P-adic lattices are not K\"ahler groups

In this note we show that any lattice in a simple p-adic Lie group is not the fundamental group of a compact Ka\"hler manifold, as well as some variants of this result.Comment: Final versio

Symmetric differentials and the fundamental group

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.Comment: 14 pages; v3: references added. To appear in Duke Math.

On the second cohomology of K\"ahler groups

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 versio

The Ax-Schanuel conjecture for variations of mixed Hodge structures

We prove in this paper the Ax-Schanuel conjecture for all admissible variations of mixed Hodge structures.Comment: 38 pages, including recall for preliminary knowledge. Comments are welcom

On the closure of the Hodge locus of positive period dimension

Given V a polarizable variation of Z-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for V⊗ is the set of closed points s of S where the fiber Vs has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for V⊗ is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for V. Under the assumption that the adjoint group of the generic Mumford–Tate group of V is simple we prove that the union of the special subvarieties for 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 themoduli space Ag 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 Ag is either a closed algebraic subvariety of S or is Zariski-dense in S.Humboldt-Universität zu Berlin (1034)Peer Reviewe