Families of K3 surfaces and Lyapunov exponents

TL;DR

This paper investigates the Lyapunov exponents of families of K3 surfaces over hyperbolic curves, proving their rationality and exploring cases of maximal exponent through modular families and Kuga-Satake construction.

Contribution

It establishes the rationality of the top Lyapunov exponent for these families and connects maximal exponents to modular families via novel proofs.

Findings

Top Lyapunov exponent is rational for families of K3 surfaces.

Maximal Lyapunov exponent cases correspond to modular families.

Uses Kuga-Satake construction and integration by parts in proofs.

Abstract

Consider a family of K3 surfaces over a hyperbolic curve (i.e. Riemann surface). Their second cohomology groups form a local system, and we show that its top Lyapunov exponent is a rational number. One proof uses the Kuga-Satake construction, which reduces the question to Hodge structures of weight 1. A second proof uses integration by parts. The case of maximal Lyapunov exponent corresponds to modular families, given by the Kummer construction on a product of isogenous elliptic curves.