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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.