Speaker:

Andrzej Weber

Monday, July 22, 2013 - from 16:00
to 16:25

Abstract:

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.