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

Abstract

Let S be a K3 surface and \(S^{[n]}\) the Hilbert scheme of length n subschemes of S. Over the cartesian square \(S^{[n]}\times S^{[n]}\) there exists a natural reflexive rank \(2n-2\) coherent sheaf E, which is locally free away from the diagonal. The fiber of E over the point \((I_{Z_1},I_{Z_2})\), corresponding to ideal sheaves of distinct subschemes \(Z_1\ne Z_2\), is \(\mathrm{Ext}^1_S(I_{Z_1},I_{Z_2})\). We prove that E is slope stable if the rank of the Picard group of S is \(\le 19\). The Chern classes of \({\mathcal E}nd(E)\) are known to be monodromy invariant. Consequently, the sheaf \({\mathcal E}nd(E)\) is polystable-hyperholomorphic.