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

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Camilla Felisetti
Université de Genève
Mon, 2020-02-17 11:00 - 12:00
MPIM Lecture Hall
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Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization $\widetilde{\mathcal{M}}^G_{Dol}$ of the moduli space $\mathcal{M}^G_{Dol}$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,co), SL(2,co)$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of $\widetilde{\mathcal{M}}^G_{Dol}$ as a direct sum of the intersection cohomology of $\mathcal{M}^G_{Dol}$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal{M}^G_{Dol}$ and prove that the mixed Hodge structure on it is pure, in analogy with what happens to ordinary cohomology in the smooth case of rank and degree coprime.

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