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Let X be an irreducible 2n-dimensional holomorphic symplectic manifold. Areflexive sheaf F is very modular, if its Azumaya algebra End(F) deforms with Xto every Kahler deformation of X. We show that if F is a slope-stable reflexivesheaf of positive rank and the obstruction map from the second Hochschildcohomology of X to $Ext^2(F,F)$ has rank 1, then F is very modular. Weassociate to such a sheaf a vector in the Looijenga-Lunts-Verbitsky lattice ofrank equal to the second Betti number of X plus 2. Three sources of examples ofsuch modular sheaves emerge. The first source consists of slope-stablereflexive sheaves F of positive rank which are isomorphic to the image of thestructure sheaf via an equivalence of the derived categories of two irreducibleholomorphic symplectic manifolds. The second source consists of such F, whichare 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 linebundles on holomorphic lagrangian submanifolds Z, such that Z deforms with X inco-dimension one in moduli and L is a rational power of the canonical linebundle of Z.

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