If X is a smooth scheme of characteristic zero and Yc X is a closed subset, we find topological conditions on the singularities of Y which determine the best possible vanishing theorem for the sheaves of local cohomology Hf(F) for all i > r and all quasicoherent F. Applications include computation of the cohomological dimension of Pn - Y for arbitrary closed subsets Y and extensions of theorems of Lefschetz and Barth to the case of singular varieties.
No takes yet. Share an insight, caveat, or question.
Arthur Ogus (1973) studied this question.