Rosu, Ioanid Equivariant elliptic cohomology and rigidity. (English) Zbl 0990.55002 Am. J. Math. 123, No. 4, 647-677 (2001). 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. Reviewer: C.B.Thomas (Cambridge) Cited in 1 ReviewCited in 17 Documents MSC: 55N34 Elliptic cohomology 58J26 Elliptic genera 55N91 Equivariant homology and cohomology in algebraic topology Keywords:oriented equivariant cohomology; elliptic genus PDFBibTeX XMLCite \textit{I. Rosu}, Am. J. Math. 123, No. 4, 647--677 (2001; Zbl 0990.55002) Full Text: DOI arXiv Link Link