We study the derived categories of coherent sheaves on some singular projective curves and give a complete description of indecomposable objects using the technique of matrix problems.

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.

We prove that any weak limit of weak solutions u_k of the degenerate nonlinear elliptic system, the so-called (perturbed) H-system

A new avatar of Milnor K-theory, for algebraically closed fields, appears in the guise of certain homology groups of orthogonal groups with twisted coefficients.