On the cohomology groups of local systems over Hilbert modular varieties via Higgs bundles

TL;DR

This paper investigates the mixed Hodge structure on the cohomology of local systems over Hilbert modular varieties using Higgs bundles, proving key isomorphisms, dimension formulas, and splitting properties.

Contribution

It introduces a method to analyze Saito's mixed Hodge structure on cohomology via Higgs bundles, including proving the Eichler-Shimura isomorphism and Hodge number formulas.

Findings

Proved the Eichler-Shimura isomorphism for these cohomology groups.

Derived a dimension formula for Hodge numbers.

Showed the mixed Hodge structure is split over the real numbers.

Abstract

Let $X$ be a Hilbert modular variety and $\mathbb{V}$ a non-trivial local system over $X$ with infinite monodromy. In this paper we study Saito's mixed Hodge structure (MHS) on the cohomology group $H^k(X,\mathbb{V})$ using the method of Higgs bundles. Among other results we prove the Eichler-Shimura isomorphism, give a dimension formula for the Hodge numbers and show that the mixed Hodge structure is split over $\mathbb{R}$. These results are analogous to Matsushima-Shimura [Annals of Mathematics 78, 1963] in the cocompact case and complement the results in Freitag [Book: Hilbert modular forms, Springer-Verlag, Berlin, 1990] for constant coefficients.