Stability of a natural sheaf over the cartesian square of the Hilbert scheme of points on a K3 surface

TL;DR

This paper proves the slope stability of a natural sheaf over the cartesian square of the Hilbert scheme of points on a K3 surface, under certain conditions on the Picard group, with implications for its hyperholomorphicity.

Contribution

It establishes the slope stability of a specific natural sheaf on the cartesian square of the Hilbert scheme of points on a K3 surface, extending understanding of sheaf stability in this geometric context.

Findings

Sheaf E is slope stable if Picard rank ≤ 19.

Chern classes of End(E) are monodromy invariant.

End(E) is polystable-hyperholomorphic.

Abstract

Let S be a K3 surface and S^[n] the Hilbert scheme of length n subschemes of S. Over the cartesian square of S^[n] there exists a natural reflexive rank 2n-2 coherent sheaf E, which is locally free away from the diagonal. The fiber of E, over a pair of ideal sheaves of distinct subschemes, is the vector space of extensions of the first ideal sheaf by the second. We prove that E is slope stable if the rank of the Picard group of S is less than or equal to 19. The Chern classes of End(E) are known to be monodromy invariant. Consequently, the sheaf End(E) is polystable-hyperholomorphic.