TL;DR
This paper investigates the rigidity property of Hirzebruch genera on complex manifolds, establishing that a genus is rigid precisely when it is a generalized Todd genus, thus characterizing rigidity.
Contribution
It proves that the only rigid Hirzebruch genera on complex manifolds are the generalized Todd genera, providing a complete characterization.
Findings
Rigidity of a genus is equivalent to being a generalized Todd genus.
The paper characterizes all rigid Hirzebruch genera on complex manifolds.
Rigidity property is exclusive to generalized Todd 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 complex manifolds. In particular, we are proving that a genus is rigid if and only if it is a generalized Todd genus.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Advanced Differential Equations and Dynamical Systems · Algebraic Geometry and Number Theory