Hyperbolicity of singular spaces

We study the hyperbolicity of singular quotients of bounded symmetric domains. We give effective criteria for such quotients to satisfy Green-Griffiths-Lang's conjectures in both analytic and algebraic settings. As an application, we show that Hilbert modular varieties, except for a few possible exceptions, satisfy all expected conjectures.


INTRODUCTION
As central objects in algebraic geometry, the geometry of quotients of bounded symmetric domains Ω/Γ has been the object of many works.Classical results state that there exist sufficiently small subgroups Γ ′ ⊂ Γ such that Ω/Γ ′ has remarkable properties: it is of general type [Mum77], it is hyperbolic modulo the boundary ( [Nad89], [Rou16]), all its subvarieties are of general type [Bru16].
Nevertheless, it turns out that these properties should be true in most cases without having to take small subgroups Γ ′ ⊂ Γ.As an example, Tsuyumine [Tsu85] has shown in the Hilbert modular case (Ω = H n and Γ the Hilbert modular group) that, except finitely many cases, Hilbert modular varieties are of general type.The main difficulty, if one wants to avoid the step of taking small subgroups, is to deal with singularities.Indeed, it is well known that hyperbolicity properties may be completely lost in singular quotients (see for example Keum's singular ball quotient [Keu08]).
The above mentioned results can be seen as illustrations of the expected following conjectures of Lang and Green-Griffiths (see [Lan86] and [GG80]).
Let Exc(X) ⊂ X denote the Zariski closure of the union of the images of all non-constant holomorphic maps C → X.
Conjecture 1.1.Let X be a complex projective manifold.Then X is of general type if and only if Exc(X) = X.
Date: October 25, 2017.E. R. was partially supported by the ANR project "FOLIAGE", ANR-16-CE40-0008.E.R. and B.T. thank the University of Sydney for the invitation at the School of Mathematics where part of this work was done.
The setting we consider in this article is the following: a quotient X = Ω Γ of a bounded symmetric domain by an arithmetic lattice such that (1.1.1)X has only cyclic quotient singularities p 1 , ..., p r , with isotropy groups G i ; (1.1.2)X admits a toroidal compactification X = X ⊔ D, where D ⊂ X is a divisor with simple normal crossings.
Denote by p : Ω −→ X the canonical projection.Let X π −→ X be a resolution with exceptional divisor E i above each Let h Berg be the Bergman metric on Ω, which we normalize to have Ric(h Berg ) = −h Berg .Let γ ∈ Q * + such that the holomorphic sectional curvature of h Berg is bounded from above by −γ.
The first result is a generalization of Nadel's theorem [Nad89] to this singular setting.
Then Exc( X) ⊂ B + (L) ∪ D ∪ E, where B + (L) denotes the augmented base locus.In particular if L is big then Exc( X) = X.
Then, we turn to algebraic versions of hyperbolicity in the case of quotients of the ball and polydiscs.

More precisely, any resolution of singularities V of such a variety has big cotangent bundle.
The proof of these two theorems is given in section 3.They are obtained using negativity properties of the Bergman metric and the methods of [Cad16].The key idea in both statements is to use sections of the line bundle L to twist the Bergman metric and obtain in this way a singular metric on the compactification with the required negative curvature properties.

More precisely, any resolution of singularities V of such a variety has big cotangent bundle.
This result is proved in section 5. We use extension properties of orbifold symmetric differential forms explained in section 4. The idea is that in this setting, sections of the line bundle L naturally induce symmetric differentials on the regular part of the quotient.Then, extension properties of these differential forms as orbifold symmetric differentials are used to construct global holomorphic symmetric differentials on the compactification.
As an application of the above results, we study in section 6 the case of Hilbert modular varieties and obtain the following version of conjecture 1.1 in this case.

Theorem D.
Let n ≥ 2.Then, except finitely many possible exceptions, Hilbert modular varieties satisfy the following properties: (1.1.3)Exc(X) = X.(1.1.4)there is a proper subvariety Z such that all subvarieties not contained in Z are of general type.
The first part of this theorem was already obtained by a different method in [RT17] while the second part is a generalization of results of Tsuyumine [Tsu86] who treated the case of codimension one subvarieties.

CYCLIC QUOTIENT SINGULARITIES
In this section, we collect some basic facts about cyclic quotient singularities.
2.1.Resolution.First, we recall how one can resolve this type of singularities.This problem has been adressed by Fujiki ([Fuj75]).
Let G a finite cyclic group with a fixed generator acting on a complex affine m-space C m = C m (z 1 , ..., z m ) by the formula: where, for each i, e i is a n th root of the unity, n > 1.Then the quotient X = C m /G has the structure of a normal affine algebraic variety whose singular locus is reduced to the single point p = π(0) where π : C m → X denotes the canonical projection.
Following [Fuj75] (theorem 1 p.303) there exists a resolution of the singularity of X, that is a pair ( X, f ) consisting of a smooth variety X, a proper birational morphism f : X → X isomorphic outside f −1 (p) with the additional following properties (2.0.1)E = f −1 (p) is a simple normal crossing hypersurface, each component of which being rational.(2.0.2)There exists an affine open covering U = (U 1 , ..., U l ) of X with ) for some k 1 , .., k t (depending on i) if it is non empty.
(2.0.4)For each i the multivaluate map f • π : C m (u i ) → C m (z) takes the following form: (2.0.1) where b lk are nonnegative rational numbers (depending on i and stricly less than 1).
Remark 2.1.Remark that z k n descends to X therefore all denominators of the rational numbers b lk must divide n.
2.2.Discrepancies.Now, we describe some basic estimates on discrepancies associated to cyclic quotient singularities.Lemma 2.2.Let X = Ω Γ be a quotient with only cyclic quotient singularities p 1 , ..., p r , with isotropy groups G i such that where the a i are the discrepancies.Then, for any i, we have a Proof.On each open chart U j , we have, for any k ∈ [|1, n|], (2.2.1) Consequently

ENTIRE CURVES AND SINGULAR BALL QUOTIENTS
This section will be devoted to the proof of Theorems A and B. The main ingredient in both proofs is the use of singular metrics built from the Bergman metric.
3.1.Singular metrics.As explained in the introduction, we consider a quotient X = Ω Γ of a bounded symmetric domain by an arithmetic lattice such that (3.0.1)X has only cyclic quotient singularities p 1 , ..., p r , with isotropy groups 2) Γ has only unipotent parabolic elements, so that X admits a toroidal compactification X = X ⊔ D, where D ⊂ X is a divisor with simple normal crossings.
Denote by p : We will also denote by h Berg the Bergman metric on the regular part X \ {p 1 , ..., p r }.It induces a natural singular metric h on X \ (D ∪ E 1 ∪ ... ∪ E r ).
We will use the notations of section 2, concerning resolutions of cyclic singularities.Now, we state a lemma which controls the singularities of h near exceptional divisors.
Lemma 3.1.On each one of the open charts U i j , there is a constant C > 0 such that, for each tangent vector field v, , for any j.Because of (2.0.1), we have so we get, after taking the norm Since the ∂ ∂z l h Berg are bounded, we get the result.
In the next lemma, we explain how to extend the metric on X using sections of an appropriate adjoint line bundle.Lemma 3.2.Let −B ≤ 0 be an upper bound for the bisectional curvature of the Bergman metric on Ω. Assume that the line bundle We will show that it has the required properties.First of all, the metric det h * Berg has at most logarithmic growth near D if we see it as a metric on π * (K X + D) (see [Mum77]).Consequently, on any Finally, we compute the curvature of h at the points where it is smooth.Because of our normalization assumptions, we find and the required bounds on the curvatures are then given by an simple computation.
3.2.Proofs.Now, we can prove our two criteria for complex hyperbolicity.
Proof of Theorem A. By contradiction, assume there exists an entire curve C j −→ X not included in B + (L) ∪ D ∪ E. By our hypothesis, and since bigness is an open condition, we can pick A < γ in Lemma 3.2.This implies that there exists a singular metric h on T X , non-degenerate on j(C), locally bounded everywhere, and with holomorphic sectional curvature bounded from above by −γ + A < 0. Thus, j * h defines a smooth metric on C outside a discrete set of points.This metric has negative curvature, bounded away from zero, and is locally bounded everywhere.We can apply the usual extension theorem for plurisubharmonic functions, to obtain a singular metric on C, with negative curvature in the sense of currents, bounded away from zero.This is absurd by Ahlfors-Schwarz lemma.
Proof of Theorem B. Let V −→ X, with V ⊂ B + (L) ∪ D ∪ E, and let V be a resolution of the singularities of V. Then the induced holomorphic map V j −→ X is generically immersive.
In Lemma 3.2, we can take B = 1 n+1 , γ = 2 n+1 .Then, choosing any positive constant A < 1 n+1 will give a metric h such that j * h is locally bounded everywhere on T V , has negative bisectional curvature and negative holomorphic sectional curvature, and is bounded from above by − 1 n+1 .By [Cad16], this implies that Ω V is big.
Remark 3.3.Because of Lemma 2.2, the line bundle π * K X − K X + (n D − E) is always effective.Consequently, Theorem A (resp.Theorem B) can be restated with

ORBIFOLD DIFFERENTIAL FORMS
Our aim in the current section is to establish an extension result for orbifold differential forms (see Proposition 4.13) which plays a key role in our strategy to prove Theorem C.
4.1.Preliminaries on orbifold differential forms.The notions we are about to introduce in the current section originated in the works of Campana, cf.[Cam04].For a more thorough account of these preliminary constructions, including background and applications, the reader could also consult [JK11], [Taj16], [CKT16] and [GT16].
Definition 4.1.We define an orbifold pair (X, D) by a normal, algebraic variety X and a divisor , where a i ∈ N + ∪ {∞} and each D i is a prime divisor.
We follows the usual convention that, for a i = ∞, we have (1 − 1 a i ) = 1.The pair (X, D) in Definition 4.1 is sometimes referred to as a pair with standard coefficients (or a classical or integral orbifold pair).
With Definition 4.1 at hand, given a pair (X, D), one can naturally associate a notion of multiplicity to each irreducible component D i of D. Definition 4.2.Let (X, D) be an orbifold pair.we define the orbifold multiplicity m D (D i ) of each prime divisor D i as follows.
Our aim in now to introduce morphisms sensitive to the orbifold structure of (X, D), but to do so we first need to define a notion for pullbacks of Weil divisors over normal varieties.Definition 4.3.Let f : Y → X be a finite morphism of normal, algebraic varieties X and Y. Let D ⊂ X be a Weil divisor.We define f * (D) by the Zariski closure of the Weil divisor defined by f * (D| X reg ).
We now turn to morphisms adapted to (X, D).Such morphisms are guaranteed to exist whenever X is smooth, thanks to Kawamata's covering constructions, cf.[Laz04, Prop.4.1.12].Definition 4.4.Let (X, D) be an orbifold pair in Definition 4.1.We call a finite morphism f : Y → X of algebraic varieties X and Y, strictly adapted We can also consider collections of charts that are adapted to the orbifold structure of a given pair (X, D).Definition 4.5 (Orbifold structures).Given an orbifold pair (X, D), let {U α } α be a Zariski open covering of X.We call a collection , an orbifold structure (or strictly adapted orbifold structure) associated to (X, D).Notation 4.1.Given a coherent sheaf F on a normal algebraic variety X, by S [•] F we denote the reflexive hull (S • F ) ∨∨ of symmetric powers S • F of F .Furthermore, for a morphism of normal algebraic varieties f : Y → X, we set We can now define a notion of sheaves of differential forms adapted to the orbifold structure of (X, D).First, we need to fix some notations.Notation 4.2.Let C α be an orbifold structure for the orbifold pair (X, D).Let {D ij X α } i,j be the collection of prime divisors in X α verifying the equality an open subset of the smooth locus of (U α , D) such that f α : X • α → U • α is surjective.Definition 4.6.In the setting of Notation 4.2, we define the orbifold cotangent sheaf Ω [1] C α by the collection of reflexive, O X α -module, coherent sheaves Ω uniquely determined as the coherent extension of the kernel of the sheaf morphism naturally defined by the residue map.
Example 4.8.Let (X, ) be an orbifold pair and f α : X α → U α a single, strictly adapted chart in Definition 4.5.Assume that the branch locus of f α is divisorial and has simple normal crossing support.With this assumption, it follows that X α is smooth.Furthermore, let us assume that U α admits a coordinate system (z 1 , . . ., z n ) such that U α ∩ D = {(z 1 , . . ., z n ) ∏ l i=1 z i = 0}.Assume that w 1 , . . ., w n is a coordinate system on X α such that Definition 4.9.Give any orbifold pair (X, D) together with an orbifold structure C α = {(U α , f α , X α )} α , we define the sheaf of symmetric orbifold differential forms C α by the collection of reflexive sheaves {S Notation 4.3.Let (X, D) be an orbifold pair.We shall denote by S X ( * D red ) the sheaf of symmetric rational differential forms with poles of arbitrary order along D red , which is defined by: Sometimes It is more convenient to work with a notion of symmetric differential forms, adapted to (X, D), as a coherent sheaf on X instead of X α .This is the purpose of the following definition.Definition 4.10.Let (X, D) be an orbifold pair equipped with an orbifold structure X ( * D red ), defined, at the level of presheaves, by the following property: Remark 4.11.One can easily check that the notion of C-differential forms in Definition 4.10 is independent of the choice of the orbifold structure C α .
Remark 4.12.Local calculations show that over X reg the sheaf S X log(D) is locally free.More precisely, for every x ∈ X reg , there exists an open neighbourhood where d i = (1 − 1 a i ).4.2.An extension theorem for symmetric orbifold differential forms.We are now ready to prove an extension result for orbifold differential forms, which can be interpreted as an orbifold version of [GKK10, Prop.3.1].Proposition 4.13.Let U ⊆ C n be a normal, algebraic subset.Assume that there is a smooth algebraic subset V ⊆ C n and a finite group G acting freely in codimension one such that U ∼ = V/G.Let π : U → U be a strong log-resolution of U with E ⊆ Exc(π) being the maximal reduced exceptional divisor.Then, for every m ∈ N, the coherent sheaf (4.13.1) Proof.Let f : V → U denote the finite map encoding the isomorphism U ∼ = V/G and set V to be the normalization of the fibre product V × U U with the resulting commutative diagram: Note that f is a composition of the finite map V × U U → U, which is of degree |G|, and its normalization.Therefore deg( f ) = |G|.Now, as the problem is local, to prove the reflexivity of the sheaf G (4.13.1), it suffices to show that the naturally defined map (4.13.2) Viewing σ as an element of Γ U, S m Ω U ( * E) , and thanks to the commutativity of the diagram above, we have (4.13.3) Now, set {E i } to be the set of irreducible components of E and let be the divisor with respect to which f : V → U is strictly adapted to the orbifold pair ( U, E ′ ) (cf.Definition 4.4).Note that, by construction, we have: (4.13.4)For each i, the inequality a i ≤ |G| holds.(4.13.5) Therefore, thanks to (4.13.3) we have, by the definition of symmetric C- In particular, it follows that the map On the other hand, thanks to the inequality a i ≤ |G| in (4.13.4), we have The surjectivity of the map (4.13.2) now follows.
We conclude this section by pointing out that one can easily verify that the extension result in Proposition 4.13 holds for any resolution of U.

QUOTIENTS OF POLYDISCS
In this section, we consider the case where Ω is the polydisk D n , and consider the same setting as above.Let Γ ⊂ Aut(D) n be a discrete subgroup, and X = D n /Γ a quotient with cyclic quotient singularities p 1 , . . ., p s , whose isotropy groups have cardinal m i .
We denote p : We first establish the following weak version of Theorem C.
Proof.Let ω be a global section of π * K ⊗m X .It gives a section of K ⊗m D n invariant under Γ.Now we consider it as a section of S mn Ω(D n ) invariant under Γ.It gives a section of S mn Ω(X reg ).Therefore from the extension property 4.13, we obtain finally a section of S mn Ω( X, ∆ + D).Starting with a section ω of a multiple of where A is an ample line bundle, one gets a section of S mn Ω( X) ⊗ A −1 , which implies that Ω X is big.
Example 5.2.Already in the case of surfaces, this statement provides interesting examples of surfaces with c 2 1 ≤ c 2 and big cotangent bundle (see [GRVR17]).One can formulate the criterion involving K X instead of K X .
Corollary 5.3.In the same setting as above, if Now, let us suppose that Γ is irreducible in the following sense: the restriction of each of the n projections p j : Aut(D) n → Aut(D) to Γ is injective.
Remark that in this setting, singularities of quotients X = D n /Γ are automatically cyclic quotient.
Then we obtain the stronger statement, Theorem C. In particular, if L is big there is a proper surbvariety Z X such that all subvarieties W ⊂ Z have big cotangent bundle.
Proof.Let W ⊂ X be a subvariety.Following the same steps and notations as in the proof of theorem 5.1, we see that the pull-backs of sections ω on W may vanish in two cases: either W ⊂ B + (L) ∪ E ∪ D or W is tangent to one of the codimension one holomorphic foliation F i induced by dz i = 0 on D n .In [RT17] (see proposition 3.1 and remark 3.3), we have established that, under the hypothesis of irreducibility of Γ, leaves passing through elliptic fixed points are Stein therefore cannot contain algebraic subvarieties.Therefore if W is tangent to one of this foliation, we have π(W) ⊂ X reg .But then the Bergman metric of the polydisk D n induces a Kähler metric on X reg with nonpositive holomorphic bisectional curvature and negative holomorphic sectional curvature.Therefore, we deduce from [BKT13] that in this case W has big cotangent bundle.
Finally, we observe that we obtain in this case another proof of Theorem A.
Theorem 5.5.In the same setting as above, let f : C → X be an entire curve.Then f (C) ⊂ Z := B + (L) ∪ D ∪ E. In particular, if L is big then Exc(X) = X.
Proof.Let f : C → X be an entire curve.From the construction above, consider global symmetric differentials vanishing on an ample divisor induced by sections of π * K X − n( ∆ + D) − A. The classical vanishing theorem (see for example corollary 7.9 in [Dem97]) implies that any entire curve f : C → X satisfies f * ω ≡ 0. Let us suppose that f (C) does not lie in Z then f has to be tangent to one of the holomorphic codimension-one foliations induced by dz i = 0.But this is impossible because proposition 3.1 of [RT17] says that these leaves are hyperbolic.

HILBERT MODULAR VARIETIES
Let K be a totally real algebraic number field of degree n > 1 over the rational number field Q, and let O K be the ring of integers in K. Then Γ = Γ K = SL 2 (O K ) acts on the product H n of n copies of the upper half plane H = {z ∈ C|ℑz > 0}: for all M = α β γ δ ∈ Γ, where N(z) = ∏ n i=1 z i .Hibert modular forms are classically interpreted in terms of differential forms: if ω = dz 1 ∧ • • • ∧ dz n and f is a Hilbert modular form of weight k then f ω ⊗k gives an invariant holomorphic top-differential forms which descends on (the smooth part of) H n /Γ.
The observation already used in the previous sections is that one can also look at Hilbert modular forms as symmetric differential forms.Indeed, in the above notations, f (dz 1 . . .dz n ) k is also invariant under Γ and therefore provides a symmetric differential on (the smooth part of) H n /Γ.
Recall that there is a natural compactification Y := H n /Γ adding finitely many cusps.Then one can take a projective resolution X → Y. Now we can apply the results of the previous section.First, a corollary of Theorem 5.1 gives the following result.
Theorem 6.1.Let n ≥ 2. Then except finitely many possible exceptions Hilbert modular varieties have a big cotangent bundle.
Proof.Let E = E e + E c be the exceptional divisor where E e = ∑ i E i e corresponds to the resolution of elliptic points and E c corresponds to the resolutin of cusps.Theorem 5.1 tells us that there are constants α i depending only on the order of the stabilizer of elliptic fixed points such that if Therefore we are reduced to prove that K X + E − ∑ i α i E i e − nE c is big except finitely many possible exceptions.Let S m k denote the space of Hilbert modular forms of weight k and vanishing order at least m over cusps.Sections of corresponds to modular forms.We have to show the maximal growth of the space of corresponding modular forms.So we have to prove that one can produce more sections than the number of conditions imposed by the vanishing along the exceptional components.We shall use the following result of [Tsu85] (Sect.4) (6.1.1)dim for even k ≥ 0, where h, d K , R, ζ K denote the class number of K, the absolute value of the discriminant, the positive regulator and the zeta function of K.In particular, there is a modular form F with ord( f )/ weight( f ) ≥ ν, if (6.1.2) If we fix n, then ζ K (2) has a positive lower bound independent of K. Since hR ∼ d 1/2 K by the Brauer-Siegel Theorem, for any constant C there are only a finite number of K such that the right hand side of (6.1.2) is smaller than C. Therefore we obtain sections with the requested order of vanishing along E c .
Let us now deal with the elliptic points.Prestel [Pre68] has obtained precise formula on the number of elliptic points of the Hilbert modular group.In particular, one can deduce (see section 6.5 of [RT17] for details) that for fixed n, there are only finitely many different type of elliptic points and the number of equivalence classes of elliptic fixed points is O(d 1 2 +ǫ K ) for every ǫ > 0. This immediately gives the maximal growth of the space of modular forms satisfying the vanishing conditions with finitely many possible exceptions.
As a consequence, we recover the main result of [Tsu85] Corollary 6.2.Let n ≥ 2. Then except finitely many possible exceptions Hilbert modular varieties are of general type.
Proof.This is an immediate application of [CP15] who prove that if the cotangent bundle is big then the canonical bundle is big.Now, we can give the proof of the two statements announced in Theorem D of the introduction as corollaries of Theorems 5.4 and 5.5.Theorem 6.3.Let n ≥ 2.Then, except finitely many possible exceptions, Hilbert modular varieties satisfy Exc(X) = X.
Proof.Let X be a Hilbert modular variety.The proof of Theorem 6.1 tells that L := K X + E − ∑ i α i E i e − nE c is big except finitely many possible exceptions.Then Theorem 5.5 tells us that Exc(X) ⊂ B + (L) ∪ E.
Finally, we obtain the second statement.Theorem 6.4.Let n ≥ 2. Then except finitely many possible exceptions Hilbert modular varieties contain a proper subvariety Z such that all subvarieties not contained in Z have big cotangent bundle and are of general type.
Proof.Let X be a Hilbert modular variety such that L := K X + E − ∑ i α i E i e − nE c is big.Then define Z := B + (L) ∪ E. Let Y ⊂ X be a subvariety not contained in Z. Theorem 5.4 gives that all subvarieties not contained in Z have big cotangent bundle and are of general type.

Theorem 5. 4 .
Let Γ ⊂ Aut(D) n be a discrete irreducible subgroup and denote L := π * K X − n( ∆ + D).Let B + (L) be the augmented base locus of L and Z := B + (L) ∪ D ∪ E. Then all subvarieties W ⊂ Z have big cotangent bundle.
, π * (dz 1 ∧ ... ∧ dz n ) can be written as P(u Since for each s, we have b sj ≤ 1 − 1 (dz 1 ∧ ... ∧ dz n ) ⊗m i vanishes at order at most m * Since bigness is an open property, and because x ∈ B + (L), there exists rational numbers