Vanishing Theorems for Flag Manifolds

Abstract

Introduction. In this paper, I prove some vanishing theorems for the cohomology groups of invertible sheaves on flag manifolds. Most of these vanishing theorems follow from Kodaira's vanishing theorem in characteristic zero. I prove these by induction. This induction involves certain subvarieties of the flag manifold, which I call Schubert manifolds. These vanishing theorems are used to deduce the Cohen-Macaulayness of Schubert varieties in Grassmannians and cones over them. They were recently proved to be Cohen-Macaulay by Eagon, Hochster and Laksov [3, 5, and 8] by more algebraic methods. I prove that the Schubert varieties have rational resolutions. This is a stronger property than Cohen-Macaulayness. The original idea for this proof was to generalize the results in [6]. The vanishing theorem has been applied by T. Svanes to show the rigidity of some Schubert varieties [12].