Eyal Markman, December 3, 2021

Title: Modular vector bundles over compact hyperkahler manifolds

Abstract: An irreducible holomorphic symplectic manifold (IHSM) is a higher dimensional analogue of a K3 surface. A vector bundle F on an IHSM X is 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 modular bundles emerge. (1) Slope-stable vector bundles F which are isomorphic to the image of the structure sheaf via an equivalence of the derived categories of two IHSMs. (2) Such F, which are isomorphic to the image of a sky-scraper sheaf via a derived equivalence. (3) Such F which are 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.