Stable sheaves and hyperholomorphic vector bundles on K3 surfaces. Modular sheaves on hyperkähler varieties, examples and first properties.

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.

I will talk about a recent joint work with Claudio Onorati, where we provide a class of formal associative DG algebras. As an application, we prove formality for derived endomorphisms of a holomorphic vector bundle admitting a projectively hyper-holomorphic connection on a hyperkähler manifold X, so that we recover Verbitsky’s quadraticity result. I will then focus on deformations of autodual connections on a complex vector bundle on X.

I will also describe the DG Lie algebra controlling this deformation problem, and prove that it is formal when the connection is hyper-holomorphic.

Stable modular vector bundles on hyperkähler varieties deformation equivalent to the Hilbert square of a K3 surface, or to a 4 dimensional generalized Kummer.

Jieao Song proposed a conjectural formula which expresses the class of a Lagrangian plane in a hyperkähler variety in terms of the class of a line on it. In this talk I will explain how to prove this conjecture if the hyperkähler variety is of K3[n]-type. This generalizes earlier results of Hassett-Tschinkel, Harvey-Hassett-Tschinkel, and Bakker-Jorza.

Let X and Y be compact hyper-Kahler manifolds deformation equivalence to the Hilbert scheme of length n subschemes of a K3 surface. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let f be a Hodge isometry of their second rational cohomologies with respect to the Beauville-Bogomolov-Fujiki pairings. We prove that f is induced by an analytic correspondence. We furthermore lift f to an analytic correspondence F between their total rational cohomologies, which is a Hodge isometry with respect to the Mukai pairings, and which preserves the gradings up to sign. When X and Y are projective the correspondences f and F are algebraic.