  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.
