Equivariant Chern Classes and Localization Theorem

Weber, Andrzej

doi:10.5427/jsing.2012.5k

Cited by 24 publications

(28 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The idea to express such sums as iterated residues at infinity came from Bérczi and Szenes [2] who used it as a computational tool for Thom polynomials. This idea was implemented also by other authors, see for example Fehér and Rimányi [5,13] and Weber [14]. One can find a different approach to the residue formulas in papers of Jeffrey and Kirwan [9,10], who consider symplectic manifolds with a hamiltonian action of a compact Lie group (not necessarily abelian).…”

Section: Introductionmentioning

confidence: 99%

Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity

Zielenkiewicz

2014

Open Mathematics

View full text Add to dashboard Cite

Using the Berline-Vergne integration formula for equivariant cohomology for torus actions, we prove that integrals over Grassmannians (classical, Lagrangian or orthogonal ones) of characteristic classes of the tautological bundle can be expressed as iterated residues at infinity of some holomorphic functions of several variables. The results obtained for these cases can be expressed as special cases of one formula involving the Weyl group action on the characters of the natural representation of the torus. MSC:14M15

show abstract

Section: Introductionmentioning

confidence: 99%

Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity

Zielenkiewicz

2014

Open Mathematics

View full text Add to dashboard Cite

show abstract

“…The local equivariant version of Chern-Schwartz-MacPherson classes is studied in [Web12]. Here are the formulas for Du Val singularities.…”

Section: Hirzebruch Classes Of Du Val Singularitiesmentioning

confidence: 99%

“…Positivity of local equivariant Chern-Schwartz-MacPherson classes for Schubert varieties was noticed in [Web12] by computer experiments. So far there is no proof.…”

Section: Introductionmentioning

confidence: 98%

Equivariant Hirzebruch Classes and Molien Series of Quotient Singularities

Donten-Bury

Weber

2017

Transformation Groups

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…This phenomenon is easy to explain: resolving the singularity of X we construct new T-varieties for which the weights of the tangent spaces are combinations of the original weights. The method of [Web12] allows to compute the Hirzebruch class of the codimension one Schubert variety in the classical Grassmannian Gr n (C 2n ). Locally this Schubert variety is equal to the determinantal variety consisting of singular quadratic matrices.…”

Section: Schubert Varieties and Cellsmentioning

confidence: 99%

Equivariant Hirzebruch class for singular varieties

Weber

2016

Sel. Math. New Ser.

Self Cite

View full text Add to dashboard Cite

Abstract. We develop an equivariant version of the Hirzebruch class for singular varieties. When the group acting is a torus we apply Localization Theorem of Atiyah-Bott and Berline-Vergne. The localized Hirzebruch class is an invariant of a singularity germ. The singularities of toric varieties and Schubert varieties are of special interest. We prove certain positivity results for simplicial toric varieties. The positivity for Schubert varieties is illustrated by many examples, but it remains mysterious.The main goal of the paper is to show how a theory of global invariants can be applied to study local objects equipped with an action of a large group of symmetries. The theory of global invariants we are going to discuss is the theory of characteristic classes, more precisely the Hirzebruch class and χ y -genus. By [Yok94, BSY10] the Hirzebruch class admits a generalization for singular varieties. Let X be an algebraic variety in a compact complex algebraic manifold M . Suppose that X is preserved by a torus T acting on M . For simplicity assume that the fixed point set M T is discrete. Then by Localization Theorem of Atiyah-Bott and Berline-Vergne the χ y -genus of X can be written as a sum of contributions coming from fixed points. The contribution of a fixed point p ∈ M T is equal to the equivariant Hirzebruch class restricted to that point and divided by the Euler class of the tangent space at pThe local contributions to χ y -genus are fairly computable and they are expressed by polynomials in characters of the torus. We will describe all the necessary components of the described construction, give various examples and we will discuss positivity property of the localized Hirzebruch class. A relation with the Bia lynicki-Birula decomposition [BB73] will be given in a subsequent paper [Web14]. The reader can find useful to look at the article [Web13], where an elementary and self-contained introduction to equivariant characteristic classes is given.

show abstract

Equivariant Chern Classes and Localization Theorem

Cited by 24 publications

References 35 publications

Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity

Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity

Equivariant Hirzebruch Classes and Molien Series of Quotient Singularities

Equivariant Hirzebruch class for singular varieties

Contact Info

Product

Resources

About