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Published online by Cambridge University Press: 16 July 2025
ABSTRACT. A differential operator D commuting with an S1-action is said to be rigid if the nonconstant Fourier coefficients of ker D and coker D are the same. Somewhat surprisingly, the study of rigid differential operators turns out to be closely related to the problem of defining Chern numbers on singular varieties. This relationship comes into play when we make use of the rigidity properties of the complex elliptic genus—essentially an infinite-dimensional analogue of a Dirac operator. This paper is a survey of rigidity theorems related to the elliptic genus, and their applications to the construction of “singular” Chern numbers.
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