Symmetric differentials on complex hyperbolic manifolds with cusps

Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cotangent bundle on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of the standard tangent bundle in the more specific case of the ball. We obtain effective ramification orders for a cover $X' \longrightarrow X$, \'{e}tale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that the standard tangent bundle of such a cover is nef if the ramification is high enough. Moreover, the ramification orders we obtain do not depend on the dimension of the quotient of the ball we consider

Hyperbolicity of varieties supporting a variation of Hodge structure

We generalize former results of Zuo and the first author showing some hyperbolicity properties of varieties supporting a variation of Hodge structure. Our proof only uses the special curvature properties of period domains. In particular, in contrast to the former approaches, it does not use any result on the asymptotic behaviour of the Hodge metric

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.Comment: Main results extended to arbitrary quotient singularities and all bounded symmetric domain

Hyperbolicity and fundamental groups of complex quasi-projective varieties

This paper investigates the relationship between the hyperbolicity of complex quasi-projective varieties $X$ and the (topological) fundamental group $\pi_1(X)$ in the presence of a linear representation $\varrho: \pi_1(X) \to {\rm GL}_N(\mathbb{C})$. We present our main results in three parts. Firstly, we show that if $\varrho$ is bigand the Zariski closure of $\varrho(\pi_1(X))$ semisimple, then for any $X^\sigma:=X\times_\sigma\mathbb{C}$ where $\sigma\in {\rm Aut}(\mathbb{C}/\mathbb{Q})$, there exists a proper Zariski closed subset $Z \subsetneqq X^\sigma$ such that any closed irreducible subvariety $V$ of $X^\sigma$ not contained in $Z$ is of log general type, and any holomorphic map from the punctured disk $\mathbb{D}^*$ to $X^\sigma$ with image not contained in $Z$ does not have an essential singularity at the origin. In particular, all entire curves in $X^\sigma$ lie on $Z$. We provide examples to illustrate the optimality of this condition. Secondly, assuming that $\varrho$ is big and reductive, we prove the generalized Green-Griffiths-Lang conjecture for $X^\sigma$. Furthermore, if $\varrho$ is large, we show that the special subsets of $X^\sigma$ that capture the non-hyperbolicity locus of $X^\sigma$ from different perspectives are equal, and this subset is proper if and only if $X$ is of log general type. Lastly, we prove that if $X$ is a special quasi-projective manifold in the sense of Campana or $h$-special, then $\varrho(\pi_1(X))$ is virtually nilpotent. We provides examples to demonstrate that this result is sharp and thus revise Campana's abelianity conjecture for smooth quasi-projective varieties. To prove these theorems, we develop new features in non-abelian Hodge theory, geometric group theory, and Nevanlinna theory. Some byproducts are obtained.Comment: v2, 98 pages, add new results on generalized Green-Griffiths-Lang conjecture and hyperbolicity of Galois conjugate varieties; v3, 99 pages, added results on orbifold base in quasi-projective setting, submitte

Subvarieties of quotients of bounded symmetric domains

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each p ≥ 1, this criterion gives a precise condition under which the subvarieties V ⊂ X with dim V ≥ p are of general type, and X is p-measure hyperbolic. Then, we give several applications related to ball quotients, or to the Siegel moduli space of principally polarized abelian varieties. For example, we determine effective levels l for which the moduli spaces of genus g curves with l-level structures are of general type.On présente un nouveau critère pour l'hyperbolicité complexe d'un quotient non compact X d'un domaine symétrique borné. Pour chaque p ≥ 1, ce critère donne une condition précise sous laquelle les sous-variétés V ⊂ X telles que dim V ≥ p sont de type général, et X est p-mesure hyperbolique. Ensuite, on donne plusieurs applications aux quotients de la boule, ou l'espace de modules de Siegel pour les variétés abéliennes principalement polarisées. Par exemple, on détermine des niveaux effectifs l pour que les espaces de modules des courbes de genre g avec une structure de niveau l soient de type général

Jet differentials on toroidal compactifications of ball quotients

International audienceWe give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one

Generalized algebraic Morse inequalities and jet differentials

We give a fully algebraic proof of an important theorem of Demailly, stating the existence of many Green-Griffiths jet differentials on a complex projective manifold of general type. To this end, we introduce a new algebraic version of the Morse inequalities, which we then use in our proof as an algebraic counterpart to Demailly’s and Bonavero’s holomorphic Morse inequalities.On présente une preuve entièrement algébrique d'un théorème important de Demailly, affirmant l'existence de beaucoup de différentielles de jets de Green-Griffiths sur une variété complexe projective de type général. A cette fin, on introduit une nouvelle version algébrique des inégalités de Morse, que l'on utilise ensuite dans notre preuve comme un pendant algébrique aux inégalités de Morse holomorphes de Demailly et Bonavero

Hyperbolicity and specialness of symmetric powers

International audienceInspired by the computation of the Kodaira dimension of symmetric powers Xm of a complex projective variety X of dimension n ≥ 2 by Arapura and Archava, we study their analytic and algebraic hyperbolic properties. First we show that Xm is special if and only if X is special (except when the core of X is a curve). Then we construct dense entire curves in (suf-ficiently hig) symmetric powers of K3 surfaces and product of curves. We also give a criterion based on the positivity of jet differentials bundles that implies pseudo-hyperbolicity of symmetric powers. As an application, we obtain the Kobayashi hyperbolicity of symmetric powers of generic projective hypersur-faces of sufficiently high degree. On the algebraic side, we give a criterion implying that subvarieties of codimension ≤ n − 2 of symmetric powers are of general type. This applies in particular to varieties with ample cotangent bundles. Finally, based on a metric approach we study symmetric powers of ball quotients