Borel-Moore homology of determinantal varieties

TL;DR

This paper computes the rational Borel-Moore homology groups of affine determinantal varieties, linking algebraic and topological methods, and explores their Hodge structures and spectral sequences.

Contribution

It provides explicit calculations of Borel-Moore homology for determinantal varieties and establishes the degeneration of the cech-de Rham spectral sequence, advancing understanding of their topological invariants.

Findings

Computed rational Borel-Moore homology groups for various determinantal varieties.

Established the degeneration of the cech-de Rham spectral sequence for these varieties.

Determined Hodge numbers of matrix orbit cohomology and closures for general matrices.

Abstract

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 \v{C}ech-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.