[Dscr ]-AFFINITY AND TORIC VARIETIES

Abstract

1. Introduction

Let k be an algebraically closed field of any characteristic. A toric variety over k is a normal variety X containing the algebraic group T=(k*)ⁿ as an open dense subset, with a group action T × X→X extending the group law of T.

On any smooth variety X over a field k, we can define the sheaf of differential operators [Dscr ], which carries a natural structure as an [Oscr ]_X-bisubalgebra of End_k([Oscr ]_X). A [Dscr ]-module on X is a sheaf [Fscr ] of abelian groups having a structure as a left [Dscr ]-module, such that [Fscr ] is quasi-coherent as an [Oscr ]_X-module. A smooth variety X is called [Dscr ]-affine if for every [Dscr ]-module [Fscr ] we have

[bull ] [Fscr ] is generated by global sections over [Dscr ],

[bull ] Hⁱ(X, [Fscr ])=0, i>0.

Beilinson and Bernstein have shown [1] that every flag variety over a field of characteristic zero is [Dscr ]-affine, from which they deduced a conjecture of Kazhdan and Lusztig. In fact, flag varieties are the only known examples of [Dscr ]-affine projective varieties. In this paper we prove that the [Dscr ]-affinity of a smooth complete toric variety implies that it is a product of projective spaces. Part of the method will be to translate a proof of the non [Dscr ]-affinity of a 2-dimensional Schubert variety, given by Haastert in [4], into the language of toric varieties.