Two applications of the decomposition theorem to moduli spaces

The decomposition theorem is a statement about the (derived) direct image of the intersection cohomology by an algebraic projective map. The decomposition theorem and more generally the theory of perverse sheaves have found many interesting applications, especially in representation theory. Usually a lot of work is needed to apply it in concrete situations, to identify the various summands. This thesis proposes two applications of the decomposition theorem. In the first we consider the moduli space of Higgs bundles of rank 2 and degree 0 over a curve of genus 2. Applying the decomposition theorem, we are able to compute the weight polynomial of the intersection cohomology of this moduli space. The second result contained in this thesis is concerned with the general problem of determining the support of a map, and therefore in line with the ”support theorem” by Ngo. We consider families C ! B of integral curves with at worst planar singularities, and the relative ”nested” Hilbert scheme C^[m,m+1]. Applying the technique of higher discriminants, recently developed by Migliorini and Shende, we prove that in this case there are no supports other than the whole base B of the family. Along the way we investigate smoothness properties of C[m,m+1], which may be of interest on their own.