Abstract
If $X$ is a smooth scheme of characteristic zero and $Y\subset X$ 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 $\underline{H}_Y^i(F)$ for all $i>r$ and all quasicoherent $F$. Applications include computation of the cohomological dimension of $\mathbf{P}^n-Y$ for arbitrary closed subsets $Y$ and extensions of theorems of Lefschetz and Barth to the case of singular varieties.
