  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).
