On Rigid Hirzebruch Genera

Abstract

The classical multiplicative (Hirzebruch) genera of manifolds have the wonderful property which is called rigidity. Rigidity of a genus h means that if a compact connected Lie group G acts on a manifold X, then the equivariant genus h G (X) is independent on G, i.e. h G (X) = h(X).In this paper we are considering the rigidity problem for stably complex manifolds. In particular, we are proving that a genus is rigid if and only if it is a generalized Todd genus.2000 MSC: 55N22, 57R77.

Hirzebruch Functional Equations and Krichever Complex Genera

“…As it noted above, a and b are symmetric, and their transposition corresponds to the replacement λ → −λ. The case λ = 0 corresponds to (2). In the case λ = 2 we have that f sing is a particular case of (2) for α = −2, β = 0.…”

“…It is well known that the function f Td (t) := e αt − e βt αe βt − βe αt (2) defines a n-multiplicative Hirzebruch genus for each n. It is called two-parametric Todd genus and naturally appears in many branches of mathematics, for example, in enumerative combinatorics and toric topology. We will see below (see lemma 2.7) that it satisfies (1).…”

Hirzebruch Class and Bia Lynicki-Birula Decomposition

“…. By [Mus11] the class p * td T y (X) is concentrated in degree zero and it is equal to χ y (X). Thus we have…”

Equivariant Hirzebruch class for singular varieties

“…We apply the localization formula to the equivariant Hirzebruch class td T y (X). By rigidity (see [Mus11]) we obtain just the class of gradation zero, the χ y -genus Here the Euler class eu(p) = w i is the product of weights w i ∈ Hom(T, C * ) = H 2 T (pt) appearing in the tangent representation T p X. Note that the χ y -genus can be written in the form χ y (X) = p∈X T 1 eu(p) td T (X) · ch T (Λ y (T * X)) |p .…”