Rigidity of Differential Operators and Chern Numbers of Singular Varieties

A differential operator D commuting with an S-action is said to be rigid if the non-constant Fourier coefficients of kerD 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 “singular” Chern numbers. 1. Rigidity of elliptic differential operators Let D : Γ(E) → Γ(F ) be an elliptic operator maping sections of a vectorbundle E to sections of F . If D commutes with a T = S1 action, then kerD and coker D are finite-dimensional S1-modules. We define the charactervalued index IndT (D) = kerD − coker D ∈ R(T ) For example, if D = d + d : Ωeven → Ωodd is the de Rham operator on a smooth manifold X with a T action, then by Hodge theory and homotopy invariance of de Rham cohomology, IndT (D) is a trivial virtual T -module of rank equal to the Euler characteristic of X. In general, if IndT (D) is a trivial T -module, we say that D is rigid. In the case where D is the de Rham operator, both kerD and coker D are independently trivial T modules. However, more interesting cases exist where D is rigid, but both kerD and coker D are nontrivial T -modules. For example, if X is a spin manifold and D : Γ(∆+) → Γ(∆) is the Dirac operator, then D is rigid. It is instructive to sketch the proof of this fact, which is due to Atiyah and Hirzebruch [3]: For simplicity, assume that T acts on X with isolated fixed points {p}, and that the action lifts to the spin bundles ∆. At each fixed point p, TpX decomposes into a sum of one-dimensional complex representations of T with weights m1(p), ...,mn(p), where 2n = dimX. If we view IndT (D) as The author supported by NSF Post-doctoral Fellowship.