Topology of Algebraic Varieties

Esnault asked whether a smooth complex projective variety with infinite fundamental group has a nonzero symmetric differential, meaning a section of some symmetric power of the cotangent bundle. We prove a partial result in this direction, using Simpson's theory of representations of the fundamental group. The result gives some information on Kobayashi hyperbolicity for these varieties. (Joint with Yohan Brunebarbe and Bruno Klingler)