Holomorphic Higgs bundles over the Teichmüller space

Abstract

We study which representations $\rho$ of the fundamental group of a compact oriented surface $X$ admit Higgs data that depend holomorphically on the Riemann surface $\Sigma =(X,J)$ via non-abelian Hodge correspondence. For representations $\rho$ into $\mathrm{SL}(2,\mathbb{C})$ we show that holomorphic dependency is equivalent to $\rho$ being unitary. For higher ranks this equivalence fails -- we show the existence of non-unitary and irreducible representations of the fundamental group into $\mathrm{SL}(n,\mathbb{C})$ admitting Higgs data that are holomorphic in $\Sigma$, for $n$ large enough.

Final version; Communications in Analysis and Geometry (to appear)