Vincent Koziarz

Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations

In this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a non-equidimensional surjective holomorphic map between compact ball quotients, our method applies to show that the set of critical values must be nonempty and of codimension 1. In the equivariant setting the line of arguments extends to holomorphic mappings of maximal rank into the complex projective space or the complex Euclidean space, yielding in the latter case a lower bound on the dimension of the singular loci of certain holomorphic maps defined by integrating holomorphic 1-forms. In another direction, we extend the nonexistence statement on holomorphic submersions to the case of ball quotients of finite volume, provided that the target complex unit ball is of dimension

The Toledo invariant on smooth varieties of general type

m \ge 2

HARMONIC MAPS AND REPRESENTATIONS OF NON-UNIFORM LATTICES OF PU(m, 1)

, giving in particular a new proof that a local biholomorphism between noncompact

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

m

Numerical character of the effectivity of adjoint line bundles

-ball quotients of finite volume must be a covering map whenever

Fonctions holomorphes sur l'espace des cycles: la methode d'intersection

m \ge 2

On the Cartwright-Steger surface

. Finally, combining our results with Hermitian metric rigidity, we show that any holomorphic submersion from a bounded symmetric domain into a complex unit ball equivariant with respect to a lattice must factor through a canonical projection to yield an automorphism of the complex unit ball, provided that either the lattice is cocompact or the ball is of dimension at least 2.

On the equidistribution of totally geodesic submanifolds in compact locally symmetric spaces and application to boundedness results for negative curves and exceptional divisors

Abstract We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: assuming that the rank of the target Lie group is not greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.