Elliptic genera of singular varieties

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The notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced, and the relation between them is studied. The elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties introduced earlier. The orbifold elliptic genus is given in terms of the fixed-point sets of the action. We show that the generating function for the orbifold elliptic $\sum {\rm Ell\sp {orb}}(X\sp n,\Sigma\sb n)p\sp n$ for symmetric groups $\Sigma\sb n$ acting on $n$-fold products coincides with the one proposed by R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde. The two notions of elliptic genera are conjectured to coincide.