Equivariant Hirzebruch class for singular varieties

Abstract

The Hirzebruch $td_y(X)$ class of a complex manifold X is a formal combination of Chern characters of the sheaves of differential forms multiplied by the Todd class. The related $\chi_y$-genus admits a generalization for singular complex algebraic varieties. The equivariant version of the Hirzebruch class can be developed as well. The general theory applied in the situation when a torus acts on a singular variety allows to apply powerful tools as the Localization Theorem of Atiyah and Segal for equivariant K-theory and Berline-Vergne formula for equivariant cohomology. We obtain a meaningful invariant of a germ of singularity. When it is made explicit it turns out to be just a polynomial in characters of the torus. We discuss a relation of the properties of a singularity germ with its local Hirzebruch class. The issue of positivity of coefficients in a certain expansion remains mysterious. The quotient singularities, toric singularities, the singularities of Schubert varieties are of special interest.