Abstract
AbstractLetCbe a smooth projective curve of genus$2$. Following a method by O’Grady, we construct a semismall desingularisation$\tilde {\mathcal {M}}_{Dol}^G$of the moduli space$\mathcal {M}_{Dol}^G$of semistableG-Higgs bundles of degree 0 for$G=\mathrm {GL}(2,\mathbb {C}), \mathrm {SL}(2,\mathbb {C})$. By the decomposition theorem of Beilinson, Bernstein and 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 pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.
Highlights
The main problem in doing that is that the moduli space is singular and many fundamental theorems like Poincare duality or Hard-Lefschetz theorem fail for ordinary cohomology groups
To do that we extend the natural C∗-action on MGDol, given by rescaling the Higgs field, to M GDol: this yields an isomorphism of mixed Hodge structures between the cohomology of M GDol and that of a compact subvariety, namely, the preimage (π ◦ χ)−1(0) of the nilpotent cone in the resolution
One can show in the same way that the Hodge structure on the intersection cohomology of moduli space of rank 2 Higgs bundles is pure in any genus
Summary
One can show in the same way that the Hodge structure on the intersection cohomology of moduli space of rank 2 Higgs bundles is pure in any genus. This is due to the fact that the resolution is obtained by blowing up along C∗-equivariant subsets, so that the above argument still works. In the smooth coprime case, the cohomology of these spaces has been widely studied: Poincare polynomials for SL(2,C) were computed by Hitchin in his seminal paper on Higgs bundles [22], for SL(3,C) by Gothen in [20] and in rank 4 by Garcia-Prada, Heinloth and Schmitt [17].
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