Equivariant characteristic classes of singular varieties

Andrzej Weber
Monday, July 22, 2013 - from 16:00 to 16:25
The Hirzebruch $td_y(X)$ class of a complex manifold $X$ is the formal combination of Chern characters of the sheaves of differential forms multiplied by the Todd class $$td_y(X)=td(X)\sum_{i=0}^{\dim X}ch(\Omega^i_X)y^i\in H^*(X)[y]$$ 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 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 will discuss a relation between the properties of the singularity with this way obtained the local Hirzebruch class. Especially the issue of positivity of coefficients in a certain expansion will be presented. The quotient singularities, toric singularities, the singularities of Schubert varieties are of special interest.