Characteristic Cohomology II: Matrix Singularities

TL;DR

This paper computes the characteristic cohomology of matrix singularities, extending previous work to various matrix varieties and revealing algebraic structures and detection criteria for these complex singularities.

Contribution

It determines the characteristic cohomology for matrix singularities across different types, linking it to classical symmetric spaces and providing explicit algebraic descriptions.

Findings

Characteristic subalgebra is the image of an exterior algebra on explicit generators.

Identification of algebraic structures in Milnor fibers, complements, and links.

Detection criteria for subalgebras using special 'unfurled kite maps'.

Abstract

Let $\mathcal{V} \subset M$ denote any of the varieties of singular $m \times m$ complex matrices which may be general, symmetric, or skew-symmetric ($m$ even), or $m \times p$ matrices, in the corresponding space $M$ of such matrices. A "matrix singularity", $\mathcal{V}_0$ of "type $\mathcal{V}$", for any of the $\mathcal{V} \subset M$ is defined as $\mathcal{V}_0 = f_0^{-1}(\mathcal{V})$ by a germ $f_0 : \mathbb C^n, 0 \to M, 0$ (appropriately transverse to $\mathcal{V}$). In part I of this paper we introduced the notion of characteristic cohomology for a singularity $\mathcal{V}_0$ of type $\mathcal{V}$ for the Milnor fiber (for $\mathcal{V}$ a hypersurface) and for the complement and link (in the general case). We determine here the characteristic cohomology for matrix singularities in all of these cases. For these singularities we had shown in another paper that the Milnor fibers and complements have "compact model submanifolds", which are classical symmetric spaces in the sense of Cartan. We show that characteristic subalgebra is the image of an exterior algebra (or in one case a module on two generators over an exterior algebra) on an explicit set of generators. We give "detection criteria"using the vanishing compact models for identifying a exterior subalgebras in the characteristic sublgebra, when $f_0$ contains a special type of "unfurled kite map". This will be valid for the Milnor fiber, complement, and link.