On the positivity of the logarithmic cotangent bundle

The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. These examples are constructed from any smooth n-dimensional complex projective varieties by considering the sum of at least n general sufficiently ample hypersurfaces

Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

In the paper we study various hyperbolicity properties for the quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero dimensional. In the first part we prove that $U$ is algebraically hyperbolic, and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite unramified cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ so that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion free lattices.Comment: 30 pages. V3, main results are improved: no monodromy assumptions are needed. Comments are very welcome

SIMPSON CORRESPONDENCE FOR SEMISTABLE HIGGS BUNDLES OVER KÄHLER MANIFOLDS

In this note we provide an elementary proof of the Simpson correspondence between semistable Higgs bundles with vanishing Chern classes and representation of fundamental groups of Kähler manifolds