INTERSECTION COHOMOLOGY OF THE MODULI SPACE OF HIGGS BUNDLES ON A GENUS 2 CURVE

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