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

Let X be an irreducible 2n-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) →Ext2 (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 b2 (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 (OX) 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 K3 surface X to get the very modular vector bundle F on the Hilbert scheme Xn associated to the equivariant vector bundle G⊠⋯⊠G on Xn 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 Gii=1n 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 Xn corresponding to the equivariant bundle ⊕σ∈SnGσ (1) ⊠⋯⊠Gσ (n) on Xn.