Modular sheaves with many moduli

O’Grady, Kieran G

doi:10.2140/gt.2026.30.203

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Modular sheaves with many moduli

Kieran G O’Grady

Geometry & Topology 30 (2026) 203–246

DOI: 10.2140/gt.2026.30.203

Abstract

We exhibit moduli spaces of slope stable vector bundles on general polarized HK varieties $(X, h)$ of type $K 3^{[2]}$ which have an irreducible component of dimension $2 a^{2} + 2$ , with $a$ an arbitrary integer greater than $1$ . This is done by studying the case $X = S^{[2]}$ where $S$ is an elliptic $K 3$ surface. We show that in this case there is an irreducible component of the moduli space of stable vector bundles on $S^{[2]}$ which is birational to a moduli space of sheaves on $S$ . We expect that if the moduli space of sheaves on $S$ is a smooth HK variety (necessarily of type $K 3^{[a^{2} + 1]}$ ) then the following more precise version holds: the closure of the moduli space of slope stable vector bundles on $(X, h)$ in the moduli space of Gieseker–Maruyama semistable sheaves with its GIT polarization is a general polarized HK variety of type $K 3^{[a^{2} + 1]}$ .

Keywords

hyperkähler varieties, stable sheaves

Mathematical Subject Classification

Primary: 14J42, 14J60

References

Publication

Received: 28 July 2024

Revised: 3 June 2025

Accepted: 11 July 2025

Published: 19 January 2026

Proposed: Arend Bayer

Seconded: Marc Levine, Mark Gross

Authors

Kieran G O’Grady
Dipartimento di Matematica Sapienza Università di Roma Rome Italy

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Volume 30, issue 1 (2026)