Geometric Variations of Local Systems and Elliptic Surfaces

TL;DR

This paper develops a formalism for geometric variations of local systems, applies it to families of elliptic surfaces, and explores their connections to Hodge structures and isomonodromic deformations.

Contribution

It introduces a new formal framework for geometric variations of local systems and applies it to elliptic surfaces, linking to Hodge theory and isomonodromic deformations.

Findings

Interpretation of twisted elliptic surface families via middle convolution

Calculation of Hodge structure variations for M_N-polarized K3 surfaces

Connection established between geometric variations and isomonodromic deformations

Abstract

Geometric variations of local systems are families of variations of Hodge structure; they typically correspond to fibrations of K\"{a}hler manifolds for which each fibre itself is fibred by codimension one K\"{a}hler manifolds. In this article, we introduce the formalism of geometric variations of local systems and then specialize the theory to study families of elliptic surfaces. We interpret a construction of twisted elliptic surface families used by Besser-Livn\'{e} in terms of the middle convolution functor, and use explicit methods to calculate the variations of Hodge structure underlying the universal families of $M_N$-polarized K3 surfaces. Finally, we explain the connection between geometric variations of local systems and geometric isomonodromic deformations, which were originally considered by the first author in 1999.