O'Grady spaces and symplectic resolution of moduli spaces of Higgs bundles

TL;DR

This paper explores the degeneration of symplectic desingularizations of Higgs bundle moduli spaces into O'Grady's exceptional holomorphic symplectic manifolds, connecting different geometric structures.

Contribution

It provides a detailed analysis of the degeneration process linking Higgs moduli spaces to O'Grady's examples, including new details missing in prior surveys.

Findings

Degeneration of Higgs moduli spaces into O'Grady's manifolds

Connection established between different symplectic structures

Clarification of previous incomplete descriptions

Abstract

We describe here a degeneration of the symplectic desingularization of the moduli spaces of topologically trivial $GL(2,\mathbb{C})$ and $SL(2,\mathbb{C})$-Higgs bundles over a hyperelliptic curve, into O'Grady's ten and six dimensional exceptional examples of irreducible holomorphic symplectic manifolds. Most of this note is a survey of work on these degenerations by Donagi-Ein-Lazarsfeld, de Cataldo-Maulik-Shen and Felissetti-Mauri, although in certain cases we provide details that were missing the previous articles.