Borel-Moore homology of determinantal varieties. (English) Zbl 1539.14109
Summary: We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is the relation with Hartshorne’s algebraic de Rham homology theory, and the calculation of the singular cohomology of matrix orbits using the methods of Cartan and Borel. We also establish the degeneration of the Čech-de Rham spectral sequence for determinantal varieties and compute explicitly the dimensions of de Rham cohomology groups of local cohomology with determinantal support, which are analogues of Lyubeznik numbers first introduced by Switala. Additionally, in the case of general matrices, we further determine the Hodge numbers of the singular cohomology of matrix orbits and of the Borel-Moore
homology of their closures, based on Saito’s theory of mixed Hodge modules.
MSC:
14M12 | Determinantal varieties |
14D07 | Variation of Hodge structures (algebro-geometric aspects) |
32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |
55N33 | Intersection homology and cohomology in algebraic topology |
55N35 | Other homology theories in algebraic topology |
57T15 | Homology and cohomology of homogeneous spaces of Lie groups |