Stable vector bundles on a hyper-Kähler manifold with a rank 1 obstruction map are modular.

Let X be an irreducible 2 n-dimensional holomorphic symplectic manifold. A reflexive sheaf F is very modular if its Azumaya algebra End (F) deforms with X to every Kähler deformation of X. We show that if F is a slope-stable reflexive sheaf of positive rank and the obstruction map HH² (X) → Ext 2 (F, F) has rank 1, then F is very modular. We associate to such a sheaf a vector in the Looijenga-Lunts-Verbitsky lattice of rank b 2 (X) + 2. Three sources of examples of such modular sheaves emerge. The first source consists of slope-stable reflexive sheaves F of positive rank that are isomorphic to the image Φ (O X) of the structure sheaf via an equivalence Φ: Db (X) → Db (Y) of the derived categories of two irreducible holomorphic symplectic manifolds. The second source consists of such F, which are isomorphic to the image of a skyscraper sheaf via a derived equivalence. The third source consists of images Φ (L) of torsion sheaves L supported as line bundles on holomorphic lagrangian submanifolds Z such that Z deforms with X in codimension 1 in moduli, and L is a rational power of the canonical line bundle of Z. An example of the first source is constructed using a stable and rigid vector bundle G on a K 3 surface X to get the very modular vector bundle F on the Hilbert scheme X [n] associated to the equivariant vector bundle G ⊠ ⋯ ⊠ G on X n via the Bridgeland-King-Reid (BKR) correspondence. This builds upon and partially generalizes results of O'Grady for n = 2. A construction of the second source associates to a set {Gi}ni = 1 of n distinct stable vector bundles in the same two-dimensional moduli space of vector bundles on a K3 surface X the very modular vector bundle F on X [n] corresponding to the equivariant bundle ⊕ σ ∈ S n [Gσ(1)...Gσ(n)] on Xn. [ABSTRACT FROM AUTHOR]