2013/08/04 by Andrzej Weber, Weber, Andrzej
Mathematics · #14C17 #14E15 #14M15 #14M25 #19L10 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.AT #math.RT #msc:14C17 #msc:14E15 #msc:14M15 #msc:14M25 #msc:19L10
paper · pdf · doi:10.48550/arxiv.1308.0788
38 pages, 3 figures
arxiv created 2015/08/08 · arxiv updated 2015/08/11
The Hirzebruch tdy(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 χy-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.