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

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 × M there exists a rank 2n − 2 reflexive hyperholomorphic sheaf EM , whose fiber over a non-diagonal point (F1, F2) is Ext 1 S (F1, F2). The sheaf EM can be deformed along some twistor path to a sheaf EX over the cartesian square X × X of every Kähler manifold X deformation equivalent to M . We prove that EX is infinitesimally rigid, and the isomorphism class of the Azumaya algebra E nd(EX ) is independent of the twistor path chosen. This verifies conjectures in [MM1, MM2] and renders the results of these two papers unconditional.

The Chow ring of hyperkähler varieties of $$K3^{[2]}$$-type via Lefschetz actions

“…A priori, M might depend on the chosen deformation, but as it turns out, κ 2 (M) does not. As in [11] we denote by M the moduli space of isomorphism classes of marked hyperkähler manifolds.…”

“…We stress that Markman requires γ to map the irreducible components of the possibly reducible curve C isomorphically onto twistor lines in the moduli space M . Now, by [11,Theorem 1.11]…”

“…Moreover, we have l = * (L) = 5 6 c 2 (T X ), where : X → X ×X is the diagonal embedding and c 2 (T X ) is the second Chern class of the tangent bundle, satisfying c 2 (T X ) = 5g 2 − 8c. Hence, the tautological subring R [11], the Azumaya algebra E nd(M) is universally defined over the moduli space of Fano varieties. Hence, by [4, Theorem 1.10], the explicit lift of B from (2.4), which clearly is universally defined as well, agrees with Markman's lift (2.1).…”

The Chow ring of hyperkähler varieties of $K3^{[2]}$-type via Lefschetz actions

“…As in [MMV19] we denote by M Λ the moduli space of isomorphism classes of marked hyperkähler manifolds.…”

“…Let M be Markman's sheaf on F (Y ) × F (Y ) as in Theorem 2.2. By [Mar16, Corollary 1.6] and [MMV19], the Azumaya algebra End(M ) is universally defined over the moduli space of Fano varieties. Hence, by [FLVS19, Theorem 1.10], the explicit lift of B from (2.7), which clearly is universally defined as well, agrees with Markman's lift (2.2).…”