  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.
