Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

TL;DR

This paper proves that invariant subbundles of the Kontsevich-Zorich cocycle respect the Hodge structure, establishing semisimplicity and polynomial variation on affine manifolds, with implications for Jacobians and endomorphisms.

Contribution

It establishes Deligne semisimplicity for the Kontsevich-Zorich cocycle and shows invariant subbundles respect the Hodge structure, extending understanding of the cocycle's algebraic and geometric properties.

Findings

Invariant subbundles respect the Hodge structure

Invariant subbundles vary polynomially on affine manifolds

Tensor powers of the cocycle have coinciding measurable and algebraic hulls

Abstract

We prove that invariant subbundles of the Kontsevich-Zorich cocycle respect the Hodge structure. In particular, we establish a version of Deligne semisimplicity in this context. This implies that invariant subbundles must vary polynomially on affine manifolds. All results apply to tensor powers of the cocycle and this implies that the measurable and real-analytic algebraic hulls coincide. We also prove that affine manifolds parametrize Jacobians with non-trivial endomorphisms. Typically a factor has real multiplication. The tools involve curvature properties of the Hodge bundles and estimates from random walks. In the appendix, we explain how methods from ergodic theory imply some of the global consequences of Schmid's work on variations of Hodge structures. We also derive the Kontsevich-Forni formula using differential geometry.