Dualization invariance and a new complex elliptic genus

TL;DR

This paper introduces a new elliptic genus called psi, establishing its algebraic properties, geometric interpretation, and relation to existing genera, with implications for complex manifolds and modular forms.

Contribution

It defines psi, proves its isomorphism properties on the bordism ring, and provides a geometric and modular form interpretation, connecting it to known genera.

Findings

psi induces an isomorphism on the bordism ring modulo a specific ideal

psi is the universal genus multiplicative over Calabi-Yau 3-folds

psi's values are holomorphic Euler characteristics for complex manifolds

Abstract

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 projective bundles and their duals 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-Hoehn's complex elliptic genus and see that their only common specializations are Ochanine's elliptic genus and the chi_y-genus.