Equivariant Hirzebruch class for quadratic cones via degenerations

Abstract

The difference between the Hirzebruch class of the generic fiber and the Hirzebruch class of the special fiber is measured by the appropriate version of Milnor class, studied in [CMSS] for hypersurfaces and in [MSS] the general case. The same phenomenon happens for the equivariant Hirzebruch class developed in [We3], compare also with [Oh, Sec.4] for the equivariant Hirzebruch class in the context of quotient stacks. We fix our attention on the varieties with torus action. If we are interested in local invariants of singularities, we study the localization of the equivariant Hirzebruch class tdy (Y → X) at a fixed point. The bottom degree of the Hirzebruch class is the equivariant fundamental class, also called the multi-degree of the variety. It does not change in the deformation class. For example, let Qn ⊂ Cn be the cone over a quadric in Pn−1, in other words, Qn in some coordinates is ∗Supported by NCN grant 2013/08/A/ST1/00804