Equivariant elliptic cohomology and rigidity (Q2747076)

One of the basic properties of the (equivariant) elliptic genus \(\Phi_G(M)\) is that it is constant as a function of \(G\), a compact connected group acting on the spin manifold \(M\). Indeed this `rigidity' is one way of defining the elliptic nature of the genus \(\Phi\). The author of the present paper provides an alternative proof exploiting one particular definition of \(S^1\)-equivariant cohomology in terms of coherent holomorphic sheaves over suitable varieties (the definition is due to I. Grojnowski), and the possibility of pushing this forward along an equivariant map \(f:X\to Y\). This is exploited to construct a section for the sheaf \(E^*_{S^1} (X)^{[\pi]}\), when the superscript refers to a certain twisting. The paper is well-written, with some attempt made to explain the technicalities. Part of its interest is to show that a ``geometric'' definition of equivariant elliptic cohomology can be used to establish the important property of rigidity.