Very modular vector bundles over compact hyperkahler manifolds

An irreducible holomorphic symplectic manifold (IHSM) is a higher dimensional analogue of a K3 surface. A vector bundle F on an IHSM X is very modular, if the projective bundle P(F) deforms with X to every Kahler deformation of X. We show that if F is a slope-stable vector bundle and the obstruction map from the second Hochschild cohomology of X to Ext^2(F,F) has rank 1, then F is modular.
Three sources of examples of such slope stable modular bundles F emerge.
(1) F is isomorphic to the image of the structure sheaf via an equivalence of the derived categories of two IHSMs.
(2) F is isomorphic to the image of a sky-scraper sheaf via a derived equivalence.
(3) F which is the image of a torsion sheaf L supported as line bundles on a holomorphic lagrangian submanifold 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.