### Abstract

Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet, Schüurmann, and Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson $L$-classes). In this paper we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds.

Original language | English (US) |
---|---|

Pages (from-to) | 1722-1769 |

Number of pages | 48 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 65 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2012 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*65*(12), 1722-1769. https://doi.org/10.1002/cpa.21427

**Equivariant Characteristic Classes of Singular Complex Algebraic Varieties.** / Cappell, Sylvain; Maxim, Laurentiu G.; Schürmann, Jörg; Shaneson, Julius L.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 65, no. 12, pp. 1722-1769. https://doi.org/10.1002/cpa.21427

}

TY - JOUR

T1 - Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

AU - Cappell, Sylvain

AU - Maxim, Laurentiu G.

AU - Schürmann, Jörg

AU - Shaneson, Julius L.

PY - 2012/12

Y1 - 2012/12

N2 - Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet, Schüurmann, and Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson $L$-classes). In this paper we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds.

AB - Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet, Schüurmann, and Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson $L$-classes). In this paper we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds.

UR - http://www.scopus.com/inward/record.url?scp=84867626077&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867626077&partnerID=8YFLogxK

U2 - 10.1002/cpa.21427

DO - 10.1002/cpa.21427

M3 - Article

VL - 65

SP - 1722

EP - 1769

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 12

ER -