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.
Categories: math.AG math.DG