Generalized Witten Genus and Vanishing Theorems

Abstract

We construct a generalized Witten genus for spin${}^c$ manifolds, which takes values in level 1 modular forms with integral Fourier expansion on a class of spin${}^c$ manifolds called string${}^c$ manifolds. We also construct a mod 2 analogue of the Witten genus for $8k+2$ dimensional spin manifolds. The Landweber-Stong type vanishing theorems are proven for the generalized Witten genus and the mod 2 Witten genus on string${}^c$ and string (generalized) complete intersections in (product of) complex projective spaces respectively.

28 pages