Integral Transforms and Deformations of K3 Surfaces

Abstract

Let X be a K3 surface and M a smooth and projective moduli space of stable sheaves on X of Mukai vector v. A universal sheaf U over X x M induces an integral transform F from the derived category D(X) of coherent sheaves on X to that on M. (1) We prove that the integral transform F is faithful. F is not full if the dimension of M is greater than 2. (2) We exhibit the full subcategory of D(M), consisting of objects in the image of F, as the quotient of a category, explicitly constructed from D(X), by a natural congruence relation defined in terms of the Mukai vector v. (3) Let C be a component of the moduli space of isomorphism classes of marked irreducible holomorphic symplectic manifolds deformation equivalent to the Hilbert scheme X^[n] of n points on a K3 surface X, n > 1. C is 21-dimensional, while the moduli of Kahler K3 surfaces is 20-dimensional. We construct a geometric deformation of the derived categories of K3 surfaces over a Zariski dense open subset of C, which coincides with D(X) whenever the marked manifold is a moduli space of sheaves on X satisfying a technical condition.

References in corpus (10)

• Semiorthogonal decomposition for algebraic varieties
• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• New derived symmetries of some hyperkähler varieties
• The Mukai pairing, I: the Hochschild structure
• The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface
• Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles
• Cusps of the Kähler moduli space and stability conditions on K3 surfaces
• Caldararu's conjecture and Tsygan's formality
• Cohomological descent theory for a morphism of stacks and for equivariant derived categories
• Ergodic complex structures on hyperkahler manifolds

Cited by in corpus (9)

• New derived symmetries of some hyperkähler varieties
• The K3 category of a cubic fourfold
• Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences
• Derived autoequivalences of generalised Kummer varieties
• Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme
• Universal functors on symmetric quotient stacks of Abelian varieties
• Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles
• Rigid hyperholomorphic sheaves remain rigid along twistor deformations of the underlying hyparkahler manifold
• Hochschild cohomology and deformations of $\mathbb{P}$-functors