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arXiv · 2006.08752

P=W conjectures for character varieties with symplectic resolution

Camilla Felisetti·Mirko Mauri
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Abstract

We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-\'{e}tale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and the projectivity of the compactification of the de Rham moduli space. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6.

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Camilla Felisetti, Mirko Mauri. 2020-06-15. P=W conjectures for character varieties with symplectic resolution. https://arxiv.org/abs/2006.08752

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