it is a mystery which groups can occur as fundamental groups of smooth complex projective varieties. it is conceivable that whenever the fundamental group is infinite, the variety has some "negative curvature" properties. we discuss a result in this direction, in terms of "symmetric differentials". there are interesting open questions even about the special case of compact quotients of the unit ball in $c^n$. (joint work with yohan brunebarbe and bruno klingler.)