Projective varieties of low codimension in characteristic 𝑝>0

Abstract

Let X be an s-dimensional closed Cohen-Macaulay subvariety of projective n-space, over an algebraically closed field of characteristic p > 0 p > 0 . Assume s ⩾ 1 2 ( n + 1 ) s \geqslant \tfrac {1}{2}(n + 1) . Then (1) every stratified vector bundle on X is trivial; (2) X is simply connected. Assertion (1) generalizes Gieseker’s result for projective space, while (2) is a strengthened analogue of results of Barth and Ogus in characteristic zero.