Hodge-to-singular correspondence

The decomposition theorem for proper morphisms of algebraic varieties 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 on the locus of reduced spectral curves. Surprisingly we relate the summands of the decomposition theorem to the singularity theory of the moduli spaces of Higgs bundles in (fixed!) degree zero. This is based on a collaboration with Luca Migliorini.