Symmetric differentials and the fundamental group

TL;DR

This paper proves that smooth complex projective varieties with certain types of infinite fundamental groups possess nonzero symmetric differentials, linking fundamental group properties to geometric structures and hyperbolicity.

Contribution

It establishes the existence of symmetric differentials for varieties with infinite fundamental groups under finite-dimensional representation conditions, extending known cases.

Findings

Varieties with infinite fundamental groups have nonzero symmetric differentials under certain conditions.

Produced many symmetric differentials on bases of variations of Hodge structures.

Results imply restrictions on rational curves, informing Kobayashi hyperbolicity.

Abstract

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.