Arakelov-Milnor inequalities and maximal variations of Hodge structure

TL;DR

This paper investigates fixed points in Higgs bundle moduli spaces, introduces a new topological invariant generalizing known invariants, and establishes bounds and rigidity results for maximal variations of Hodge structure.

Contribution

It introduces a generalized Toledo invariant for Hodge bundles, providing bounds and rigidity results that extend classical inequalities and known cases.

Findings

Established a bound on the generalized Toledo invariant.

Proved rigidity for maximal variations of Hodge structure.

Unified inequalities for Hermitian and classical Hodge variations.

Abstract

In this paper we study the $\mathbb{C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. A main result of this paper is a bound on this invariant which generalizes both the Milnor-Wood inequality of the Hermitian case and the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case.