Dualization invariance and a new complex elliptic genus

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.