Equivariant Riemann-Roch for \(G\)-quasi-projective varieties. I
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.
scientific article; zbMATH DE number 1309207