Vikram B. Mehta

On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic p.

This paper is about sheaf cohomology for varieties (schemes) in characteristic

Simply connected projective manifolds in characteristic p >0 have no nontrivial stratified bundles

p>0

Restriction of stable sheaves and representations of the fundamental group

. We assume the presence of a Frobenius splitting. (See V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is that a non-zero higher direct image under a proper map of the ideal sheaf of a compatibly Frobenius split subvariety can not have a support whose inverse image is contained in that subvariety. Earlier vanishing theorems for Frobenius split varieties were based on direct limits and Serres vanishing theorem, but our theorem is based on inverse limits and Grothendiecks theorem on formal functions. The result implies a Grauert--Riemenschneider type theorem.

A characterization of rational singularities

We show that simply connected projective manifolds in characteristic p>0 have no nontrivial stratified bundles. This gives a positive answer to a conjecture by D. Gieseker (Ann. Sc. Norm. Super. Pisa, 4 Sér. 2(1):1–31, 1975). The proof uses Hrushovski’s theorem on periodic points.