A smooth Birman-Hilden theory for hyperkähler manifolds

Abstract

This article grew out of an effort to understand the smooth mapping class groups of certain 4-manifolds in a geometric manner. We prove a smooth analog of the Birman-Hilden theorem for manifolds that admit a hyperkähler structure. This allows us to probe the smooth mapping class groups associated to certain manifolds with nontrivial fundamental groups. Along the way, and of independent interest, we prove a global Torelli theorem for generalized Enriques manifolds. The techniques used are analogous to Teichmüller-theoretic methods in the classical theory of mapping class groups. We apply this hyperkähler Birman-Hilden theorem to obtain results regarding smooth, metric, and complex Nielsen realization on Enriques surfaces.

31 pages. To appear in Crelle's Journal