Topological calculation of local cohomological dimension
Thomas Reichelt, Morihiko Saito, and Uli Walther
Journal of Singularities
volume 26 (2023), 13-22
Received: 2 November 2022. Received in revised form: 28 June 2023.
Abstract:
We show that the sum of the local cohomological dimension and the rectified 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 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.
2020 Mathematical Subject Classification:
Primary 32C36, 32S60; Secondary 14B15, 14F10
Key words and phrases:
local cohomological dimension, homological depth, de Rham depth, t-structure, holonomic D-module
Author(s) information:
Thomas Reichelt
Lehrstuhl für Algebraische Geometrie
Universität Mannheim
B6 26, 68159 Mannheim, Germany
email: reichelt@math.uni-mannheim.de
Morihiko Saito
RIMS Kyoto University
Kyoto 606-8502, Japan
email: msaito@kurims.kyoto-u.ac.jp
Uli Walther
Dept. of Mathematics
Purdue University
150 N. University St.
West Lafayette, IN 47907, USA
email: walther@math.purdue.edu