Equivariant genera of complex algebraic varieties

Sylvain Cappell, Laurentiu Maxim, Julius L. Shaneson

Research output: Contribution to journalArticle

Abstract

Equivariant Hirzebruch genera of a variety X acted upon by a finite group of algebraic automorphisms are defined by combining the group action with the information encoded by the Hodge filtration in cohomology. For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. While for a projective manifold equivariant genera can by computed by the Atiyah-Singer holomorphic Lefschetz theorem, we derive a Atiyah-Meyer-type formula for such genera even when X is not necessarily smooth or compact, but just fibers equivariantly (in the complex topology) over an algebraic manifold. These results apply to computing Hirzebruch invariants of orbit spaces. We also obtain results comparing equivariant genera of the range and domain of an equivariant morphism in terms of its singularities.

Original languageEnglish (US)
Pages (from-to)2313-2337
Number of pages25
JournalInternational Mathematics Research Notices
Volume2009
Issue number11
DOIs
StatePublished - Feb 2009

Fingerprint

Algebraic Variety
Equivariant
Michael Francis Atiyah
Genus
Orbit Space
Smooth Manifold
Monodromy
Group Action
Morphism
Filtration
Cohomology
Automorphisms
Finite Group
Signature
Fiber
Singularity
Topology
Invariant
Computing
Theorem

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Equivariant genera of complex algebraic varieties. / Cappell, Sylvain; Maxim, Laurentiu; Shaneson, Julius L.

In: International Mathematics Research Notices, Vol. 2009, No. 11, 02.2009, p. 2313-2337.

Research output: Contribution to journalArticle

Cappell, Sylvain ; Maxim, Laurentiu ; Shaneson, Julius L. / Equivariant genera of complex algebraic varieties. In: International Mathematics Research Notices. 2009 ; Vol. 2009, No. 11. pp. 2313-2337.
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