Mathematics – Algebraic Geometry
Scientific paper
2002-02-14
Duke Math. J. 121, no. 2 (2004), 231-267
Mathematics
Algebraic Geometry
30 pages, Latex, uses xypic; revised, the appendix by D. Keeler has been expanded
Scientific paper
10.1215/S0012-7094-04-12122-0
The primary goal of this paper is to systematically exploit the method of Deligne-Illusie to obtain Kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties. The key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the Frobenius or F-amplitude. As the terminology indicates, the definition is most natural over a field of positive characteristic, however it can be forced into characteristic 0 by standard tricks. The F-amplitude enters into the statement of the basic vanishing theorem, and this leads to the problem of calculating, or at least estimating, this number. Most of the work in this paper is devoted to doing this various situations. The key result in this direction is a bound on the F-amplitude of an ample vector bundle. When combined with the vanishing theorem, this has some reasonably down to earth corollaries (both old and new).
Arapura Donu
Keeler Dennis S.
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