# Lyapunov exponents of the Brownian motion on a K\"ahler manifold

## TL;DR

This paper introduces a method to compute Lyapunov exponents for flat bundles over Kähler manifolds using harmonic measures, establishing inequalities relating these exponents to bundle degrees and exploring conditions for equality.

## Contribution

It defines the Lyapunov spectrum for flat bundles on Kähler manifolds and provides a novel computational approach via harmonic measures, along with new inequalities linking exponents and bundle degrees.

## Key findings

- Lyapunov exponents can be computed using harmonic measures on foliated spaces.
- Established inequalities relating Lyapunov exponents to degrees of holomorphic subbundles.
- Discussed conditions under which the inequalities become equalities.

## Abstract

If E is a flat bundle of rank r over a K\"ahler manifold X, we define the Lyapunov spectrum of E: a set of r numbers controlling the growth of flat sections of E, along Brownian trajectories. We show how to compute these numbers, by using harmonic measures on the foliated space P(E). Then, in the case where X is compact, we prove a general inequality relating the Lyapunov exponents and the degrees of holomorphic subbundles of E and we discuss the equality case.