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

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.