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
Endk([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  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
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
in , into the language of toric varieties.