  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.
