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In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. $2)$ We prove that the period map associated to any pure polarized variation of integral Hodge structures $\mathbb{V}$ on a smooth complex quasi-projective variety $S$ is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. $3)$ As a corollary of $2)$ and of Peterzil-Starchenko's o-minimal Chow theorem we recover that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$, a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable $SL_2$-orbit theorem of Cattani-Kaplan-Schmid., 23 pages, final version. arXiv admin note: substantial text overlap with arXiv:1803.09384

Given $${{\mathbb {V}}}$$ V a polarizable variation of $${{\mathbb {Z}}}$$ Z -Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ V s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$ V . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ V is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ V whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ A g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ A g is either a closed algebraic subvariety of S or is Zariski-dense in S.

1.1. Bi-algebraic geometry and the Ax-Lindemann-Weierstras property. — Let X and S be complex algebraic varieties and suppose π : Xan −→ San is a complex analytic, nonalgebraic, morphism between the associated complex analytic spaces. In this situation the image π(Y) of a generic algebraic subvariety Y ⊂ X is usually highly transcendental and the pairs (Y ⊂ X,V ⊂ S) of irreducible algebraic subvarieties such that π(Y) = V are rare and of particular geometric significance. We will say that an irreducible subvariety Y ⊂ X (resp. V ⊂ S) is bi-algebraic if π(Y) is an algebraic subvariety of S (resp. any analytic irreducible component of π−1(V) is an irreducible algebraic subvariety of X). Notice that V ⊂ S is bi-algebraic if and only if any analytic irreducible component of π−1(V) is bi-algebraic.

While Margulis’ superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real Lie groups, almost nothing is known concerning the representation theory of complex hyperbolic lattices. The main result of this paper (Theorem 1.3) is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices. It relies on the following two ingredients: Theorem 1.6 showing that the representations of the topological fundamental group of a compact Kahler manifold X are controlled by the global symmetric differentials on X. An arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms, in particular deep results of Clozel on base change (Theorem 1.11).

We prove general local rigidity results for cocompact complex hyperbolic lattices using Hodge theory.

Let Γ be a locally compact second countable group, F a local field of characteristic zero and G an F-almost-simple F-algebraic group. In this paper we study the space X(Γ,G) of Zariski-dense representations ρ : Γ → G = G(F) using the natural morphism of cohomological functors ρ* : H*(G, ·) → H*(Γ, ·) (where H denotes the continuous cohomology).

The motivation of this work comes from the study of lattices in real simple Lie groups. The famous Margulis’s superrigidity theorem claims that finite dimensional reductive representations of any lattice of a real simple Lie group of real rank ≥2 are superrigid. As a corollary such a lattice is arithmetic. These results extend to the real rank one case for lattices in Sp(n,1) and F4(-20) by the work of Corlette and Gromov-Schoen. On the other hand Mostow and Deligne-Mostow exhibited arithmetic lattices with non-superrigid representations as well as non-arithmetic lattices in the unitary group PU(2,1). A natural question is then to find simple sufficient conditions for superrigidity or arithmeticity of lattices in PU(2,1). Rogawski conjectured the following: let Γ be a torsion-free cocompact lattice in PU(2,1) such that the hyperbolic quotient M=Γ\B2ℂ verifies the cohomogical conditions b1(M)=0 and H1,1(M,ℂ)∩H2(M,ℚ)≃ℚ. Then Γ is arithmetic. In this paper we consider a smooth complex projective surface M verifying the above cohomological assumptions and study Zariski-dense representations of the fundamental group π1(M) in a simple k-group H of k-rank ≤2 (where k denotes a local field). Our main result states that there are strong restrictions on such representations, especially when k is non-archimedean (Theorem 5). We then consider some cocompact lattices in PU(2,1) of special geometric interest: recall that a “fake P2ℂ” is a smooth complex surface (distinct from P2ℂ) having the same Betti numbers as P2ℂ. “Fake P2ℂ” exist by a result of Mumford and are complex hyperbolic quotients Γ\H2ℂ by Yau’s proof of the Calabi conjecture. They obviously verify the hypotheses of Rogawski’s conjecture. In this case we prove that every Zariski-dense representation of Γ in PGL(3) is superrigid in the sense of Margulis (Theorem 3). As a corollary every “fake P2ℂ” is an arithmetic quotient of the ball B2ℂ.

Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let \( {\rm M} = \Gamma \backslash X \) be a quotient of X by a torsion-free discrete subgroup \( \Gamma \) of G such that M is of finite volume in the canonical metric. Then, due to the G-equivariant Borel embedding of X into its compact dual Xc, the locally symmetric structure of M can be considered as a special kind of a \( (G_{\Bbb C} , X_c) \)-structure on M, a maximal atlas of Xc-valued charts with locally constant transition maps in the complexified group \( {\rm G}_{\Bbb C} \). By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the \( ({\rm G}_{\Bbb C} , X_c) \)-structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung.

Esnault asked whether every smooth complex projective variety with infinite fundamental group has a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). In a sense, this would mean that every variety with infinite fundamental group has some nonpositive curvature. We show that the answer to Esnault's question is positive when the fundamental group has a finite-dimensional representation over some field with infinite image. This applies to all known varieties with infinite fundamental group. Along the way, we produce many symmetric differentials on the base of a variation of Hodge structures. One interest of these results is that symmetric differentials give information in the direction of Kobayashi hyperbolicity. For example, they limit how many rational curves the variety can contain., 14 pages; v3: references added. To appear in Duke Math. J

In this paper we prove, assuming the Generalized Riemann Hypothesis, the Andr?e-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional assumption, we prove the conjecture without assuming the GRH., Comment: Submitted to Annals of Mathematics, Version of September 2013

Let $\Gamma \stackrel{i}{\hookrightarrow} L$ be a lattice in the real simple Lie group $L$. If $L$ is of rank at least 2 (respectively locally isomorphic to $Sp(n,1)$) any unbounded morphism $\rho: \Gamma \longrightarrow G$ into a simple real Lie group $G$ essentially extends to a Lie morphism $\rho_L: L \longrightarrow G$ (Margulis's superrigidity theorem, respectively Corlette's theorem). In particular any such morphism is infinitesimally, thus locally, rigid. On the other hand, for $L=SU(n,1)$, even morphisms of the form $\rho : \Gamma \stackrel{i}{\hookrightarrow} L \longrightarrow G$ are not infinitesimally rigid in general. Almost nothing is known about their local rigidity. In this paper we prove that any {\em cocompact} lattice $\Gamma$ in SU(n,1) is essentially locally rigid (while in general not infinitesimally rigid) in the quaternionic groups $Sp(n,1)$, SU(2n,2) or SO(4n,4) (for the natural sequence of embeddings $SU(n,1) \subset Sp(n,1) \subset SU(2n,2) \subset SO(4n,4))$., Comment: 24 pages

Carlson and Toledo conjectured that any infinite fundamental group $\Gamma$ of a compact K\"ahler manifold satisfies $H^2(\Gamma,\R)\not =0$. We assume that $\Gamma$ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ($\C$-VHS) on the K\"ahler manifold. We prove the conjecture under some assumption on the $\C$-VHS. We also study some related geometric/topological properties of period domains associated to such $\C$-VHS., Comment: 21 pages. Exposition improved. Final version

This thesis is divided in two independent parts. In the first part, we study the strata of holomorphic differentials from the point of view of bialgebraic geometry and Hodge theory. We define a bialgebraic structure on the strata and we study their bialgebraic subvarieties: in particular, we provide a partial classification of bialgebraic curves in the minimal stratum. We then focus on the distribution of the special points in the strata. We prove that they are dense in each stratum, for the euclidean topology. In contrast to the classical cases of commutative group varieties and Shimura varieties, there exist ℚ-bialgebraic subvarieties containing only finitely many special points. Lastly, we study the ``tameness'' properties of certain local coordinates of the strata (the period coordinates) and we prove that they are definable, in the sense of o-minimality. In the second part, we focus on the topology of complex algebraic varieties: more precisely, we study the cohomology jump loci in the Betti moduli space of singular complex algebraic varieties. We obtain results on the structure of the cohomology jump loci in rank one for normal varieties and for projective varieties whose first cohomology group H¹(X,ℚ) is pure of weight one. We then prove the compatibility of the cohomology jump loci with the Hodge theory of the completed local rings in the representation variety (for any rank) of a smooth projective variety. In rank one, we do not need the hypotheses of smoothness and projectivity, and we allow normal quasi-projective varieties. Lastly, we study certain jump loci associated to the weight filtration and to the Hodge filtration in the cohomology of unitary local systems of rank one.; La thèse est divisée en deux parties indépendantes. Dans le premier partie, nous étudions les strates de différentielles holomorphes du point de vue de la géométrie bialgébrique et de la théorie de Hodge. Nous définissons une structure bialgébrique sur les strates et nous étudions les sous-variétés bialgébriques: nous donnons en particulier une classification partielle des courbes bialgébriques dans la strate minimale. Nous étudions ensuite la distribution des points spéciaux dans les strates. Notamment, ils sont denses dans chaque strate pour la topologie euclidienne. Contrairement aux cas classiques des variétés en groupes commutatifs et des variétés de Shimura, on trouve des sous-variétés ℚ-bialgébriques ne contenant qu'un nombre fini de points spéciaux. En dernier lieu, on s'intéresse aux propriétés de ``modération'' des coordonnées locales des strates et nous prouvons leur définissabilité au sens de l'o-minimalité. Dans la deuxième partie, on s'intéresse à la topologie des variétés algébriques complexes: plus précisément, nous étudions les lieux de saut pour la cohomologie dans l'espace de modules de Betti associé aux variétés algébriques complexes singulières. Nous obtenons des énoncés de structure pour les lieux de saut en rang un pour les variétés normales et pour les variétés projectives dont le premier groupe de cohomologie H¹(X,ℚ) est pur de poids un. Ensuite, nous démontrons la compatibilité des idéaux des lieux de saut avec la théorie de Hodge des anneaux locaux complétés dans la variété de représentations (en tout rang), pour une variété projective lisse. En rang un, nous n'avons pas besoin de l'hypothèse de projectivité ou de lissité et on admet des variétés quasi-projectives normales. En dernier lieu, nous traitons certains lieux de saut pour la filtration du poids et la filtration de Hodge dans la cohomologie des systèmes locaux unitaires de rang un.