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KODAIRA–AKIZUKI–NAKANO VANISHING: A VARIANT

Published online by Cambridge University Press:  01 March 2000

KIRTI JOSHI
KIRTI JOSHI
Affiliation:
Department of Mathematics, University of Arizona, 617 N. Santa Rita, P.O. Box 210089, Tucson, AZ 85721, USA School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai (Bombay) 400005, India
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Abstract

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 Pn 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.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 32 , Issue 2 , March 2000 , pp. 171 - 176
Copyright
© The London Mathematical Society 2000

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