Symplectic structures on moduli spaces of sheaves via the Atiyah class

By Alexander Kuznetsov, D. Markushevich. 2007

85 citations recorded by the source corpus.

Abstract It is proven that the composition of the Yoneda coupling with the semiregularity map is a closed 2-form on moduli spaces of sheaves. Two examples are given when this 2-form is symplectic. Both of them are moduli spaces of torsion sheaves on the cubic 4-fold Y . The first example is the Fano scheme of lines in Y . Beauville and Donagi showed that it is symplectic but did not construct an explicit symplectic form on it. We prove that our construction provides a symplectic form. The other example is the moduli space of torsion sheaves which are supported on the hyperplane sections H ∩ Y of Y and are cokernels of the Pfaffian representations of H ∩ Y .

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/67bafda4b385d403ac1cd77a5bc093e20b9758f0

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Hodge theory and derived categories of cubic fourfolds (2012, unknown): “Finally we could instead try to use F ( X t ) [9], the Fano variety of lines on X t , which is also a moduli space of objects in A X t (namely the projection into A X t of the structure sheaves of the lines [27, § 5]).”
Fano varieties of cubic fourfolds containing a plane (2009, background): “As remarked in [28], while Il 6∈ TY , the (stable) sheaf Fl ∈ Coh(Y ) defined by the short exact sequence”
Semiorthogonal decompositions in algebraic geometry (2014, background): “Therefore, any smooth moduli space of objects in AY should be hyperkähler, and the Fano scheme of lines can be realized in this way, see [40].”
Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces (2018, methodology): “ain (−2)-classes. This means that there is no class v∈ He(Ku(W),Z) such that v2 = −2. This will be enough for the application to the Torelli Theorem for cubic fourfold. Let Lbe a line in W. Following [KuMa09], consider the kernel of the evaluation map FL := Ker O⊕4 W ։ IL(1) , which is a torsion-fre”
Stability conditions on Kuznetsov components (2017, background): “, several recent constructions of hyperkähler varieties associated to moduli spaces of sheaves on the cubic fourfold are induced by the projection to the K3 category Ku(X) [KM09] (where moduli spaces naturally come with a holomorphic symplectic structure, due to the fact that Ku(X) is a K3 category)”