Monodromy action on character varieties for Lefschetz pencils

Abstract

Given a Riemann surface {\Sigma}, let {\Gamma} {\subseteq} Mod({\Sigma}) denote the monodromy subgroup of a family of complex curves homeomorphic to {\Sigma}, arising from a sufficently ample Lefschetz pencil. We establish that the group {\Gamma} acts with Zariski dense orbits or ergodically on certain character varieties for {\pi_1}({\Sigma}). This answers a version of a Conjecture of Katzarkov, Pantev, and Simpson appearing in [KPS03].