Dualization invariance and a new complex elliptic genus

Details Dualization invariance and a new complex elliptic genus Dualization invariance and a new complex elliptic genus

2011-09-25

arxiv.org/abs/1109.5394v1

Mathematics

Algebraic Topology

33 pages

Scientific paper

We define a new elliptic genus \psi\ on the complex bordism ring. With coefficients in Z[1/2], we prove that it induces an isomorphism of the complex bordism ring modulo the ideal which is generated by all differences P(E)-P(E*) of a projective bundle and the dual projective bundle onto a polynomial ring on 4 generators in degrees 2, 4, 6 and 8. As an alternative geometric description of \psi, we prove that it is the universal genus which is multiplicative in projective bundles over Calabi-Yau 3-folds. With the help of the q-expansion of modular forms we will see that for a complex manifold M, the value \psi(M) is a holomorphic Euler characteristic. We also compare \psi\ with Krichever-H\"ohn's complex elliptic genus and see that their only common specializations are Ochanine's elliptic genus and the chi_y-genus. In addition, we discuss general relations between a projective bundle, the dual projective bundle and the trivial projective bundle. As a consequence we see that the well-known description of the chi_y-genus as the universal multiplicative genus on the rational complex bordism ring also holds with integral coefficients.

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