On smooth projective D-affine varieties Adrian Langer December 17, 2019 ADDRESS: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland e-mail:
[email protected] Abstract We show various properties of smooth projective D-affine varieties. In particular, any smooth pro- jective 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 P2 or P1 P1. In positive characteristic, a basic tool in the proof is a new generalization of Miyaoka’s generic semipositivity× theorem. Introduction Let X be a scheme defined over some algebraicallyclosed field k. Let DX be thesheafof k-linear differential operators on X. A DX -module is a left DX -module, which is quasi-coherent as an OX -module. X is called D-quasi-affine if every DX -module M is generated over DX by its global sections. X is called D-affine if it i is D-quasi-affine and for every DX -module M we have H (X,M)= 0 for all i > 0. In [2] Beilinson and Bernstein proved that every flag variety (i.e., a quotient of a reductive group by some parabolic subgroup) in characteristic 0 is D-affine. This fails in positive characteristic (see [17]), although some flag varieties are still D-affine (see, e.g., [12], [22] and [36]). However, there are no known examples of smooth projective varieties that are D-affine and that are not flag varieties.