A remark on bigness of the tangent bundle of a smooth projective variety and D-simplicity of its section rings

Abstract

We point out a connection between bigness of the tangent bundle of a smooth projective variety X over ℂ and simplicity of the section rings of X as modules over their rings of differential operators. As a consequence, we see that the tangent bundle of a smooth projective toric variety or a (partial) flag variety is big. Some other applications and related questions are discussed.