Supports of the Hitchin fibration: singularities and zonotopes

The key idea behind Hodge theory is that rich geometric structures on a complex manifold (e.g. holonomy U(n)) induce a decomposition of its cohomology, whose building blocks have a clear geometric meaning (e.g. sheaf cohomology of holomorphic differential forms, or spaces of algebraic cycles). In the same spirit, the decomposition theorem for proper morphisms grants that the cohomology of the domain splits in elementary summands. However, in general, it is a subtle task to determine explicitly these summands. We prove that this is in fact possible in the case of Hitchin fibrations for Higgs bundles of arbitrary degree. Surprisingly we relate the summands of the decomposition theorem to the singularity theory of the moduli spaces of Higgs bundles in (fixed!) degree zero. We also provide a combinatorial version of the decomposition theorem via counts of lattice points in zonotopes. This is based on a collaboration with Luca Migliorini and Roberto Pagaria.