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 X[n] 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 {Gi}i=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 X[n] corresponding to the equivariant bundle ⊕σ∈Sn[Gσ(1)⊠⋯⊠Gσ(n)] on Xn.