Lyapunov spectrum of invariant subbundles of the Hodge bundle

TL;DR

This paper investigates the Lyapunov spectrum of the Kontsevich--Zorich cocycle on invariant subbundles of the Hodge bundle, deriving formulas relating exponents to geometric structures and exploring their properties in specific cases.

Contribution

It provides new formulas for sums of Lyapunov exponents in terms of the second fundamental form and analyzes the structure of invariant subbundles in the Hodge bundle.

Findings

Formulas for partial sums of Lyapunov exponents in terms of the second fundamental form.

Relations between the central Oseledets subbundle and the kernel of the second fundamental form.

Illustrations of theoretical results in two special cases.

Abstract

We study the Lyapunov spectrum of the Kontsevich--Zorich cocycle on $SL(2,\mathbb{R})$-invariant subbundles of the Hodge bundle over the support of a $SL(2,\mathbb{R})$-invariant probability measure on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (or Kodaira--Spencer map) of the Hodge bundle with respect to Gauss--Manin connection and investigate the relations between the central {Oseldets} subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.