Rigid hyperholomorphic sheaves remain rigid along twistor deformations of the underlying hyparkahler manifold

2017/06/28 by Eyal Markman, Sukhendu Mehrotra, Misha Verbitsky
Mathematics · #math.AG

paper · pdf · doi:10.1007/s40879-019-00323-w

published as Eur. J. Math. 5 (2019), no. 3, 964-1012 · 39 pages

arxiv created 2017/06/28 · arxiv updated 2021/09/20

Abstract

Let S be a K3 surface and M a smooth and projective 2n-dimensional moduli space of stable coherent sheaves on S. Over M x M there exists a rank 2n-2 reflexive hyperholomorphic sheaf E_M, whose fiber over a non-diagonal point (F,G) is Ext¹(F,G). The sheaf E_M can be deformed along some twistor path to a sheaf E_X over the cartesian square of every Kahler manifold X deformation equivalent to M. We prove that E_X is infinitesimally rigid, and the isomorphism class of the Azumaya algebra End(E_X) is independent of the twistor path chosen. This verifies conjectures in arXiv:1310.5782 and arXiv:1507.03108 on non-commutative deformations of K3 surfaces and renders the results of these two papers unconditional.

Citations

• A global Torelli theorem for rigid hyperholomorphic sheaves
• Integral Transforms and Deformations of K3 Surfaces