Geometric Variations of Local Systems

Abstract

The formalism of variations of local systems is applied in a geometric setting to define a notion of geometric variation of local systems; this provides a natural framework with which to study families of fibrations of Kahler manifolds. We apply this formalism in various contexts, starting with an examination of the moduli space of rational elliptic surfaces with four singular fibres. From there, we use the quadratic twist operation to construct families of K3 surfaces and examine the resulting geometric variations of local systems. We then proceed to study families of K3 surface fibrations. Specifically, we study families of M-polarized K3 surface fibrations and M_n-polarized K3 surface fibrations in the context of geometric variations of local systems; in particular, we are able to show how to obtain the fourteenth-case of integral variation of Hodge structures from the Doran-Morgan classification in this setting. Finally, we explain the connection to geometric isomononodromic deformations and, more generally, to solutions of the Schlesinger equations.