Equivariant Riemann-Roch for \(G\)-quasi-projective varieties. I
type: Item
instance of: scholarly article
title: Equivariant Riemann-Roch for \(G\)-quasi-projective varieties. I
zbMATH Open document ID: 0926.19002
DOI: 10.1023/A:1007766614969
author: Roy Joshua
published in: \(K\)-Theory
publication date: 1999-11-10T00:00:00Z
review text: The main result of the paper is the following equivariant Riemann-Roch type theorem. Let \(G\) be a complex linear algebraic group and \(f:X \rightarrow Y\) a \(G\)-equivariant proper map between two \(G\)-quasi-projective complex \(G\)-varieties. Then the natural transformation from the Grothendieck group of equivariant coherent modules to equivariant Atiyah-Segal \(K\)-homology, the natural transformation from equivariant Atiyah-Segal \(K\)-homology to equivariant topological \(K\)-homology, and the Chern-Todd character from equivariant topological \(K\)-homology to equivariant homology (with rational coefficients) commute with the corresponding push-forward homomorphisms \(f_*\). The main part of the paper deals with the construction of the various equivariant (\(K\)-)homology theories and natural transformations. For instance, the definition of equivariant Atiyah-Segal \(K\)-homology invokes the generalized equivariant homology theories as developed by \textit{L. G. Lewis jun., J. P. May} and \textit{M. Steinberger} [``Equivariant stable homotopy theory'', Lect. Notes Math. 1213 (1986; Zbl 0611.55001)], and the definition of the natural transformations is based on Poincaré-Lefschetz duality isomorphism (established in the paper) and the deformation to the normal cone. The main application of the above Riemann-Roch theorem concerns convolution in \(K\)-theory as considered by Lusztig, Ginzburg and others. The last section and the appendix discuss the following techniques used in the paper: reduction to the action by a maximal torus, existence of generic slices for torus actions; equivariant Fredholm complexes.
reviewed by: Bernhard Köck
Mathematics Subject Classification ID: 19L10
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 19L47
zbMATH DE Number: 1309207
zbMATH Keywords: equivariant Riemann-Roch
zbMATH Keywords: equivariant \(K\)-theory
zbMATH Keywords: Atiyah-Segal \(K\)-theory
zbMATH Keywords: Fredholm complexes
zbMATH Keywords: topological \(K\)-theory
zbMATH Keywords: equivariant cohomology
zbMATH Keywords: Chern character
zbMATH Keywords: Todd homomorphism
zbMATH Keywords: Poincaré-Lefschetz duality
zbMATH Keywords: convolution
MaRDI profile type: Publication
full work available at URL: a:1007766614969
OpenAlex ID: W1990764239
description: scientific article; zbMATH DE number 1309207