# On smooth projective D-affine varieties

## TL;DR

This paper investigates properties of smooth projective D-affine varieties, establishing their topological and geometric characteristics, classifying surfaces in characteristic zero, and introducing a generalization of Miyaoka's theorem for positive characteristic.

## Contribution

It provides new insights into the structure of D-affine varieties, including their simple connectivity, behavior under fibrations, and a classification of surfaces in characteristic zero.

## Key findings

- D-affine varieties are algebraically simply connected.
- In characteristic zero, D-affine varieties are uniruled.
- Classification of D-affine surfaces as ${
m P}^2$ or ${
m P}^1 imes {
m P}^1$ in most characteristics.

## 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 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.