Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

Cappell, Sylvain E.; Maxim, Laurenţiu; Schürmann, Jörg; Shaneson, Julius L.

doi:10.1002/cpa.21427

Cited by 24 publications

(44 citation statements)

References 39 publications

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“…After that we review [CMSS12], adapting the notation to our purposes. These strengthened versions of Lefschetz-RiemannRoch allows us to compute the full Hirzebruch class.…”

Section: Lefschetz-riemann-rochmentioning

confidence: 99%

“…These strengthened versions of Lefschetz-RiemannRoch allows us to compute the full Hirzebruch class. The proof of Theorem 1 is just checking that the methods of [BFQ79] and [CMSS12] apply in the equivariant case and interpreting the result for Y = C n .…”

Section: Lefschetz-riemann-rochmentioning

confidence: 99%

“…We remark that the proof given in [CMSS12] uses K-theory of mixed Hodge modules of M. Saito [Sai00].…”

Section: Lrr For the Hirzebruch Classmentioning

confidence: 99%

“…The proof is based on a version of the Lefschetz-Riemann-Roch theorem proved in [CMSS12]. It is given in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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Equivariant Hirzebruch Classes and Molien Series of Quotient Singularities

Donten-Bury

Weber

2017

Transformation Groups

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Abstract. We study properties of the Hirzebruch class of quotient singularities C n /G, where G is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of G under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of G. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4-dimensional symplectic quotient singularities.

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“…After that we review [CMSS12], adapting the notation to our purposes. These strengthened versions of Lefschetz-RiemannRoch allows us to compute the full Hirzebruch class.…”

Section: Lefschetz-riemann-rochmentioning

confidence: 99%

Section: Lefschetz-riemann-rochmentioning

confidence: 99%

See 2 more Smart Citations

Equivariant Hirzebruch Classes and Molien Series of Quotient Singularities

Donten-Bury

Weber

2017

Transformation Groups

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“…We let cat G (X) be a category of G-equivariant objects on X in the underlying category cat(X) (e.g., see [11,28]), which in this paper refers to one of the following examples: coherent sheaves Coh(X ), algebraically constructible sheaves of complex vector spaces Constr (X ), and (algebraic) mixed Hodge modules MHM (X) on X. We denote by K 0 (cat G (X)) the corresponding Grothendieck groups of these Q-linear abelian categories.…”

Section: Introductionmentioning

confidence: 99%

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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Characteristic Classes of Singular Toric Varieties

Maxim

Schürmann

2014

Comm Pure Appl Math

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Abstract. In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As important special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus-orbit decomposition and the motivic properties of the motivic Chern and resp. homology Hirzebruch classes to express the latter in terms of dualizing sheaves and resp. the (dual) Todd classes of closures of orbits. This method even applies to torus-invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes, yielding generalized Pick-type formulae. Secondly, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz-Riemann-Roch theorem of Edidin-Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd and mock L-classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.

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Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

Cited by 24 publications

References 39 publications

Equivariant Hirzebruch Classes and Molien Series of Quotient Singularities

Equivariant Hirzebruch Classes and Molien Series of Quotient Singularities

Equivariant characteristic classes of external and symmetric products of varieties

Characteristic Classes of Singular Toric Varieties

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