Equivariant Hirzebruch class for singular varieties

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.Comment: 38 pages, 3 figure

Purity of boundaries of open complex varieties

We study the boundary of an open smooth complex algebraic variety $U$. We ask when the cohomology of the geometric boundary $Z=X\setminus U$ in a smooth compactification $X$ is pure with respect to the mixed Hodge structure. Knowing the dimension of singularity locus of some singular compactification we give a bound for $k$ above which the cohomology $H^k(Z)$ is pure. The main ingredient of the proof is purity of the intersection cohomology sheaf.Comment: 12 page

Weights in the cohomology of toric varieties

We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complex $IH_T^*(X)\otimes H^*(T)$. We also describe the weight filtration in $IH^*(X)$.Comment: 14 pages. Final version to appear in Central European Journal of Mathematic