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arXiv · 2406.00395

Holomorphic symplectic manifolds from semistable Higgs bundles

Roland Abuaf·Riccardo Carini
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Abstract

Let $\mathcal{M}_{C}(2, 0)$ be the moduli space of semistable rank two and degree zero Higgs bundles on a smooth complex hyperelliptic curve $C$ of genus three. We prove that the quotient of $\mathcal{M}_{C}(2, 0)$ by a twisted version of the hyperelliptic involution is an 18-dimensional holomorphic symplectic variety admitting a crepant resolution, whose local model was studied by Kaledin and Lehn to describe O'Grady's singularities. Similarly, by considering the moduli space of Higgs bundles with trivial determinant $\mathcal{M}_C(2, \mathcal{O}_{C})\subseteq \mathcal{M}_C(2, 0)$, we show that the quotient of $\mathcal{M}_C(2, \mathcal{O}_{C})$ by the hyperelliptic involution is a 12-dimensional holomorphic symplectic variety admitting a crepant resolution.

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Roland Abuaf, Riccardo Carini. 2024-06-01. Holomorphic symplectic manifolds from semistable Higgs bundles. https://arxiv.org/abs/2406.00395

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