Symmetric groups and the cup product on the cohomology of Hilbert schemes

By M. Lehn, Christoph Sorger. 2000

106 citations recorded by the source corpus.

Let C(Sn) be the Z-module of integer valued class functions on the symmetric group Sn. We introduce a graded version of the con- volution product on C(Sn) and show that there is a degree preserving ring isomorphism C(Sn) ! H ∗ (Hilb n (A 2); Z) to the cohomology of the Hilbert scheme of points in the complex affine plane.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/848be4e1c58ec81aa73dc952adfaa32ea8a4b659

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In Geometry and Topology

1. R. Pandharipande
2. A. Okounkov
3. T. Graber
4. V. Ginzburg
5. Jim Bryan
6. Barbara Fantechi
7. Y. Ruan
8. Pavel Etingof

In Mathematical Physics

1. R. Pandharipande
2. A. Okounkov
3. T. Graber
4. V. Ginzburg
5. Y. Ruan
6. Jim Bryan
7. Pavel Etingof
8. Barbara Fantechi

Citations in the corpus

• Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism (2000, result): “the Hilbert scheme of n points on the plane, and the corresponding result on the algebra structure of H • Hilbn(C2) � has been recently proved in [ LSo ] and [Va] (independently).”
• The local Donaldson–Thomas theory of curves (2005, background): “see Lehn [14], Lehn–Sorger [15], Li et al [20] and Okounkov–Pandharipande [31].”
• The cup product of the Hilbert scheme for K3 surfaces (2000, background): “A G ] 1 8 D ec of generalised n-tuples of any smooth projective surface X with trivial canonical bundle.”
• Orbifold cohomology for global quotients (2001, background): “tive in Theorem 3.10 can be replaced by Scompact, since every compact complex surface with torsion canonical class can be deformed to a projective one. In fact, Theorem 3.10 is also true for S= A2 by [LS1]. If Sis a not necessarily compact complex surface which is also an abelian group (e.g. an abel”
• Orbifolds and Stringy Topology (2007, background): “Special cases of this conjecture were proved by Lehn and Sorger [92] for symmetric products of C2, by Lehn and Sorger [93], Fantechi and Göttsche [52], Uribe [153] for symmetric products of K3 or T 4, by Li, Qin, and Wang [100] for symmetric products of the cotangent bundle T ∗ of a Riemann surface,”
• Cohomology ring of crepant resolutions of orbifolds (2001, methodology): “Based on an important observation by Frenkel and Wang [FW], Lehn-Sorger [ LS1 ] determined the cohomology ring of (C2)[n].”
• The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^2$ (2009, background): “ult extends to the Hilbert scheme of an arbitrary smooth quasiprojective surface. One has to replace H by a Heisenberg-Clifford algebra modeled over the homology lattice H∗(S,Z). See also [LQW], [L1], [LS], [V], etc, for other works in this direction. In the well-documented analogy between the Hilber”
• Poisson deformations of symplectic quotient singularities (2002, methodology): “ in our notation, to a linear bijection γ∗: HBM d−•(X,Q) −→∼ grq(ZG), where d= dimR X. The multiplication structure in the cohomology was first computed independently by Vasserot [Vas] and Lehn-Sorger [LS] (cf. also [LQW]) in the case when Xis the Hilbert scheme of npoints on C2. At the same time, Va”
• The cup product of Hilbert schemes for K3 surfaces (2003, background): “In this sense this paper is related to [16] as our paper [15] to the last section of [14].”
• The Crepant Resolution Conjecture (2006, methodology): “onjecture for the nonequivariant theory it suffices to compare the ring structure on the ordinary cohomology of the Hilbert scheme with the orbifold cohomology of the symmetric product. This is done in [28, 20] (see also [16, 26]). The nontrivial, fully equivariant genus 0 Gromov-Witten theory of Y is”