Theta groups and projective models of hyperk\"ahler varieties

TL;DR

This paper introduces the theta group associated with certain sheaves on hyperk"ahler manifolds, computes their properties, and aims to describe families of polarized varieties of Kummer or OG6 type explicitly.

Contribution

It defines theta groups for specific sheaves on hyperk"ahler manifolds and computes their commutator pairings, advancing understanding of polarized hyperk"ahler varieties.

Findings

Computed commutator pairings of theta groups for line bundles and rank 4 vector bundles.

Found trivial commutator pairing for the tangent bundle.

Provided explicit descriptions of theta groups in specific cases.

Abstract

We define the theta group associated to a simple coherent sheaf $\cal F$ on a hyperk\"ahler manifold $X$ of Kummer type or OG6 type, provided $g^{*}({\cal F})$ is isomorphic to $\cal F$ for every automorphism $g$ of $X$ acting trivially on $H^2(X)$. Note that this condition is satisfied if $\cal F$ is invertible, if $\cal F$ is one of the rank $4$ stable vector bundles on general polarized HK fourfolds with certain discrete invariants constructed in arXiv:2203.03987, or if $\cal F$ is the tangent bundle. We compute the commutator pairings of theta groups of line bundles and the rank $4$ modular vector bundles of arXiv:2203.03987 (the commutator pairing of the tangent bundle is trivial). We have been motivated by the quest for an explicit description of locally complete families of polarized varieties of Kummer (or OG6) type.