodge metric ℎ over ( , )| -which is adapted to ( , ). Since is a complex semi-simple Lie group, the Hodge representation ′ : → (det ) induced by has image contained in (det ) = 1. Hence ′ is trivial. Note that det = × det , which is thus a trivial line bundle on . Hence ∫ ℎ 2 ( ) • 1 ( ) dim -2 = 0, it follows from Proposition 1.12 that ( , )| -admits a Hodge metric ℎ over ( , )| - which is adapted to ( , ).

Let us show that | : → ( ) is faithful. By (2.0.7), one has ( 0 ) ⊂ ( , ℎ ). Since is the complexification of 0 and | 0 : 0 → ( ) is assumed to be faithful, one concludes that | : → ( ) is also faithful. Let us now recall some Tannakian arguments. The representation induces a representation , : → ( , ) for any , ∈ N, where , := Hom( ⊗ , ⊗ ). Since | : → ( ) is faithful, we can consider as a reductive algebraic subgroup of ( ). There is a one dimensional complex subspace 1 ∈ , for some ( , ) ∈ N 2 so that

= { ∈ ( ) | , ( )( 1 ) = 1 }. (3.0.1)
Since is reductive, there is a complementary subspace 2 of , for 1 which is invariant under .

By Lemma 2.10, the Hodge representation , and ( , ) gives rise to a system of log Hodge bundles ( × , , , := , ( )) over ( , ), which is nothing but , ( , ). Recall that , ( )( 1 ) = 1 and , ( )( 2 ) = 2 . Consider the log Higgs bundles

( 1 , 1 ) := ( × 1 , , ( )) and
( 2 , 2 ) := ( × 2 , , (
)) over ( , ). Note that , ( , ) = ( 1 , 1 ) ⊕ ( 2 , 2 ). By Theorem 1.8, , ( , ) is -polystable with ∫ 1 ( , ( )) • 1 ( ) dim -1 = 0 with respect to an arbitrary polarization . Since 1 ( , ( )) = 1 ( 1 ) + 1 ( 2 ), by the polystability of , ( , ), we conclude that ( 1 , 1 ) and ( 2 , 2 ) are both -polystable. By Proposition 1.12, each ( | -, | -) admits a harmonic metric ℎ which is adapted to ( , ). Moreover, ℎ coincides with ℎ 1 ⊕ ℎ 2 up to some obvious ambiguity.

In the rest of the proof, any object which appears is restricted over -. Let us first enlarge the structure group of by defining ( ) := × ( ) via the faithful representation | : → ( ). This is the holomorphic principal (frame) bundle associated to . We can consider = × ⊂ ( ) as a reduction of ( ) . The metric ℎ for gives rise to a reduction ( ,ℎ) of ( ) with the structure group ( , ℎ ). Indeed, note that = ( ) × ( ) and thus the metric ℎ for induces a family of hermitian metrics ℎ for parametrized by ∈ ( ) . It has the obvious relation ℎ • = * ℎ for any ∈ ( ). We define

( ,ℎ) := { ∈ ( ) | ℎ = ℎ } (3.0.2)
and it is obvious that if ∈ ( ,ℎ) , then • ∈ ( ,ℎ) if and only if ∈ ( , ℎ ). Hence the structure group of ( ,ℎ) is ( , ℎ ).

Let us define := ∩ ( ,ℎ) whose structure group is ( , ℎ ) ∩ ⊃ 0 by (2.0.7). Since 0 is the maximal compact subgroup of and ( , ℎ ) ∩ is also compact, one has moreover ( , ℎ ) ∩ = 0 . Hence ⊂ is a metric reduction with the structure group 0 . Obviously, if we follow Lemma 2.10 to define a new metric ℎ ′ for by setting ℎ ′ (( , ), ( , )) := ℎ ( , )

for any ∈ and for any , ∈ , then Let ∈ C ∞ ( ( ) , * ( ) ⊗ ( )) be the Chern connection 1-form for the principal bundle ( ) induced by the Chern connection ℎ for ( , ℎ). Fix a base point ∈ ⊂ ( ) , and we denote by : → the projection map. Recall that , ( , ℎ) = ( 1 , ℎ 1 ) ⊕ ( 2 , ℎ 2 ), and = × .

ℎ ′ = ℎ (3.
Hence the holonomy ( , ) ∈ ( ) with respect to the connection along any smooth loop based at ( ) satisfies that , ( , ) ( ) ⊂ for = 1, 2. By (3.0.1), one has ( , ) ∈ . Hence the restriction of to is 1-form with values in . In other words, is induced by a connection on .

On the other hand, by the definition of the Chern connection, is also induced by a connection on ( ,ℎ) ; in other words, the restriction of to ( ,ℎ) is 1-form with values in Lie( ( , ℎ )), where Lie( ( , ℎ )) denotes the Lie algebra of ( , ℎ ). Since 0 = ∩ Lie( ( , ℎ )), there is a connection 0 ∈ C ∞ ( , * ⊗ 0 ) for the smooth principal 0 -fiber bundle := ( ,ℎ) ∩ which induces the connection . 0 is moreover the Chern connection with respect to the reduction of by our construction. Let us define ∈ A 1,1 ( × 0 0 ) to be the curvature form of the connection 0 + + over the smooth principal 0 -bundle × 0 0 , , where is the adjoint of with respect to the metric reduction ⊂ . Recall that := ( ). By (3.0.3), one has ℎ = ( ). Hence

( ) = ( ℎ + + ℎ ) 2 = ℎ ( ) = 0 (3.0.4)
where ℎ is the Chern connection for ( , ℎ). Since : 0 → ( ) is assumed to be injective. By (3.0.4) this implies that = 0. In conclusion, ( | -, | -, ) is a principal variation of Hodge structures on -.

U

This section is devoted to the proof of Theorem A. In § 4.2 we shall prove a basic result for the extension of plurisubharmonic functions. This lemma will be used in the proof of Theorem A. We shall also give an application of this fact in Hodge theory: we can give a much simpler proof of the negativity of kernel of Higgs fields for tame harmonic bundles originally proven by Brunebarbe [START_REF]Semi-positivity from Higgs bundles[END_REF] (see also [START_REF]On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applications[END_REF] for systems of log Hodge bundles). With all the tools developed above, we are able to prove Theorem A in § 4.3.

4.1.

Adaptedness to log order and acceptable metrics. We recall some notions in [Moc07, §2.2.2]. Let be a C ∞ -manifold, and be a C ∞ -vector bundle with a hermitian metric ℎ. Let v = ( 1 , . . . , ) be a C ∞ -frame of E. We obtain the ( )-valued function (ℎ, v),whose ( , )-component is given by ℎ( , ).

Let us consider the case = Δ , and = ℓ =1 with = ( = 0). We have the coordinate ( 1 , . . . , ). Let ℎ, and v be as above.

A frame v is called adapted up to log order, if the following inequalities hold over --1 (-

ℓ =1 log | |) -≤ (ℎ, v) ≤ (- ℓ =1 log | |)
for some positive numbers and .

Definition 4.1. Let ( , ) be a log pair, and let be a holomorphic vector bundle on . A hermitian metric ℎ for | -is adapted to log order if for any point ∈ , there is an admissible coordinate ( ; 1 , . . . , ), a holomorphic frame v for | which is adapted up to log order. Definition 4.2 (Acceptable metric). Let ( , ) be a log pair and let ( , ) be a log Higgs bundle over ( , ). We say that the metric ℎ for | -is acceptable, if for any ∈ there is an admissible coordinate ( ; 1 , . . . , ) around , so that the norm | ℎ | ℎ, ≤ for some > 0 over -. Such triple ( , , ℎ) is called an acceptable bundle on ( , ).

One can easily check that acceptable metrics and adaptedness to log order defined above are invariant under bimeromorphic transformations.

Lemma 4.3. Let ( , ) be a log pair, and let : ˜ → be a bimeromorphic morphism so that -1 ( ) = ˜ . For a log Higgs bundle ( , ) over ( , ), one can define a log Higgs bundle ( ˜ , ˜ ) on ( ˜ , ˜ ) by setting ˜ = * and ˜ to be the composition * * --→ * ( ⊗ Ω 1 (log )) → * ⊗ Ω 1 ˜ (log ˜ ). If the metric ℎ for ( , )| -is acceptable or adapt to log order, so is the metric * ℎ for ( ˜ , ˜ )| ˜ -˜ .

Extension of psh functions and negativity of kernel of Higgs fields.

In this subsection we shall prove a result on the extension of plurisubharmonic (psh for short) functions, which will be used in the proof of Theorem A and Proposition 5.6. As a byproduct, we give a very simple proof of the negativity of kernels of Higgs fields of tame harmonic bundles by Brunebarbe [Bru17, Theorem 1.3], which generalizes the earlier work by Zuo [START_REF]On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applications[END_REF] for system of log Hodge bundles. = ( = 0). Let be a psh function on * . We assume that for any > 0, there is a positive constant so that

( ) ≤ ℓ =1 (-log | | 2 )) + on * .
Then extends uniquely to a psh function on .

Proof. Define := + ℓ =1 (log | | 2 ) for any > 0. Then for each > 0, is locally bounded from above, which thus extends to a psh ˜ on the whole by the well-known fact in pluripotential theory. By the maximum principle, for any 0 < < 1, there is a point ∈ (0, ) × • • • (0, ) so that sup

∈Δ(0, )ו••×Δ(0, ) ( ) ≤ ( ) ≤ ( ) where (0, ) := { ∈ Δ | | | = }. Note that the compact set (0, ) × • • • (0, ) is contained in -. Since is psh on -, there exists 0 ∈ (0, ) × • • • (0, ) so that sup ∈ (0, )ו•• (0, ) ( ) ≤ ( 0 ) < +∞.
Hence is uniformly locally bounded from above.

We define the upper envelope ˜ := sup >0 ˜ , and define the upper semicontinuous regularization of ˜ by ˜ ★ ( ) := lim →0 + sup B( , ) ˜ ( ), where B( , ) is the unit ball of radius centered at . Then by the well-known result in pluripotential theory [Dem12b, Chapter 1, Theorem 5.7], ˜ ★ is a psh function on . By our construction, ˜ ★ ( ) = ( ) on -. This proves our result.

A direct consequence of the above lemma is the following extension theorem of positive currents.

Lemma 4.5. Let ( , ) be a log pair and let be a line bundle on . Assume that ℎ is a smooth hermitian metric for | -, which is adapted to log order. Assume further that the curvature form √ -1 ℎ ( | -) ≥ 0. Then ℎ extends to a singular hermitian metric h for with zero Lelong numbers so that the curvature current √ -1 h ( ) is closed and positive. In particular, is a nef line bundle.

Let us show how to apply Lemma 4.4 to reprove the negativity of kernels of Higgs fields of tame harmonic bundles.

Theorem 4.6 (Brunebarbe). Let be a compact Kähler manifold and let be a simple normal crossing divisor on . Let ( , , ℎ) be a tame harmonic bundle on -, and let ( ⋄ , ) be the prolongation defined in [Moc02, §4.1]. Let F be any coherent torsion free subsheaf of ⋄ which lies in the kernel of the Higgs field : ⋄ → ⋄ ⊗ Ω 1 (log ), namely (F ) = 0. Then (i) the singular hermitian metric ℎ| F for F , is semi-negatively curved in the sense of [PT18, Definition 2.4.1]. (ii) The dual F * of F is weakly positive over • -in the sense of Viehweg, where

• ⊂ is the Zariski open set so that F | • → ⋄ | • is a subbundle. (iii) If the harmonic metric ℎ is adapted to log order and F is a subbundle of ⋄ so that (F ) = 0, then the line bundle O P(F * ) (1) admits a singular hermitian metric with zero Lelong numbers so that the curvature current √ -1 (O P(F * ) (1)) ≥ 0; in particular, F * is a nef vector bundle. Proof. By [PT18, Definition 2. 4.1], it suffices to prove that for any open set and any ∈ F ( ), log | | 2 ℎ extends to a psh function on . Pick any point ∈ . By the definition of ⋄ , for any > 0, there are an admissible coordinate ( ; 1 , . . . , ) centered at , and a positive constant so that Let us prove the last statement. Since F is a subbundle of ⋄ , one has • = . Since ℎ is assumed to be adapted to log order, the singular hermitian metric for O P(F * ) (1) thus has zero Lelong numbers everywhere. This implies the nefness of the vector bundle F * . 4.3. Characterization of non-compact ball quotient. Let us state and prove our first main theorem in this paper.

log | | 2 ℎ ≤ ℓ =1 (-log | | 2 )) + on -. Recall that ℎ ( ) + [ , ℎ ] = ℎ ( ) = 0. Since ( ) = 0, we have √ -1 log | | 2 ℎ ≥ - √ -1{ , } | | 2 ℎ - √ -1{ ℎ , ℎ } | | 2 ℎ = - √ -1{ ℎ , ℎ } | | 2 ℎ ≥ 0.
Theorem 4.7. Let be an -dimensional complex projective manifold and let be a simple normal crossing divisor on . Let be an ample polarization on . For the log Hodge bundle (Ω 1 (log ) ⊕ O , ) on ( , ) with defined in (0.1.1), we assume that it is -polystable. Then one has the following inequality

2 2 (Ω 1 (log )) -+ 1 1 (Ω 1 (log )) 2 • 1 ( ) -2 ≥ 0. (4.3.1)
When the above equality holds,

(i) if is smooth, then -≃ B
Γ for some torsion free lattice Γ ⊂ ( , 1) acting on B . Moreover, is the (unique) toroidal compactification of B Γ , and each connected component of is the smooth quotient of an Abelian variety by a finite group acting freely on .

(ii) If is not smooth, then the universal coverofis not biholomorphic to B , though there exists a holomorphic map -→ B which is locally biholomorphic.

In both cases, + is big, nef and ample over -.

Proof. Denote the log Hodge bundle ( , ) = ( 1,0 ⊕ 0,1 , ) by 1,0 := Ω 1 (log ), 0,1 := O .

By [Moc06, Theorem 6.5] we have the following Bogomolov-Gieseker inequality for ( , )

2 2 (Ω 1 (log )) -+ 1 1 (Ω 1 (log )) 2 • 1 ( ) -2 = (4.3.2) 2 2 ( ) - rank -1 rank 1 ( ) 2 • 1 ( ) -2 ≥ 0
This shows the desired inequality (4.3.1).

The rest of the proof will be divided into three steps. In Step 1, we shall construct a uniformizing variation of Hodge structures onso that the corresponding period map defined in (2.0.4) induces a holomorphic map (so-called period map in Remark 2.7) from the universal cover ofto B which is locally biholomorphic. By (2.0.5), this period map is moreover an isometry if we equipwith hermitian metric induced by the Hodge metric. In Step two we will prove that, when is smooth, the hermitian metric oninduced by the Hodge metric is complete. Together with arguments in Remark 2.7, this proves that the above period map is indeed a biholomorphism. In Step three we shall prove Theorem 4.7.(ii) and the positivity of + .

Step 1. We apply Proposition 2.11 to the above system of log Hodge bundles ( 1,0 ⊕ 0,1 , ). Then there is a principal system of log Hodge bundles ( , ) on ( , ) with the structure group = ( ( 1,0 ) × ( 0,1 )) with rank 1,0 = rank 1,0 = , and rank 0,1 = rank 0,1 = 1. Here we use the notations in Example 2.1. Then by Proposition 2.11 the Hodge group relative to ( , ) is 0 = ( , 1), and 0 = ∩ 0 = ( ( ) × (1)) = ( ). For the complexified group = ( ) of 0 , its adjoint representation : → ( ) = ( ( )) is faithful. By Example 2.9, this is a Hodge distinguish the sections of log Higgs bundles and log forms, we write 1 := log 1 and = for = 2, . . . , . Denote by 0 = 1 the constant section of O . Let us introduce a new metric h on ( , )| * as follows.

| 1 | 2 h := (-log | 1 | 2 ); , h := 0 for ≠ ; | | 2 h := 1 for = 2, . . . , ; | 0 | 2 h := (-log | 1 | 2 ) -1 .
Write ℎ := | | 2 h, and h ( ) := h ( ) ⊗ * ⊗ . Then for , = 2, . . . , , one has

h ( ) 11 = h ( ) 10 = h ( ) 01 = h ( ) 0 = h ( ) 0 = 0 h ( ) = (-log | 1 | 2 ) -1 ¯ ∧ h ( ) 1 = 1 (-log | 1 | 2 ) 2 ¯ 1 ¯ 1 ∧ h ( ) 1 = 1 (-log | 1 | 2 ) 1 ¯ ∧ 1 h ( ) 00 = =2 (-log | 1 | 2 ) -1 ∧ ¯ .
In conclusion, there is a constant 1 > 0 so that one has

| h ( )| 2 ℎ, = 0≤ , ≤ | h ( ) ⊗ * ⊗ | 2 ℎ, ≤ 1 (-log | 1 | 2 ) 3 | 1 | 2 (4.3.4) over * ( 1 2 ) (notation defined in Definition 1.3), where = √ -1 =1 ∧ ¯ is the Euclidean metric on * .
We abusively denote by h the induced metric on (E, )| * := (End( ), ( ) )| * , which is adapted to log order on ( , ∩ ) in the sense of Definition 4.1 by our construction. Then

h (E) = h ( ) ⊗ * + ⊗ h * ( * ) = h ( ) ⊗ * -⊗ h ( ) †
where h ( ) † is the transpose of h ( ). Hence

h (E)( ⊗ * ) = ,ℓ ( ℓ h ( ) - h ( ) ℓ )( ⊗ * ℓ )
for 0 ≤ , , , ℓ ≤ . It then follows from (4. 3.4) that

| h (E)| 2 ℎ, ≤ 2 (-log | 1 | 2 ) 3 | 1 | 2 (4.3.5) over * ( 1
2 ) for some constant 2 > 0. Consider the identity map for E, which can be seen as a holomorphic section of End(E, E). We denote by (F , Φ) := (End(E, E), (E) ) the induced Higgs bundle by (E, ). One can check that Φ( ) = 0. (4. 3.6) We equip F | * with the metric ℎ F := h ⊗ ℎ * , where ℎ is the harmonic metric for (E, )| -constructed in Step one. Note that

ℎ F (F ) = h (E) ⊗ E * + E ⊗ ℎ * (E * ) = h (E) ⊗ E *
By (4.3.5), there is a constant 0 > 0 so that one has

| ℎ F (F )| ℎ F , ≤ 0 (-log | 1 | 2 ) 3 2 | 1 | (4.3.7) over * ( 1 2 ). Then √ -1 log | | 2 ℎ F ≥ - √ -1{ ℎ F , } | | 2 ℎ F = - √ -1{Φ , Φ } | | 2 ℎ F - √ -1{Φ ℎ F , Φ ℎ F } | | 2 ℎ F - √ -1{ ℎ F (F ) , } | | 2 ℎ F ≥ - √ -1{ ℎ F (F ) , } | | 2 ℎ F .
Here the third inequality follows from (4.

Δ ≥ -| ℎ F (F )| ℎF , ≥ - 0 (-log | 1 | 2 ) 3 2 | 1 | =:
where 0 is some uniform constant which does not depend on . Note that

2 := ∫ 0<| 1 |< 1 2 | ( 1 )| 2 1 ¯ 1 < 4 (4.3.8)
for some constant 4 > 0. For any fixed 0 ≤ 2 , . . . , ≤ 1 2 , consider the Dirichlet problem

= on { 1 | | 1 | = 1 2 } Δ = on { 1 | 0 < | 1 | < 1 2 } (4.3.9)
By (4. 3.8) and the elliptic estimate, one has sup

0<| 1 |< 1 2 | ( 1 )| ≤ 5 ( 2 + sup | 1 |= 1 2 ). (4.3.10) over { 1 | 0 < | 1 | < 1
2 } for some uniform positive constant 5 which does not depending on . Hence Δ( -) ≥ 0 over

{ 1 | 0 < | 1 | < 1 2 }.
Since both ℎ and h are adapted to log order, so is ℎ F . Hence there is a constant 6 > 0 so that

log | | 2 ℎ F ≤ 6 log(- ℓ =1 log | |)
over * ( 1 2 ). By Lemma 4.4, we conclude thatextends to a subharmonic function on

{ 1 | | 1 | < 1 2 }. Note that ( 1 ) -( 1 ) = 0 when | 1 | = 1 2 .
Hence by maximum principle,

( 1 ) ≤ ( 1)

for any 0 < | 1 | < 1 2 . Let 7 := sup | 1 |= 1 2 ,0≤ 2 ,..., ≤ 1 2 ( 1 )
which is finite. By (4. 3.8) and (4.3.10), we have sup

0<| 1 |< 1 2 ,0≤ 2 ,..., ≤ 1 2 log | | 2 ℎ F ( 1 , . . . , ) ≤ 5 ( 4 + 7 ).
This implies that ℎ ≥ 8 • h over * ( 1 2 ) for some constant 8 > 0. By (4.3.5), one has

| h * (E * )| 2 ℎ * , ≤ 0 (-log | 1 | 2 ) 3 | 1 | 2 .
Hence if we use the metric ℎ⊗ h * for F and do the same proof, we can prove that ℎ ≤ 9 • h over * ( 1 2 ) for some constant 9 > 0. Therefore, h and ℎ are mutually bounded on * ( 1 2 ). By 3.12) we obtain the norm estimate for the metric

( 1 1 ) = * 1 ⊗ 0 (4.3.11) ( ) = * ⊗ 0 for = 2, . . . , , (4.
* ℎ ∼ * h = √ -1 1 ∧ ¯ 1 | 1 | 2 (log | 1 | 2 ) 2 + =2 √ -1 ∧ ¯ -log | 1 | 2 (4.3.13)
Though * ℎ is strictly less than the Poincaré metric near , one can easily prove that it is still a complete metric. Therefore, the hermitian metric * ℎ = * ℎ onis also complete. Based on Remark 2.7, we conclude thatis uniformized by the complex unit ball of dimension , namely, there is a torsion free lattice Γ ⊂ ( , 1) so that -≃ B Γ . By (4. 3.11) and (4.3.12), the canonical Kähler-Einstein metric := * ℎ for (-log )| is adapted to log order. It follows from Theorem A.7 that is the unique toroidal compactification for the non-compact ball quotient B Γ . We accomplish the proof of Theorem 4.7.(i).

Step 3. Assume now is not smooth. By (4. 3.3), the period map -→ B is locally biholomorphic. Assume by contradiction that it is an isomorphism. Since ℎ is adapted to log order, the canonical Kähler-Einstein metric := * ℎ for (-log )| is also adapted to log order. It follows from Theorem A.7 that cannot be singular. The contradiction is obtained, and thus the period map is not a uniformizing mapping. We proved Theorem 4.7.(ii).

Let us show that + is big, nef and ample over -. Note that the metric det -1 for ( + )| is adapted to log order, and that √ -1

2 det -1 (( + )| ) = ( + 1) .
By Lemma 4.5, the hermitian metric det -1 extends to a singular hermitian metric ℎ + for + with zero Lelong numbers. Hence + is nef. Since √ -1 ℎ + ( + ) > 0 on -, + is thus big and ample over -. We finish the proof of the theorem.

Remark 4.8. Note that the asymptotic behavior of the metric (4. 3.13) is exactly the same as that of the Kähler-Einstein metric for the ball quotient near the boundary of its toroidal compactification (see [Mok12, eq. (8) on p. 338]). This is indeed the hint for our construction of h. Remark 4.9. We expect that Theorem 4.7.(ii) cannot happen. This is the case when dim = 2. Indeed, when the Miyaoka-Yau type equality in (0.1.2) holds, together with the conclusion that + is big, nef and ample overin Theorem 4.7, it follows from [START_REF]Einstein-Kähler V-metrics on open Satake V-surfaces with isolated quotient singularities[END_REF] thatis uniformized by B 2 , which is a contradiction to Theorem 4.7.(ii).

H

In this section, we will prove Theorem B. § § 5.1 and 5.2 are technical preliminaries. In § 5.3 we prove that a log Higgs bundle ( , ) on a compact Kähler log pair is slope polystable with respect to some polarization by big and nef cohomology (1, 1)-class, if ( , ) admits a Hermitian-Yang-Mills metric with "mild singularity" near the boundary divisor. In § 5.4 we use the Bergman metric for quotients of complex unit balls by torsion free lattices to construct such Hermitian-Yang-Mills metric. This proves Theorem B. 5.1. Notions of positivity for curvature tensors. We recall some notions of positivity for Higgs bundles in [DH19, §1.3].

Let ( , ) be a Higgs bundle endowed with a smooth metric ℎ. For any ∈ , let 1 , . . . , be a frame of at , and let 1 , . . . , be its dual in * . Let 1 , . . . , be a local coordinate centered at . We write

ℎ ( ) = ℎ ( ) + [ , ℎ ] = ¯ ∧ ¯ ⊗ ⊗ Set ¯ ¯ := ℎ ¯ ¯ , where ℎ ¯ = ℎ( , ). ℎ ( ) is called Nakano semi-positive at if , , , ¯ ¯ ≥ 0 for any = , ⊗ ∈ ( 1,0 ⊗ ) . ( , , ℎ) is called Nakano semipositive if ℎ (
) is Nakano semi-positive at every ∈ . When = 0, this reduces to the same positivity concepts in [Dem12b, Chapter VII, §6] for vector bundles.

We write

ℎ ( ) ≥ Nak ( ⊗ ) for ∈ R if , , , ( ¯ ¯ - ¯ ℎ ¯ )( ) ≥ 0
for any ∈ and any = , ⊗ ∈ ( 1,0 ⊗ ) . Let us recall the following lemma in [DH19, Lemma 1.8].

Lemma 5.1. Let ( , , ℎ) be a Higgs bundle on a Kähler manifold ( , ). If there is a positive constant so that | ℎ ( )| ℎ, ≤ for any ∈ , then

⊗ ≥ Nak ℎ ≥ Nak - ⊗ .
The following easy fact in [DH19, Lemma 1.9] will be useful in this paper.

Lemma 5.2. Let ( 1 , 2 , ℎ 1 ) and ( 2 , 2 , ℎ 2 ) are two metrized Higgs bundles over a Kähler manifold ( ,

) such that | ℎ 1 ( )| ℎ 1 , ≤ 1 and | ℎ 2 ( )| ℎ 2 , ≤ 2 for all ∈ .
Then for the hermitian vector bundle ( 1 ⊗ 2 , ℎ 1 ℎ 2 ), one has

| ℎ 1 ⊗ℎ 2 ( )| ℎ 1 ⊗ℎ 2 , ≤ 2 2 2 1 + 2 1 2 2
for all ∈ . Here := rank .

Some pluripotential theories.

In this subsection we recall some results of deep pluripotential theories in [START_REF]Monge-Ampère equations in big cohomology classes[END_REF][START_REF]Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor[END_REF]. The results in this subsection will be used in the proof of Proposition 5.6. Let us first recall the definitions of big or nef cohomology (1, 1)-classes in [Dem12a, §6].

Definition 5.3. Let ( , ) be a compact Kähler manifold. Let ∈ 1,1 ( , R) be a cohomology (1, 1)-class of . The class is nef (numerically eventual free) if for any > 0, there is a smooth closed (1, 1)-form ∈ so that ≥ -. The class is big if there is a closed positive (1, 1)-current ∈ so that ≥ for some > 0. Such a current will be called a Kähler current.

Let be a complex manifold of dimension and let ⊂ be a Zariski open set of . Pick a smooth hermitian form on . For any smooth differential form of degree on so that

∫ | | ∧ < +∞,
one can trivially extend to a current on of degreeby setting

, := ∫ ∧ (5.2.1)
where is the any test form of degree which has compact support. In general, might not be closed even if is closed.

Let ( , ) be a compact Kähler manifold of dimension . Let 1 , . . . , be big cohomology classes. Let ∈ be positive closed (1, 1)-currents whose local potential is locally bounded outside a closed analytic subvariety of (a particular case of small unbounded locus of [BEGZ10, Definition 1.2]). In this celebrated work by Boucksom-Eyssidieux-Guedj-Zariahi [START_REF]Monge-Ampère equations in big cohomology classes[END_REF], they defined non-pluripolar product for these currents 1 ∧ • • • ∧ which is a closed positive ( , )-current, and does not charge on any closed proper analytic subsets. Therefore, if we assume further that is smooth overwhere is a closed analytic subvariety of , then

1 ∧ • • • ∧ is nothing but the trivial extension of the ( , )-form ( 1 ∧ • • • ∧ )| -to .
Following [BEGZ10, Definition 1.21], for a big class , a positive (1, 1)-current ∈ has full Monge-Ampère mass if ∫ = Vol( ).

The set of such positive currents in with full Monge-Ampère mass is denoted by E ( ).

We will not recall the definition of the volume of big classes by Boucksom in [START_REF]On the volume of a line bundle[END_REF].

We just mention that when the class is big and nef, one has

Vol( ) = .

The following lemma will be used in § 5.3.

Lemma 5.4. Let ( , ) be a compact Kähler manifold and let be a simple normal crossing divisor on . Let be a closed positive (1, 1)-current on so that | -is a smooth (1, 1)-form overwhich is strictly positive at one point and has at most Poincaré growth near . Then the cohomology class := { } is big and nef, and ∈ E ( ).

Proof. Let be the Kähler current on constructed in Remark 1.5. Since | -has at most Poincaré growth near , there exists a constant 1 > 0 so that 1 -≥ 0. Pick any point ∈ . Then there exists some admissible coordinates ( ; 1 , . . . , ) centered at so that the local potential of satisfies that

≥ -1 log(- ℓ =1 log | 1 | 2 ) -2
for some constant 2 > 0. Hence has zero Lelong numbers everywhere and thus is nef. Since is strictly positive at one point on -, it is big by [START_REF]On the volume of a line bundle[END_REF]. It follows from [Gue14, Proposition 2.3] that ∈ E ( ). The lemma is proved.

Let us recall an important theorem in [BEGZ10].

Theorem 5.5 ([BEGZ10, Corollary 2.15]). Let ( , ) be a compact Kähler manifold of dimension . Let 1 , . . . , be big and nef classes on . For ∈ E ( ) which are all smooth outside a closed proper analytic subset , one has

∫ - 1 ∧ • • • ∧ = ∫ 1 ∧ • • • ∧ = 1 • • • .

5.3.

Hermitian-Yang-Mills metric and stability. Let ( , ) be a compact Kähler manifold and let be a simple normal crossing divisor on . For applications of birational geometry, one usually considers more general polarization by big and nef line bundles.

In this subsection, we will prove that a log Higgs bundle ( , ) on ( , ) is -polystable if ( , )| -admits a Hermitian-Yang-Mills metric whose growth at infinity is "mild", where is certain big and nef cohomology class. When dim = 1 or = and the polarization is Kähler, this has been proved by Simpson [START_REF]Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization[END_REF][START_REF]Harmonic bundles on noncompact curves[END_REF]. As we have seen in Theorem 1.8, when is projective and both the first and second Chern classes of vanish and the polarization is an ample line bundle, this result has been proved by Mochizuki.

We start with the following technical result, which is strongly inspired by the deep result of Guenancia [Gue16, Proposition 3.8].

Proposition 5.6. Let ( , 0 ) be a compact Kähler manifold and let be a simple normal crossing divisor on . Let ( , ) be a log Higgs bundle on ( , ). Let be a big and nef cohomology (1, 1)-class containing a positive closed (1, 1)-current ∈ so that | - is a smooth Kähler form and has at most Poincaré growth near . Assume that there is a hermitian metric ℎ for ( , )| -which is adapted to log order (in the sense of Definition 4.1) and is acceptable (in the sense of Definition 4.2). Then for any saturated Higgs subsheaf ⊂ , one has

1 ( ) • -1 = ∫ -- ( √ -1 ℎ ( )) ∧ -1 (5.3.1)
where is the analytic subvariety of codimension at least two so that | -⊂ | -is a subbundle, and ℎ is the metric on induced by ℎ.

Proof. By Remark 1.5, one can construct a Kähler current

:= 0 - √ -1 log(- ℓ =1 log | • | 2 ℎ ), (5.3.2)
over , whose restriction onis a complete Kähler form , which has the same Poincaré growth near . Here is the section 0 ( , O ( )) defining , and ℎ is some smooth metric for the line bundle O ( ). Since we assume that ℎ is acceptable, (after rescaling by multiplying a constant) one thus has

| ℎ ( )| ℎ, ≤ 1. By Lemma 5.1, one has -⊗ ≤ ℎ ( ) ≤ ⊗ over -.
We first consider the case that is an invertible saturated subsheaf of which is invariant under . Then the metric ℎ of induces a singular hermitian metric ℎ for defined on the whole , which is smooth on on • := --. The curvature current √ -1 ℎ ( ) is a closed (1, 1)-current on -, which is a smooth (1, 1)-form on • . Define by : | • → | • the orthogonal projection with respect to ℎ and ⊥ : | • → ⊥ | • the projection to its orthogonal complement. By the Chern-Weil formula (see for example [Sim88, Lemma 2.3]), over • , we have

ℎ ( ) = ℎ ( ) = ℎ ( )| + ℎ ∧ -∧ ℎ (5.3.3)
where ℎ ( )| is the orthogonal projection of ℎ ( ) on Hom( , )| • = O • , and ∈ A 1,0 ( • , Hom( , ⊥ )) is the second fundamental form, and

∈ A 1,0 ( • , Hom( ⊥ , )) is equal to | ⊥ . Hence √ -1 ℎ ( ) ≤ √ -1 ℎ ( )| . For any local frame of | • , note that | | 2 ℎ • √ -1 ℎ ( )| = √ -1 ℎ ( )( ), ℎ ≤ ⊗ , ℎ = | | 2 ℎ • Hence √ -1 ℎ ( )| -
is a semi-negative (1, 1)-form on • , and thus over • one has

- √ -1 ℎ ( ) + ≥ - √ -1 ℎ ( )| ≥ 0
Since we assume that ( , ℎ) is adapted to log order, ( -1 | -, ℎ -1 | -) is thus adapted to log order for the log pair ( -, -). By Lemma 4.5 and (5.3.2), -√ -1 ℎ ( ) + extends to a closed positive (1, 1)-current on -. Since is of codimension at least two, a standard fact in pluripotential theory shows that -√ -1 ℎ ( ) + extends to a positive closed (1, 1)-current on the whole .

Denote by ∈ 0 ( , ⊗ -1 ) the section defining the inclusion → . We fix a smooth hermitian metric ℎ 0 for and we define a function :

= | | 2 ℎ•ℎ -1 0 = ℎ • ℎ -1 0 on -. Then √ -1 log = √ -1 ℎ 0 ( ) - √ -1 ℎ ( ). (5.3.4)
Hence there is a constant 0 > 0 so that √ -1 log + 0 ≥ . (5.3.5) By Lemma 5.4, ∈ E ( ). Since √ -1 ℎ 0 ( ) is a smooth (1, 1)-form on , it follows from Theorem 5.5 that ∫

• √ -1 ℎ 0 ( ) ∧ -1 = 1 ( ) • -1 .
To prove (5.3.1), by (5.3.4) and the above equality it suffices to prove that ∫

• √ -1 log ∧ -1 = 0. (5.3.6)
We will pursue the ideas in [Gue16, Proposition 3.8] to prove this equality.

Let us take a log resolution : ˜ → of the ideal sheaf I defined by ∈ 0 ( , ⊗ -1 ), with O ˜ (-) = * I and ˜ := -1 ( ) a simple normal crossing divisor. Let us denote by ( ˜ , ˜ ) the induced log Higgs bundle on ( ˜ , ˜ ) by pulling back ( , ) via .

Then the metric h := * ℎ for ( ˜ , ˜ )| ˜ -˜ is also adapted to log order and acceptable by Lemma 4.3. Note that Supp(O /I ) = . Write ˜ := * . There is a nowhere vanishing section

˜ ∈ 0 ( ˜ , ˜ ⊗ ˜ -1 ⊗ O ˜ (-))
so that * = ˜ • , where is the canonical section in 0 ( ˜ , O ˜ ( )) which defines the effective exceptional divisor .

Fix a Kähler form ˜ on ˜ , as Remark 1.5 we construct another Kähler current

˜ := ˜ - √ -1 log(- =1 log | • ˜ | 2 h ), (5.3.7)
over ˜ , whose restriction on ˜ -˜ is a complete Kähler form, which has the same Poincaré growth near ˜ . Here ˜ is the section 0 ( , O ( ˜ )) defining ˜ , and h is some smooth metric for the line bundle O ˜ ( ˜ ).

Let us fix a smooth hermitian metric ℎ for O ˜ ( ).

Write ˜ := | ˜ | 2 h• * ℎ -1 0 •ℎ -1 .
Since h is adapted to log order and ˜ is nowhere vanishing, there is a constant 1 , 2 > 0 so that log ˜ ≥ 1 -2 , (5. 3.8) where we denote by := -log(-

ℓ =1 log | • ˜ | 2 h ).
Since h := * ℎ for ( ˜ , ˜ )| ˜ -˜ is acceptable, by same arguments as those for (5.3.5), one can show that √ -1 log ˜ + 3 ˜ ≥ ˜ over ˜ -˜ for some constant 3 > 0. Note that the local potential of √ -1 log ˜ + 3 ˜ is bounded from below by ( 1 + 3 ) according to (5. 3.8). By [Gue14, Proposition 2.3], one has √ -1 log ˜ + 3 ˜ ∈ E ({ 3 ˜ }). One can check that * ≤ 4 ˜ for some constant 4 > 0. By Lemma 5.4 again, * ∈ E ( * ). Hence by Theorem 5.5 one has

∫ -1 ( • ) ( √ -1 log ˜ + 3 ˜ ) ∧ * -1 = { 3 ˜ } • * -1 .
Recall that ˜ ∈ E ({ ˜ }) by Lemma 5.4. Hence ∫

-1 ( • )

3 ˜ ∧ * -1 = { 3 ˜ } • * -1 .
One thus has

∫ -1 ( • ) √ -1 log ˜ ∧ * -1 = 0. (5.3.9) Note that over ˜ -˜ , one has √ -1 log ˜ + [ ] - √ -1 ℎ ( ) = * √ -1 log
where [ ] is the current of integration of . Hence over √ -1 ℎ ( ) ∧ * -1 = 1 ( ) • * -1 = 0, (5. 3.11) where the last equality follows from the fact that is -exceptional. (5.3.9), (5.3.10) together with (5. 3.11) shows the desired equality (5.3.6). We finish the proof of (5.3.1) when rank = 1.

-1 ( • ) ≃ • , one has √ -1 log ˜ - √ -1 ℎ ( ) = * √ -1 log . (5.
Assume that rank = . We replace ( , , ℎ) by the wedge product ( ˜ , ˜ , h) := Λ ( , , ℎ). By Lemma 5.2, the induced metric h is also acceptable and one can easily check that it is also adapted to log order. Note that det is also invariant under ˜ , and that det → Λ . We then reduce the general cases to rank 1 cases. The proposition is thus proved.

Let us state and prove the main result in this section.

Theorem 5.7. Let be a compact Kähler manifold and let be a simple normal crossing divisor on . Let be a big and nef cohomology (1, 1)-class containing a positive closed (1, 1)-current ∈ so that | -is a smooth Kähler form and has at most Poincaré growth near . Let ( , ) be a log Higgs bundle on ( , ). Assume that there is a hermitian metric ℎ on ( , )| -such that • it is adapted to log order (in the sense of Definition 4.1); • it is acceptable (in the sense of Definition 4.2); • it is Hermitian-Yang-Mills:

Λ ℎ ( ) ⊥ = 0. Then ( , ) is -polystable.

Proof. We shall use the same notations as those in Proposition 5.6. Let be any saturated Higgs-subsheaf ⊂ , and denote by the analytic subvariety of codimension at least two so that | -⊂ | -is a subbundle. By the Chern-Weil formula again, over

• := --we have

Λ ℎ ( ) = Λ ( ℎ ( )) rank ⊗ + Λ ( ℎ ∧ -∧ ℎ ).
where ∈ A 1,0 ( • , Hom( , ⊥ )) is the second fundamental form of in with respect to the metric ℎ, and

∈ A 1,0 ( • , Hom( ⊥ , )) is equal to | ⊥ . Hence ∫ • ( √ -1 ℎ ( )) ∧ -1 = ∫ • rank rank ( √ -1 ℎ ( )) ∧ -1 -(| | 2 ℎ + | | 2 ℎ ) .
By Proposition 5.6 together with the above inequality, one concludes the slope inequality

( ) ≤ ( )
and the equality holds if and only if ≡ 0 and ≡ 0. We shall prove that if the above slope equality holds, is a sub-Higgs bundle of , and we have the decomposition

( , ) = ( , | ) ⊕ ( , )
where ( , ) is another sub-Higgs bundle of . Set rank = and rank = . We first prove that is a subbundle of . It is equivalent to show that det → Λ is a subbundle, and we thus reduce the problem to the case that rank = 1. Assume that ( ) = ( ) and thus ≡ 0 and ≡ 0. By (5.3.3), over

• one has √ -1 ℎ ( ) = √ -1 ℎ ( )| ≥ -| •, (5.3.12)
where is the Kähler current defined in (5.3.2). By Lemma 4.5, √ -1 ℎ ( ) + extends to a closed positive (1, 1)-current on -, and thus to the whole .

Assume now 0 ∈ is a point where ( / ) 0 is not locally free. Take a local holomorphic frame of on some open neighborhood ( ; 1 , . . . , ) of , and a holomorphic frame 1 , . . . , of . Then = =1 ( ) , where ∈ O( ) so that 1 ( 0 ) = • • • = ( 0 ) = 0. By the asssumption that ℎ is adapted to log order, one concludes that 3.13) the cohomology class ( (-log )) ∈ , ( ). Let us first prove (0.1.3), which is indeed an easy computation.

log | | 2 ℎ ≤ 1 log(| 1 | 2 + • • • + | | 2 ) + 2 log(-log( ℓ =1 | | 2 )) (5.
For any 0 ∈ -, we take a normal coordinate system ( 1 , . . . , ) centered at 0 so that

= √ -1 1≤ℓ, ≤ ℓ ℓ ∧ ¯ - , ,ℓ, ℓ ¯ + (| | 3 )
where ℓ is the coefficients of the Chern curvature tensor

( ) = , ,ℓ, ℓ ∧ ¯ ⊗ ( ℓ ) * ⊗ .
By [Mok89, p. 177], one has

ℓ ( 0 ) = -( ℓ + ℓ ). (5.4.1)
The main motivation of this appendix is to provide one building block for Theorem 4.7.(ii). Our main result, Theorem A.7, says that there is no other smooth compactification for non-compact ball quotient than the toroidal one, so that the Bergman metric grows "mildly" near the boundary. Besides its own interests, this result is applied to show that the smoothness of in Theorem A is necessary if one would like to characterize non-compact ball quotients.

A.1. Toroidal compactifications of quotients by non-neat lattices. In this section, we recall a well known way of constructing the toroidal compactifications of ball quotients in the case where the lattice has torsion at infinity. The reader will find more details about the natural orbifold structure on these compactifications in [START_REF]Orbifold Kähler Groups related to arithmetic complex hyperbolic lattices[END_REF]. For our purposes, the basic result given in Proposition A.1 will be sufficient.

Recall that we say that a lattice Γ ⊂ ( , 1) is neat (cf. [Bor69]) if for any ∈ Γ, the subgroup of C * generated by the eigenvalues of is torsion free. This implies that Γ is torsion free and that all parabolic elements of Γ are unipotent, so that the toroidal compactifications of B Γ provided by [START_REF]Smooth compactifications of locally symmetric varieties[END_REF][START_REF]Projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume[END_REF] are smooth (there is no "torsion at infinity"). Note that by [Bor69, Proposition 17.4] in the arithmetic case, and [Bor63], or [Rag72, Theorem 6.11] in general, any lattice in ( , 1) admits a finite index neat normal sublattice. Before explaining how to prove Proposition A.1, let us recall the construction of ′ as it is defined in [START_REF]Projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume[END_REF] (see also [Cad16] for a similar discussion).

Each component of ′ -′ is associated to a certain Γ ′ -orbit of points of B , whose points are called the Γ ′ -rational boundary components of (cf. [AMRT10, Chapter 3] or [Mok12, §1.3]). Let ∈ B be such a point, and let ⊂ ( , 1) be its stabilizer. This is a maximal parabolic real subgroup of ( , 1); let us denote by its unipotent radical. This group is an extension 1 → → → → 1, where C -1 , and R is the center of . Let = . This reductive group can be embedded as a Levi subgroup in , so that = • . Moreover, we have a further decomposition = ( -1) × R. (all this description can be obtained e.g. by specializing the discussion of [BB66, Section 1.3] or [AMRT10, Section 4.2] to the case of the ball).

This Lie theoretic description of can be understood more easily by expressing the action of the previous groups on the horoballs tangent to . Let ( ( ) ) ≥0 be the family of these horoballs. Each ( ) ⊂ B can be described as an open subset in a Siegel domain of the third kind, as follows:

(A.1.1) ( ) ≃ {( ′ , ) ∈ C -1 × C | Im > || ′ || 2 + }.
We have (0) B , and when = (0, ..., 0, 1), the change of coordinates between the two descriptions of the ball is given by the Cayley transform 1 -, 1 + 1 -) ∈ (0) (0,...,0,1) .

Since has klt singularities, the work of Hacon-McKernan [HM07] implies that every fiber of is rationally connected. Thus, ( -1 ( )) is a point since abelian varieties do not contain rational curves. This gives a contradiction.

Remark A.5. If we make the more restrictive hypothesis that has at most quotient singularities, we can replace the use of [HM07] by the work of Kollar [Kol93], which implies that each fiber of is simply connected. Asis a disjoint union of abelian varieties, this also implies that the image of ˜ : -1 ( ) → -must be a point.

Let us introduce a natural class of pairs under which our rigidity theorem will hold.

Definition A.6. Let ( , ) be a pair consisting of normal algebraic variety and a reduced divisor. We say that ( , ) has algebraic quotient singularities if it admits a finite affine cover ( ) ∈ , such that each ( , ∩ ) is the quotient of a smooth SNC pair ( , ) by a finite group leaving invariant.

Note that for any lattice Γ ⊂ Aut(B ), if is the toroidal compactification of = B Γ described in Section A.1, then ( , -) has algebraic quotient singularities.

We can now state our main result as follows.

Theorem A.7. Let := B Γ be an -dimensional ball quotient by a torsion free lattice Γ ⊂ ( , 1). Let be a klt compactification of , and let := -. Let (1) ⊂ be the divisorial part of . If the Kähler-Einstein metric on the bundle (-log (1) )| is adapted to log order near the generic point of any component of (1) , then ( , ) identifies with the toroidal compactification of .

Remark A.8.

(1) Under the more restrictive assumption that ( , ) has algebraic quotient singularities, the use of Lemma A.4 in our proof below can be made without appealing to the difficult result of [HM07] (see Remark A.5).

(2) As an easy consequence of Theorem A.7, we can remark that there is no klt compactification of such thathas codimension ≥ 2.

Corollary A.9. With the same assumptions as in Theorem A.7, if is smooth and has simple normal crossings, then is in fact smooth, and each component is a smooth quotient of an abelian variety by some finite group acting freely on .

Let us prove Theorem A.7. Let Γ ′ ⊂ Γ be a subgroup of finite index so that all parabolic elements of Γ ′ are unipotent. Writing ′ := B Γ ′, this gives a finite étale surjective morphism ′ → .

Let ′ be the normalization of in the function field of ′ : this is a normal projective variety ′ compactifying ′ , with a compatible finite surjective morphism : ′ → (see e.g. [AHCG11, Chapter 12, §9]). Since klt singularities are preserved under finite surjective morphisms, the variety ′ has at most klt singularities (see [START_REF]Birational geometry of algebraic varieties[END_REF]Corollary 5.20

]).

Remark A.10. If ( , ) has algebraic quotient singularities, one sees easily that this is also the case for ′ . To see this, form the fiber product ′ = × ′ , where → is an affine covering as in Definition A. 6. By [Kol07, Theorem 2.23], the variety ′ , endowed with it natural boundary divisor, has algebraic quotient singularities. Finally, Lemma A.14 shows that ′ → ′ is a quotient map, which gives the result.

Let ′ be the toroidal compactification of ′ , so that the boundary := ′ -′ is a smooth divisor.

Lemma A.11. The identity map on ′ extends as an isomorphism : ′ → ′ . In particular, there is a finite surjective morphism : ′ → , which identifies with the étale and surjective map ′ → over -.

Proof. Since ′ is klt, Lemma A.4 shows that the identity map of ′ extends to a birational morphism : ′ → ′ . Assume by contradiction that is not an isomorphism. As ′ is smooth, it follows from [KM98, Corollary 2.63] that the exceptional set Ex( ) is of pure codimension one. Thus, the birational morphism must contract an irreducible divisorial component of the boundary ′ := ′ -′ .

Denote by sing the singular locus of , and let ′ := * , be the canonical Kähler Einstein metric on ′ . Lemma A.12 below shows that ′ is adapted to log-order for ′• (-log • ), where ′• := -1 ( -sing ), and • := ′• ∩ . We are going to derive a contradiction with the fact the is contracted. Let 1 be the component of containing ( ). We can take admissible coordinates (W; 1 , . . . , ) and (U; 1 , . . . , ) centered at some well-chosen ′ ∈ ∩ ′• and := ( ′ ) ∈ 1 respectively so that (W) ⊂ U, and | : → ( ) is smooth at ′ . Denote by ( 1 ( ), . . . , ( )) the expression of within these coordinates. Then if the admissible coordinates are chosen properly, one has

( 1 ( ), . . . , ( )) = ( 1 1 1 ( ), . . . , 1 ( ), +1 , . . . , ) where 1 ( ), . . . , ( ) are holomorphic functions defined on W so that ( ) ≠ 0 and ≥ 1 for = 1, . . . , . Since is exceptional, one has ≥ 2. By the norm estimate in [Mok12, eq. (8) on p. 338], the Kähler-Einstein metric for (-log )| is adapted to log order. More precisely, one has

| 2 | 2 -1 ∼ (-log | 1 | 2 ).
Since * log 2 = 2 log 1 + log 2 ( ), one thus has the following norm estimate

| log 1 | 2 ′-1 ≥ 1 2 2 * | log 2 | 2 -1 - 1 2 2 * | 2 2 | 2 -1 ≥ (-log | 1 | 2 ) | 1 | 2 2
for some constants > 0. Since log 1 is a local nowhere vanishing section for Ω 1 ′ (log ′ ), we conclude that the metric ′-1 for Ω 1 ′• (-log ′• ) is not adapted to log order, and so is ′ for ′• (-log ′• ). This gives a contradiction, and ends the proof of the lemma.

Lemma A.12. With the notations of the proof of Lemma A.11, the metric ′ is adapted to log-order for ′• (-log • ).

Proof. Write W := -1 (V). Since | W-′ : W -′ → V -is a finite unramified cover, the image of ( | W-′ ) * 1 (W -′ ) is a subgroup of 1 (W -) ≃ Z of index . Letting ( 1 , • • • , ) = ( 1 , 2 , . . . , ), one has thus the following commutative diagram

Δ * × Δ -1 W Δ V | Δ * ×Δ -1 ℎ • | W ≃
so that ℎ • Δ * ×Δ -1 : Δ * × Δ -1 → W ∩ ′ is an isomorphism. By the Riemann removable singularities theorem, ℎ extends to a holomorphic map ℎ : Δ → W, which is easily seen to be surjective with finite fibers. Hence ℎ is moreover biholomorphic. (W; 1 , . . . , ; ℎ) is therefore an admissible coordinate centered at ′ with ( 1 = 0) = W ∩ ′ so we can now identify with . Hence, * log 1 = log 1 , * 2 = 2 , . . . , * = , and the frame ( log 1 , 2 , . . . , ) for Ω 1 ′ (log ′ )| W is adapted to log order. This shows that the metric ′ is adapted to log order for ′• (-log ′• ).

  0.3) by (3.0.2). We shall prove that ( | -, | -, ) is a principal variation of Hodge structures onfollowing the elegant arguments in [Mau15, Proposition 3.7].

  Lemma 4.4. Let = Δ , and = ℓ =1 with

over - .

 . Hence log | | 2 ℎ is a psh function on -. By Lemma 4.4, we conclude that log | | 2 ℎ extends to a psh function on . This proves that (F , ℎ) is negatively curved in the sense of Păun-Takayama. The metric ℎ induces a negatively curved singular hermitian metric ℎ 1 (in the sense of [PT18, Definition 2.2.1]) on the subbundle F | • . By Lemma 4.5, ℎ 1 induces a singular metric for the line bundle O P(F * | • ) (1) so that √ -1 (O P(F * | • ) (1)) ≥ 0. Note that -• is a codimension at least two subvariety. The second statement then follows from Hörmander's 2 -techniques in [PT18, Proof of Theorem 2.5.2].

  Proposition A.1. Let Γ ⊂ ( , 1) be a torsion free lattice, and let Γ ′ ⊂ Γ be a finite index normal neat sublattice. Let = B Γ , ′ = B Γ ′, and denote by ′ the smooth toroidal compactification of ′ = B Γ ′ as constructed in [AMRT10, Mok12]. Then the natural action of the finite group = Γ Γ ′ on ′ extends to ′ , and the quotient = ′ is a normal projective space, with boundarymade of quotient of abelian varieties by finite groups. Moreover, when Γ is arithmetic, coincides with the toroidal compactification of constructed in [AMRT10].

(

  1 , ..., -1 , ) ∈ B ↦ → ( ′ , ) = (

  3.6). For any = ( 2 , . . . , ) with 0 ≤ 2 , . . . , ≤ Then the above inequality together with (4.3.7) implies that

1 2 , we define a smooth function over Δ * parametrized by by

( 1 ) := log | | 2 ℎ F ( 1 , 2 , . . . , ).

( ) = 0. Since we assume that ( , ) is -polystable with

This work is supported by "le fond Chern" à l'IHES..

Conversely, one can associate a system of log Hodge bundles with a principal one as follows. The following result shall be applied in the proof of Theorem A.

Proposition 2.11. Let ( , ) = (⊕ + = , , ) be a system of log Hodge bundles on a log pair ( , ). Then there is a principal system of log Hodge bundles ( , ) with the structure group associated to ( , ), where is the semi-simple Lie group in Example 2.1. Moreover, any hermitian metric (not necessarily harmonic) ℎ := ⊕ + = ℎ for | -gives rise to a metric reduction for | -with the structure group 0 defined in Example 2.1.

Proof. We shall adopt the same notions as those in Example 2.1. Denote by := rank , , := + = and set ℓ := ≥ . We consider the following frame bundle ˜ . The fiber of ˜ over a point is the set of all ordered bases 1 , . . . , (or say frames) for such that ℓ -+1 , . . . , ℓ is a basis for , . The structure group of ˜ is thus ( , C), which is the subgroup of ( , C). ˜ can be equipped with the holomorphic structure induced by . Consider the homomorphism :

( , C) → ( , C) =: , and set = ( , C) to be the image of ( , C) under . Set to be the holomorphic -fiber bundle obtained by extending the structure group of ( , C) using . Note that × -1,1 = ⊕ + = Hom( , , -1, +1 ). Let us define := . The pair ( , ) is a principal system of log Hodge bundles on the log pair ( , ).

Recall that the metric ℎ for the Hodge bundle ( , )| -is a direct sum ℎ = ⊕ + = ℎ . We take a sesquilinear form of defined by ( , ) := ( √ -1) -ℎ( , ) for , ∈ , . We take ˜ to be a reduction of ˜ | -consisting of unitary frames with respect to . In other words, The fiber of ˜ over a point is the set of frames 1 , . . . , for such that ℓ -+1 , . . . , ℓ is an orthonormal basis for ( , , ℎ ). Hence the structure group of ˜ is ˜ 0 := + = ( ). Define 0 := + = ( ) , which is the image ( ˜ 0 ).

Set

to be the smooth principal 0 -fiber bundle onobtained by extending the structure group of ˜ using : → 0 . Then ⊂ -is also a metric reduction. The Hodge group 0 will be ( 0 , 0 ) where 0 := and 0 := , and := ( , C) is the complexification of 0 . The proposition is proved.

T

In this section, we shall construct principal variation of Hodge structures over quasiprojective manifolds. Its proof is based on Proposition 1.12 together with some Tannakian considerations in [START_REF]Harmonic bundles on noncompact curves[END_REF][START_REF] Kobayashi | Hitchin correspondence for tame harmonic bundles and an application[END_REF][START_REF]Higgs bundles and representations of complex hyperbolic lattices[END_REF].

Theorem 3.1. Let ( , ) be a projective log pair endowed with an ample polarization . Let ( , ) be a principal system of log Hodge bundles on ( , ), and let be a Hodge representation : → ( ) for some polarized Hodge structure ( = ⊕ + = , , ℎ ) so that | 0 : 0 → ( ) is faithful and : 0 → ( ) is injective. If the system of log Hodge bundles ( := × , := ( )) defined in Lemma 2.10 is -polystable with ∫ ℎ 2 ( ) • 1 ( ) dim -2 = 0, then there exists a metric reduction for | -so that the triple ( | -, | -, ) is a principal variation of Hodge structures on -. Moreover, YA DENG representation. By Lemma 2.10, such Hodge representation induces a system of log Hodge bundles ( × , ( )( )) over ( , ). It follows our construction of ( , ) that ( × , ( )( )) = (End( ) ⊥ , ( ) ⊥ ). where End( ) ⊥ is the trace-free subbundle of End( ), and ( ) ⊥ is the induced Higgs field from ( , ).

On the other hand, an easy computation shows that 1 (End( )) = 0, and

= 2 1 ( + ) -2( + 1) 2 (Ω 1 (log )) = 0 since the equality in (4. 3.2) holds by our assumption. Since we assume that ( , ) ispolystable, by Theorem 1.9, (End( ), ( ) ) is also -polystable. We now apply Proposition 1.12 to find a Hodge metric ℎ for the system of log Hodge bundle (End( )| -, ( ) | -) which is adapted to (End( ), ( ) ). Since (End( ), ( ) ) = (End( ) ⊥ , ( ) ⊥ ) ⊕ (O , 0), we conclude that ℎ = ℎ 1 ⊕ℎ 2 , where ℎ 1 is the harmonic metric for (End( ) ⊥ | -, ⊥ ( ) | -) which is adapted to the log Higgs bundle (End ⊥ ( ), ⊥ ( ) ), and ℎ 2 is the canonical metric for the trivial Higgs bundle (O , 0).

We now apply Theorem 3.1 to conclude that ℎ 1 induces a reduction for | -with the structure group 0 = ( ( ) × (1)) ≃ ( ), which is compatible with ℎ 1 such that ( | -, | -, ) is a principal variation of Hodge structures on -. Note that

) is moreover a uniformizing variation of Hodge structures overin the sense of Definition 2.6. By Remark 2.7, it gives rise to a holomorphic map, the so-called period map, 3.3) defined in (2.0.4), which is locally biholomorphic. Hereis the universal cover of -.

Note that the reduction together with the hermitian metric ℎ in (2.0.1) gives rise to a natural metric ℎ over × | -defined in (2.0.3). By Remark 2.7 again, if the pull back * ℎ is a complete metric on -, thenis uniformized by 0 0 = ( , 1) ( ) which is the complex unit ball of dimension , denoted by B . It follows from (3.0.3) that ℎ 1 = ℎ . It now suffices to show that * ℎ is complete if we want to prove thatis uniformized by B , where we recall

In next step, we will apply similar ideas by Simpson [Sim90, Corollary 4.2] to prove this. Note that until now we made no assumption on the smoothness of .

Step 2. Throughout Step 2, we will assume that is smooth. Consider now the system of log Hodge bundles (E, ) := (End( ), ( ) ). We first mention that the above Hodge metric ℎ for (E, )| -is adapted to log order in the sense of Definition 4.1. Indeed, it follows from [Moc02, Corollary 4.9] that the eigenvalues of monodromies of the flat connection := ℎ + + ℎ around the divisor are 1. By the "weak" norm estimate in [Moc02, Lemma 4.15], we conclude that ℎ is adapted to log order 2 .

We first give an estimate for * ℎ. For any point ∈ , consider an admissible coordinates ( ; 1 , . . . , ) centered at as Definition 1.3 so that ∩ = ( 1 = 0). To 2 Indeed, a strong norm estimate has already been obtained by Cattani-Kaplan-Schmid in [CKS86].

Here we only need to know that ℎ is adapted to log order, which is a bit easier to obtain using Andreotti-Vesentini type results by Simpson [START_REF]Harmonic bundles on noncompact curves[END_REF] and Mochizuki [START_REF]Asymptotic behaviour of tame nilpotent harmonic bundles with trivial parabolic structure[END_REF]Lemma 4.15].

for some positive constants 1 and 2 . On the other hand, by (5.3.12) on we have

By the construction of , we conclude that

for some positive constants 3 and 4 . This contradicts with (5. 3.13). Hence we conclude that when the slope equality holds, is a subbundle of .

We now find the desired decomposition of ( , ). By the above argument, when the slope equality holds, ( , | ) is a Higgs subbundle of ( , ) (not assumed to be rank 1 now), and ≡ 0 and ≡ 0. This means that the orthogonal projection :

We shall prove that extends to a morphism ˜ : → so that • = . For any point 0 ∈ , we pick an admissible coordinate ( ; 1 , . . . , ) centered at 0 and a holomorphic fame ( 1 , . . . , ) for | adapted to log order so that ( 1 , . . . , ) is a holomorphic fame for | . Write ( | -) = =1 ( ) , where ( ) ∈ O( -). For = 1, . . . , , one has ( | -) = and it extends naturally. For > and some 1 < < 1, over * ( ) one has

for some , > 0, where the second inequality is due to the fact that is the orthogonal projection with respect to ℎ, and the last inequality follows from the fact that ℎ is adapted to log order. Hence each | | must be locally bounded from above on , and it thus extends to a holomorphic function on . We conclude that extends to a morphism ˜ : → , whose rank is constant and ˜ • = , where : → denotes the inclusion. Let us define by := ker ˜ , which is a subbundle of so that = ⊕ . Note that | -= ⊥ . By (5.3.14) together with the continuity propery we conclude that is a sub-Higgs bundle of ( , ), and that ( , ) = ( , | ) ⊕ ( , | ). Since ℎ| (resp. ℎ| ) is a Hermitian-Yang-Mills metric for ( , | ) (resp. ( , | )) satisfying the three conditions in the theorem, we can argue in the same way as above to decompose ( , | ) and ( , | ) further to show that ( , ) is a direct sum of -stable log Higgs bundles with the same slope. Hence ( , ) is -polystable. We prove the theorem. Γ has a unique structure of a quasi-projective complex algebraic variety (see for example [BU20, Theorem 3.1.12]). When the parabolic subgroups of Γ are unipotent, by the work of Ash et al. [START_REF]Smooth compactifications of locally symmetric varieties[END_REF] and Mok [Mok12, Theorem 1], B Γ admits a unique smooth toroidal compactification, which we denote by . Let us denote by := -B Γ the boundary divisor, which is a disjoint union of abelian varieties. Let be the Bergman metric for B , which is complete, invariant under ( , 1) and has constant holomorphic sectional curvature -1. Hence it descends to a metric on -. If we consider as a metric for (-log )| -, by [To93, Proposition 2.1] it is good in the sense of Mumford [Mum77, Section 1]. Therefore, by for any ≥ 1, it follows from [Mum77, Theorem 1.4] that the trivial extension of the Chern form ( -, ) onto defines a ( , )-current [ ( -, )] on , which represents One can check that 1 ( -, ) 2 -2( + 1) 2 ( -, ) ≡ 0. We thus conclude that the Chern classes (Ω 1 (log )) satisfies

For the log Hodge bundle ( , ) = ( 1,0 ⊕ 0,1 , ), given by 1,0 := Ω 1 (log ), 0,1 := O with the Higgs field defined in (0.1.1), we shall prove that it is -polystable for the big and nef polarization in Theorem 5.7. We equipped ( 1,0 ⊕ 0,1 )| -with the metric

where ℎ is the canonical metric on O -so that |1| ℎ = 1. Recall that the curvature ℎ ( ) of the connection ℎ :

where ℎ ( ) is the Chern curvature of ( , ℎ). An easy exercise shows that √ -1 ℎ ( ) = ⊗ .

In particular, ℎ is a Hermitian-Yang-Mills metric for ( , )| -. We shall show that it satisfies the three conditions in Theorem 5.7. Indeed, we only have to check the first two conditions since √ -1 ℎ ( ) ⊥ ≡ 0. We first note that has at most Poincaré growth near in the sense of Definition 1.4. Indeed, this follows easily from the Ahlfors-Schwarz lemma (see for example [Nad89, Lemma 2.1]) since the holomorphic sectional curvature of is -1. Hence for any admissible coordinate system ( ; 1 , . . . , ) as in Definition 1.3, one has | ℎ ( )| ℎ, ≤ , where is the Poincaré metric on * . By the following result, we see that ℎ is adapted to log order.

Lemma 5.8 ( [Mok12, eq. (8) on p. 338]). Let ( , ) be as above. Then for any ∈ , there is an admissible coordinate ( ; 1 , . . . , ) at so that the frame 1 1 , 2 , . . . , -1 , is adapted to log order (in the sense of § 4.1) with respect to the above metric .

Therefore, the metric ℎ for ( , )| -satisfies the three conditions in Theorem 5. 7. In conclusion, ( , ) is -polystable for the big and nef class in Theorem 5.7

To finish the proof of Theorem B, we have to show that 1 ( + ) can be made as a polarization in Theorem 5.7, which follows from the following result.

YA DENG Lemma 5.9 ( [Mok12, Proposition 1]). The Kähler form ( +1) 2 ondefined above extends to a closed positive (1, 1)-current ∈ 1 ( + ) with zero Lelong numbers. In particular, + is big and nef.

Proof of Corollary C. We shall show how to apply Theorems A and B to derive Corollary C.

Proof of Corollary C. We first assume that parabolic subgroups of Γ are unipotent. By [Mok12, Theorem 1], there is a toroidal compactification for the ball quotient := B Γ , so that := -is a smooth divisor. Moreover, is projective. Fix any ample polarization on . By Theorem B, the log Higgs bundle ( ,

Let us denote by and the conjugate varieties of and under . Hence is a smooth projective variety and is a smooth divisor on . For any coherent sheaf E on , we denote by E its conjugate under , which is also a coherent sheaf on . Note that the conjugate action induces a canonical isomorphism between cohomology groups

and that Chern classes of vector bundles over are preserved under in the sense that Φ ( ( )) = ( ) for any holomorphic vector bundle over . Since Ω 1 (log ) = Ω 1 (log ), (5.5.1) also holds for the log cotangent bundle Ω 1 (log ). Moreover, the conjugate of ( , ) under is the log Higgs bundle ( , ) := (Ω 1 (log ) ⊕ O , ) on ( , ) defined in (0.1.1). Hence for any Higgs subsheaf F of ( , ), F is also a Higgs subsheaf of ( , ) with the slope inequality (F ) = (F ). Hence ( , ) is -polystable. By Theorem A, -is also a ball quotient, with its toroidal compactification.

For a general torsion free lattice Γ ⊂ ( , 1), there is a finite index subgroup Γ ′ ⊂ Γ so that parabolic subgroups of Γ ′ are unipotent (see for example [BU20, §3.3]). Denote by := B Γ and := B Γ ′. Since the base change of an étale morphism is étale, we conclude that → is also a finite étale surjective morphism. By the above result, is the ball quotient, and thus so is . Corollary C is proved.

Remark 5.10. In the above proof we show that a toroidal compactification of a ball quotient is also a toroidal compactification of another ball quotient. As pointed out by the referee, this fact follows from Corollary C directly. His/her elegant argument is as follows: since an Abelian variety is simply an algebraic variety with a group law defined by regular functions, a Galois conjugate of the Abelian variety as a component of the compactifying divisor is by itself an Abelian variety, which sits in the compactifying divisor of the ball quotient obtained as the conjugate of the original one according to Corollary C.

Now, if ∈ , we can write = ( , ) accordingly to the decomposition = × ( R, C -1 ), and we have, for any ( ′ , ) ∈ ( ) :

We check easily that ( ) is preserved by . Also, for any ∈ ≃ ( -1) × R, we can write = ( , ), and we then have

We are now ready to describe the quotients of ( ) by the action of Γ ′ ∩ . Note first that since Γ ′ is neat, we have Γ ′ ∩ ⊂ . Then, by the discussion above, we obtain a decomposition as sets

can be written as ( , , Id, 0). It also follows from [START_REF]Projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume[END_REF] that Γ ′ ∩ = Z for some ∈ ≃ R. This last fact permits to form the quotient ( ) = ( ) ∩ Γ ′; using (A.1.1), we can also express the latter quotient as an open subset of C -1 × C * :

and the quotient is then realized by the map

The group Λ := (Γ ′ ∩ ) ⊂ C -1 is an abelian lattice, acting on

We can form the quotient

Λ

which is then isomorphic to a tubular neighborhood of the abelian variety C -1 Λ in some negative line bundle. Finally, the toroidal compactification ′ can be obtained by glueing the open varieties Ω ( ) to ′ (as runs among a system of representatives of the rational boundary components, and is large enough).

Our claims about can be derived from the following lemma.

Lemma A.2. Let ∈ B be a Γ ′ -rational boundary component, and let ∈ Γ. Then the point ′ = • is also Γ ′ -rational, and there exists , ′ > 0, for which induces an isomorphism

As for our second claim, since the set of horoballs is preserved by the action of ( , 1), we may find , ′ such that induces a isomorphism ( ) → ( ) ′ . Let ( ′ , ) (resp. ( ′ , )) be standard coordinates on ( ) (resp.

( ′ ) ) as in (A.1.1), chosen so that ( ′ , ) = ( ′ , ) • for some ∈ ( ) satisfying • ′ = . Then ∈ , and by (A.1.2) and (A.1.3), we have ( ′ , ) • = ( ′ , ) for some affine map .

Since normalizes Γ ′ , we have (Γ ′ ∩ ) -1 = Γ ′ ∩ ′ , so the map ( ) → ( ′ ) ′ passes to the quotient to give a map :

. Using an explicit expression for the affine map , we find an (a priori multivaluate) expression for as

′ for some ∈ -1 (C), some vectors , , ′ 0 ∈ C -1 and , ∈ C. The formula above induces a well-defined, invertible map ( ) → ( ′ ) ′ , so we have = 0, = ±1. This implies that extends holomorphically to : ( ) → ( ′ ) ′ . Finally, as normalizes Γ ′ , passes to the quotient by Λ and Λ ′ , giving a biholomorphism Ω ( ) → Ω Note that even though this construction of is well adapted to our purposes, it should not be used to define as an orbifold, as it has the drawback of producing artificial ramification orders along the boundary components of . As explained in [START_REF]Orbifold Kähler Groups related to arithmetic complex hyperbolic lattices[END_REF], a better way of proceeding is to construct directly open neighborhoods of the components ofas stacks, before glueing them to .

A.2. Main results. Let us first begin with the following lemma.

Lemma A.4. Let be the toroidal compactification of the ball quotient := B Γ by a torsion free lattice Γ ⊂ ( , 1) whose parabolic isometries are all unipotent. Let be another projective compactification of , and assume that has at most klt singularities.

Then the identity map of extends to a birational morphism : → .

Proof. The identity map of defines a birational map : . Assume by contradiction that is not regular. One can take a resolution of indeterminacies : ˜ → for so that | -1 ( ) : -1 ( ) ∼ -→ is an isomorphism: ˜ ˜ By the rigidity result (see [Deb01, Chapter 3, Lemma 1.15]), there is a fiber -1 ( ) with ∈ which is not contracted by ˜ . Clearly, we have ˜ ( -1 ( )) ⊂ -.

We can now conclude the case discussed in Corollary A.9, where ( , ) is assumed to be a smooth log-pair. Since the boundary of ′ -′ is smooth, this implies that must also be smooth. Moreover, for each connected component of , there is a connected component of so that | : → is a finite surjective morphism, which is also étale by the local description of given in the proof of Lemma A.12. Hence in this case, is a smooth quotient of an abelian variety by the free action of some finite group . This suffices to establish Corollary A.9.

The proof of Theorem A.7 will be complete with the following lemma.

Lemma A. 13. The variety identifies with the quotient of ′ by the natural action of = Γ Γ ′. In particular, .

Proof. This result comes right away from Lemma A.14 below, taking = ′ , = , and = G. Remark that we have ( ′ ) = ( ′ ) = ( ) = ( ) since = ′ . For the second statement, remark that by Proposition A.1, the toroidal compactification of also identifies with the quotient ′ . Thus, there is an isomorphism compatible with the identity on . Theorem A.7 is proved.

Lemma A.14. Let : → be a finite surjective morphism between two normal reduced schemes. Assume that is acted upon by a finite groupoid G, and that is G-invariant. Suppose in addition that ( ) G = ( ), where ( ), ( ) are the rings of rational functions on , . Then is the quotient of by G.

Proof. It suffices to show that * (O ) G = O . This is a local statement on the base, so we may assume that = Spec , = Spec , with is integrally closed. We have G ⊂ ( ) G = ( ) by assumption. Since ⊂ is finite, and is integrally closed, this implies G ⊂ , as required.