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

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

2010-09-10

arxiv.org/abs/1009.2011v2

Mathematics

Algebraic Geometry

30 pages

Scientific paper

Let $X$ be a Hilbert modular variety and $\mathbb{V}$ a nontrivial local system over $X$. In this paper we study Deligne-Saito's mixed Hodge structure (MHS) on the cohomology group $H^*(X,\mathbb{V})$ using the method of Higgs bundles. Among the various results we 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.

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