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.