Vanishing theorems of generalized Witten genus for generalized complete intersections in flag manifolds

Abstract

We propose a potential function [Formula: see text] for the cohomology ring of partial flag manifolds. We prove a formula expressing integrals over partial flag manifolds by residues, which generalizes [E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in Geometry, Topology, Physics (International Press, 1995), pp. 357–422]. Using this formula, we prove a Landweber–Stong type vanishing theorem for generalized [Formula: see text] complete intersections in flag manifolds, which serves as evidence for the [Formula: see text] version of Stolz conjecture [Q. Chen, F. Han and W. Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom. 88(1) (2011) 1–39].