Naturality of the hyperholomorphic sheaf over the cartesian square of a manifold of $K3^{[n]}$-type

TL;DR

This paper proves the slope-stability of a naturally arising reflexive sheaf over the product of a hyperkähler manifold of K3^{[n]}-type and its deformations, establishing its canonical nature and independence from specific choices.

Contribution

It demonstrates the stability and canonical deformation properties of a natural sheaf over hyperkähler manifolds of K3^{[n]}-type, extending previous results to all Kahler classes.

Findings

Sheaf E is slope-stable with respect to every Kahler class on M.

Deformed sheaf E' remains slope-stable on all deformations X of M.

E' is shown to be a canonical sheaf depending only on the isomorphism class of X.

Abstract

Let M be a 2n-dimensional smooth and compact moduli space of stable sheaves on a K3 surface S and U a universal sheaf over S x M. Over M x M there exists a natural reflexive sheaf E of rank 2n-2, namely the first relative extension sheaf of the two pullbacks of U to M x S x M. We prove that E is slope-stable with respect to every Kahler class on M. The sheaf E is known to deform to a sheaf E' over X x X, for every manifold X deformation equivalent to M, and we prove that E' is slope-stable with respect to every Kahler class on X. This triviality of the stability chamber structure combines with a result of S. Mehrotra and the author to show that the deformed sheaf E' is canonical. Consequently, the pretriangulated K3 category associated to the pair (X x X,E') in our earlier work with S. Mehrotra depends only on the isomorphism class of X.