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

Abstract

Let X be an irreducible $2n$-dimensional holomorphic symplectic manifold. A reflexive sheaf F is very modular if its Azumaya algebra $\mathcal{E}nd(F)$ deforms with X to every Kähler deformation of X. We show that if F is a slope-stable reflexive sheaf of positive rank and the obstruction map $H{H}^2(X)\to {\mathrm{Ext}}^2(F,F)$ has rank 1, then F is very modular. We associate to such a sheaf a vector in the Looijenga–Lunts–Verbitsky lattice of rank $b_2(X)+2$. Three sources of examples of such modular sheaves emerge. The first source consists of slope-stable reflexive sheaves F of positive rank that are isomorphic to the image $\mathrm{\Phi }({\mathcal{O}}_X)$ of the structure sheaf via an equivalence $\mathrm{\Phi }:D^b(X)\to D^b(Y)$ 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 skyscraper sheaf via a derived equivalence. The third source consists of images $\mathrm{\Phi }(L)$ of torsion sheaves L supported as line bundles on holomorphic lagrangian submanifolds Z such that Z deforms with X in codimension 1 in moduli, and L is a rational power of the canonical line bundle of Z.

An example of the first source is constructed using a stable and rigid vector bundle G on a $K3$ surface X to get the very modular vector bundle F on the Hilbert scheme $X^{[n]}$ associated to the equivariant vector bundle $G⊠\cdots ⊠G$ on $X^n$ via the Bridgeland–King–Reid (BKR) correspondence. This builds upon and partially generalizes results of O’Grady for $n=2$. A construction of the second source associates to a set ${\{G_i\}}_{i=1}^n$ of n distinct stable vector bundles in the same two-dimensional moduli space of vector bundles on a $K3$ surface X the very modular vector bundle F on $X^{[n]}$ corresponding to the equivariant bundle ${\oplus }_{\mathit{\sigma }\in {\mathfrak{S}}_n}[G_{\mathit{\sigma }(1)}⊠\cdots ⊠G_{\mathit{\sigma }(n)}]$ on $X^n$.