Intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve

Abstract

Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization ${\tilde{\mathcal{M}}}_{Dol}^G$ of the moduli space ${\mathcal{M}}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,\mathbb{C}),SL(2,\mathbb{C})$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of ${\tilde{\mathcal{M}}}_{Dol}^G$ as a direct sum of the intersection cohomology of ${\mathcal{M}}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of ${\mathcal{M}}_{Dol}^G$ and prove that the mixed Hodge structure on it is actually pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

Final version, to appear in Journal of the Institute of Mathematics of Jussieu

References in corpus (1)

• Arithmetic of singular character varieties and their $E$-polynomials

Cited by in corpus (1)

• $P=W$ phenomena in algebraic and enumerative geometry