Theta groups and projective models of hyperk\"ahler varieties

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.