Topological calculation of local cohomological dimension

Abstract

We show that the sum of the local cohomological dimension and the rectified $\mathbb Q$-homological depth of a closed analytic subspace of a complex manifold coincide with the dimension of the ambient manifold. The local cohomological dimension is then calculated using the cohomology of the links of the analytic space. In the algebraic case the first assertion is equivalent to the coincidence of the rectified $\mathbb Q$-homological depth with the de Rham depth studied by Ogus, and follows essentially from his work. As a corollary we show that the local cohomological dimension of a quasi-projective variety is determined by that of its general hyperplane section together with the link cohomology at 0-dimensional strata of a complex analytic Whitney stratification.