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 $p > 0$. Assume $s \geqslant \tfrac {1}{2}(n + 1)$. 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.