2015/07/11 by Eyal Markman, Markman, Eyal, Sukhendu Mehrotra +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coherent sheaf #FOS: Mathematics #Functor #Geometry and complex manifolds #Hilbert scheme #Holomorphic function #Isomorphism (crystallography) #K3 surface #Mathematical analysis #Mathematics #Moduli space #Pure mathematics #Quotient #Sheaf #Vector bundle #math.AG
paper · pdf · doi:10.48550/arxiv.1507.03108
arxiv created 2015/07/11 · openalex publication_date 2015/07/11 · arxiv updated 2015/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.