Topological calculation of local cohomological dimension

TL;DR

This paper establishes a topological method to compute the local cohomological dimension of complex analytic and algebraic spaces using link cohomology and depth invariants, linking algebraic and topological perspectives.

Contribution

It introduces a topological approach to calculating local cohomological dimension via link cohomology and depth invariants, connecting algebraic and analytic cases.

Findings

Local cohomological dimension equals the sum of depth and cohomological dimension.

Cohomology of links determines local cohomological dimension.

Dimension of a quasi-projective variety relates to hyperplane sections and link cohomology.

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.