  We show various properties of smooth projective D-affine varieties. In
particular, any smooth projective D-affine variety is algebraically simply
connected and its image under a fibration is D-affine. In characteristic zero
such D-affine varieties are also uniruled.
  We also show that (apart from a few small characteristics) a smooth
projective surface is D-affine if and only if it is isomorphic to either
${\mathbb P}^2$ or ${\mathbb P}^1\times {\mathbb P}^1$. In positive
characteristic, a basic tool in the proof is a new generalization of Miyaoka's
generic semipositivity theorem.
