2,565 research outputs found
Equivariant Hirzebruch class for singular varieties
The Hirzebruch 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 -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 . We ask
when the cohomology of the geometric boundary in a smooth
compactification is pure with respect to the mixed Hodge structure. Knowing
the dimension of singularity locus of some singular compactification we give a
bound for above which the cohomology 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 . We also describe the weight filtration in .Comment: 14 pages. Final version to appear in Central European Journal of
Mathematic
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