  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 Kahler deformation of X. We show that if F is a slope-stable reflexive
sheaf of positive rank and the obstruction map from the second Hochschild
cohomology of X to $Ext^2(F,F)$ has rank 1, then F is very modular. We
associate to such a sheaf a vector in the Looijenga-Lunts-Verbitsky lattice of
rank equal to the second Betti number of X plus 2. Three sources of examples of
such modular sheaves emerge. The first source consists of slope-stable
reflexive sheaves F of positive rank which are isomorphic to the image of the
structure sheaf via an equivalence 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 sky-scraper sheaf via a derived equivalence.
The third source consists of images of torsion sheaves L supported as line
bundles on holomorphic lagrangian submanifolds Z, such that Z deforms with X in
co-dimension one in moduli and L is a rational power of the canonical line
bundle of Z.
