We show that any smooth complex projective variety whose fundamental group has a complex representation with infinite image must have a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). Along the way, we produce many symmetric differentials on the base of a variation of Hodge structures.
"Symmetric differentials and the fundamental group." Duke Math. J. 162 (14) 2797 - 2813, November 2013. https://doi.org/10.1215/00127094-2381442