The local cohomological dimension of a closed analytic subspace Y of a complex manifold X is the top degree in which algebraic local cohomology along Y of the structure sheaf is nonzero, and the rectified Q-homological depth of Y is the bottom degree in which the perverse cohomology of the constant sheaf on Y is nonzero. These two invariants, one sheaf-theoretic and one topological, always sum to the dimension of the ambient manifold, so an algebraic invariant of the embedding turns out to depend only on the topology of Y. The practical payoff is a computation rule: read the invariant off the reduced cohomology of the links of the strata of a Whitney stratification, which also shows it is unchanged under topologically equisingular deformation and recovers classical vanishing criteria such as Hartshorne-Lichtenbaum as small cases.
For a reduced closed analytic subspace Y of a complex manifold X, the local cohomological dimension lcd(Y,X) is the largest degree in which the algebraic local cohomology of the structure sheaf along…