Hilbert schemes and symmetric products: a dictionary

Qin, Zhenbo; Wang, Weiqiang

doi:10.1090/conm/310/05407

Cited by 21 publications

(30 citation statements)

References 19 publications

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“…Other applications of the abstract generating series formula (37) in the framework of orbifold cohomology and, resp., localized K-theory are explained in Section 3.1. In this way, we reprove and generalize some results from [35] and, resp., [41]. Moreover, in Section 4, we give another application of (37) to canonical constructible functions and orbifold-type Chern classes of symmetric products, reproving some results of Ohmoto [34].…”

Section: ) Hmentioning

confidence: 68%

“…This will make the object of future work by the authors. Results of this type for Chern classes already appeared in [34], while for degree versions see, e.g., [35,41,42,44].…”

Section: 3mentioning

confidence: 99%

“…The corresponding orbifold index theorem was developed in [22], by using for the first time the G-equivariant cohomology of the inertia space IX (as in (3)) of a smooth G-space X, described in terms of differential forms. The corresponding restriction and induction functors were also studied in this classical context in [35,44].…”

Section: Delocalized Equivariant Theoriesmentioning

confidence: 99%

“…(3) Grothendieck group of coherent sheaves K 0 (Coh(X )) ⊗ R with R-coefficients. Other possible choice would be: usual R-homology in even degrees H ev (X) ⊗ R. Since in this section we only need functoriality with respect to isomorphisms, we could also work with cohomological theories, such as the even degree (compactly supported) R-cohomology H ev (c) (X) ⊗ R or the Grothendieck group of algebraic vector bundles K 0 (X) ⊗ R with R-coefficients, as used in [35,44]. In this case, the corresponding covariant transformation g * , as used in this paper, is given by (g * ) −1 , the inverse of the induced pullback under g. If X is smooth, this fits with the folowing Poincaré duality isomorphisms:…”

Section: Delocalized Equivariant Theoriesmentioning

confidence: 99%

“…Here we work with H(X) := H ev (X) ⊗ Q, the (even degree) rational cohomology functor. For X smooth, our notion of H G (X) corresponds to the even degree orbifold cohomology H 2 * orb (X/G), as used for example in [35]. For Z a quasi-projective complex variety, and for a given γ ∈ H(Z), let b r := γ, for all r ≥ 1.…”

Section: Generating Series For Symmetric Group Actions On External Prmentioning

confidence: 99%

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Equivariant characteristic classes of external and symmetric products of varieties

Maxim

Schürmann

2017

Geom. Topol.

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Abstract. We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasi-projective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel-Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.

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Section: ) Hmentioning

confidence: 68%

“…This will make the object of future work by the authors. Results of this type for Chern classes already appeared in [34], while for degree versions see, e.g., [35,41,42,44].…”

Section: 3mentioning

confidence: 99%

Section: Delocalized Equivariant Theoriesmentioning

confidence: 99%

Section: Delocalized Equivariant Theoriesmentioning

confidence: 99%

Section: Generating Series For Symmetric Group Actions On External Prmentioning

confidence: 99%

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Equivariant characteristic classes of external and symmetric products of varieties

Maxim

Schürmann

2017

Geom. Topol.

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Discrete torsion, symmetric products and the Hubert scheme

Kaufmann

2004

Frobenius Manifolds

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IntroductionRecently the understanding of the cohomology of the Hilbert scheme of points on K3 surfaces has been greatly improved by Lehn and Sorger [18]. Their approach uses the connection of the Hilbert scheme to the orbifolds given by the symmetric products of these surfaces. We introduced a general theory replacing cohomology algebras or more generally Frobenius algebras in a setting of global quotients by finite groups [14]. This is our theory of group Frobenius algebras, which are group graded non-commutative algebras whose non-commutativity is controlled by a group action. The action and the grading turn these algebras into modules over the Drinfel'd double of the group ring. The appearance of the Drinfel'd double is natural from the orbifold point of view (see also [17]) and can be translated into the fact that the algebra is a G-graded G-module algebra in the following sense: the G action acts by conjugation on the grading while the algebra structure is compatible with the grading with respect to left multiplication (cf. [16,20]).In the special case of the symmetric group, we recently proved existence and uniqueness for the structures of symmetric group Frobenius algebras based on a given Frobenius algebra [15], providing explicit formulas for the multiplication in the algebra.This uniqueness has to be understood up to the action of two groups of symmetries on group Frobenius algebras called discrete torsion and super-twisting [16]. The set of G-Frobenius algebras is acted upon by both of these groups. This action only changes some defining structures of a Frobenius algebra in a projective manner while keeping others fixed.Applying this result to the global orbifold cohomology of a symmetric product, where there is a canonical choice for the discrete torsion and super-twists, we obtain its uniqueness.Our latest results on this topic [16] explain the origin of these discrete degrees of freedom. In the special case of the Hilbert scheme as a resolution of a symmetric product the choice of sign for the metric specifies a discrete torsion cocycle that in turn changes the multiplication by a much discussed sign.Assembling our results which we review we obtain: * Partially supported by NSF grant #0070681. 1 2 RALPH M. KAUFMANNTheorem.The cohomology of Hilb [n] , the Hilbert scheme of n-points for a K3 surface, is the S n invariant part of the S n -Frobenius Algebra associated to the symmetric product of the cohomology of the surface twisted by a discrete torsion. Or in other words the unique S n -Frobenius Algebra structure for the extended global orbifold cohomology twisted by the specific discrete torsion which is uniquely determined by the map of [18]. In general, the sequence of spaces Hilb [n] gives rise to the twisted second quantization of the underlying surface on the cohomological (motivic) level.Here the term associated refers to the uniqueness result of [15] stated above. This result follows from a series of considerations which we will review. The logic is roughly as follows:The theoretical backg...

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