Kodaira–Akizuki–Nakano Vanishing: a Variant

In this note we wish to prove a purely characteristic p > 0 variant of the Kodaira–Akizuki–Nakano vanishing for smooth complete intersections of dimension at least two in projective space. This has some interesting applications; in particular, we show that all Frobenius pull‐backs of the tangent bundle of any complete intersection of general type and of dimension at least three in P n are stable . We also show (see Remark 3.4) that a small modification of the techniques of [ 5 ] and a theorem of Mehta and Ramanathan (see [ 3 ]) together allow us to extend this stability result to smooth projective hypersurfaces of degree d , where ( n +1)/2< d < n +1 (that is, to some Fano hypersurfaces). It is well known that behaviour of stability under Frobenius pull‐backs is a subtle problem of the theory of vector bundles in characteristic p >0, and hence this result is not without interest. We end with an obvious conjectural form of our variant for a general class of varieties. 1991 Mathematics Subject Classification 14J760.