Elliptic Genera of Level N for Complex Manifolds

Abstract

My lecture at the Como Conference was a survey on the theory of elliptic genera as developed by Ochanine, Landweber, Stong and Witten. A good global reference are the Proceedings of the 1986 Princeton Conference [1]. In this contribution to the Proceedings of the Como Conference I shall not reproduce my lecture, but rather sketch a theory of elliptic genera of level N for compact complex manifolds which I presented in the last part of my course at the University of Bonn during the Wintersemester 1987/88. For a natural number N > 1 the elliptic genus of level N of a compact complex manifold M of dimension d is a modular form of weight d for the group Γ₁(N). In the cusps of Γ₁(N) the genus degenerates either to χ_y(M)/(1+y)^d where -y is an N^th-root of unity different from 1 or to χ(M,K^k/N) where K is the canonical line bundle and 0 < k < N. Here ${\chi _y}\left( M \right) = \sum\limits_{p = 0}^d {{\chi ^p}} \left( M \right){y^p}$ with ${\chi ^p}\left( M \right) = \chi \left( {M,{\Omega ^p}} \right) = \sum\limits_{q = 0}^d {{{\left( { - 1} \right)}^q}{h^{p,q}}}$ is the χ_y-genus introduced in [13] and χ(M.K^k/N) is the genus with respect to the characteristic power series

$$\frac{x}{{1 - {e^{ - x}}}}\cdot{e^{ - {{(k/N)}^ \bullet }x}} $$

which equals the holomorphic Euler number of M with coefficients in the line bundie L^k provided K = L^N (see [13]).