Ideals of the cohomology rings of Hilbert schemes and their applications

Li, Wei Ping; Qin, Zhenbo; Wang, Weiqiang

doi:10.1090/s0002-9947-03-03422-6

Cited by 15 publications

(14 citation statements)

References 18 publications

(47 reference statements)

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“…This theorem has been proved earlier when n = 2, 3 [ELQ, LQ], when K X is trivial [FG, LS], and when X is a smooth toric surface [Che]. We also refer to [LQW4,MO,OP,QW,Zho] for discussions when X is quasi-projective.…”

Section: Introductionmentioning

confidence: 78%

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The cohomological crepant resolution conjecture for the Hilbert–Chow morphisms

Li¹,

Qin²

2016

J. Differential Geom.

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In this paper, we prove that Ruan's Cohomological Crepant Resolution Conjecture holds for the Hilbert-Chow morphisms. There are two main ideas in the proof. The first one is to use the representation theoretic approach proposed in [QW] which involves vertex operator techniques. The second is to prove certain universality structures about the 3-pointed genus-0 extremal Gromov-Witten invariants of the Hilbert schemes by using the indexing techniques from [LiJ], the product formula from [Beh2] and the co-section localization from [KL1, KL2, LL]. We then reduce Ruan's Conjecture from the case of an arbitrary surface to the case of smooth projective toric surfaces which has already been proved in [Che]. Hilbert schemes of points on surfacesLet X be a smooth projective complex surface with the canonical class K X and the Euler class e X , and X [n] be the Hilbert scheme of points in X. An element in X [n] is represented by a length-n 0-dimensional closed subscheme ξ of X. It is well known that X [n] is smooth. For a subset Y ⊂ X, define M n (Y ) = {ξ ∈ X [n] | Supp(ξ) = {x} for some x ∈ Y }.

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Section: Introductionmentioning

confidence: 78%

“…Theorem 2.9 in [LQW4] expresses a Heisenberg monomial class in terms of a polynomial of the classes G k (γ, n). The following lemma is a special case.…”

Section: Hilbert Schemes Of Points On Surfacesmentioning

confidence: 99%

The cohomological crepant resolution conjecture for the Hilbert–Chow morphisms

Li¹,

Qin²

2016

J. Differential Geom.

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“…not the identity) in the seed cohomology ring of M . If we divide the symmetric orbifold cohomology ring by that ideal, the quotient ring is isomorphic to the cohomology ring of the Hilbert scheme of the complex plane [51]. Thus, the latter captures part of the ring structure for any manifold M .…”

Section: Jhep10(2020)201mentioning

confidence: 99%

The topological symmetric orbifold

Troost

2020

J. High Energ. Phys.

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We analyze topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by Hurwitz numbers. This proves a conjecture in the physics literature on extremal correlators. Moreover, it allows to leverage results on the combinatorics of the symmetric group to compute more structure constants explicitly. We recall that the full orbifold chiral ring is given by a symmetric orbifold Frobenius algebra. This construction enables the computation of topological genus zero and genus one correlators, and to prove the vanishing of higher genus contributions. The efficient description of all topological correlators sets the stage for a proof of a topological AdS/CFT correspondence. Indeed, we propose a concrete mathematical incarnation of the proof, relating Gromow-Witten theory in the bulk to the cohomology of the Hilbert scheme on the boundary.

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“…In the past few years, the theory of vertex operators (cf. [FLM]) has found remarkable applications in the study of the Hilbert schemes X [n] of points on a surface X (see [Na2,Le,LQW1,LQW2,Ru,Vas,Wa1] and the references therein). To a large extent, the construction of Heisenberg algebra by Nakajima [Na1] (also cf.…”

Section: Introductionmentioning

confidence: 99%

Hilbert schemes of points on the minimal resolution and soliton equations

Qin¹,

Wang²

2007

Lie Algebras, Vertex Operator Algebras and Their Applications

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Abstract. The equivariant and ordinary cohomology rings of Hilbert schemes of points on the minimal resolution C 2 //Γ for cyclic Γ are studied using vertex operator technique, and connections between these rings and the class algebras of wreath products are explicitly established. We further show that certain generating functions of equivariant intersection numbers on the Hilbert schemes and related moduli spaces of sheaves on C 2 //Γ are τ -functions of 2-Toda hierarchies.

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Ideals of the cohomology rings of Hilbert schemes and their applications

Cited by 15 publications

References 18 publications

The cohomological crepant resolution conjecture for the Hilbert–Chow morphisms

The cohomological crepant resolution conjecture for the Hilbert–Chow morphisms

The topological symmetric orbifold

Hilbert schemes of points on the minimal resolution and soliton equations

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