The primary goal of this paper is to systematically exploit the
method of Deligne and 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 that
provides a cohomological measure of the positivity of a coherent
sheaf called the Frobenius or F-amplitude. 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 in
various situations.