On Smooth Projective D-Affine Varieties

Abstract

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 0 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.

Counterexamples to the MMP for 1-foliations in positive characteristic

“…We are now ready to introduce the main example of this note by taking p -cyclic covering of abelian surfaces. A similar construction has been used in [ 24 , Sect. 5.1] to construct counterexamples to Miyaoka’s semipositivity theorems in positive characteristic.…”

“…This construction, introduced by Ekedahl [ 13 , page 145] and revisited by Kollár in [ 22 ], gives a counterexample to the Bogomolov–Sommese vanishing theorem in positive characteristic. It has recently been revisited by Langer to give counterexamples to Miyaoka’s generic semipositivity [ 24 , Sect. 5.1] and by Graf to construct counterexamples to the logarithmic extension theorem for differential forms ( [ 14 , Theorem 1.6]).…”

D-affinity and rational varieties

“…There are further notions of D-affinity and derived D-affinity in positive characteristic when instead of Grothendieck differential operators, either small differential operators [8,11,12,19] or crystalline differential operators [3,4,16] are studied. These are not covered by the present paper, although some of our methods may prove useful for these unusual differential operators.…”