Characteristic cohomology II: Matrix singularities

For a germ of a variety $V,0\subset C^N,0$, a singularity $V_0$ of “type $V$”  is given by a germ $f_0:C^n,0\to C^N,0$, which is transverse to $V\setminus \left\{0\right\}$ in an appropriate sense, such that $V_0=f_0^{-1}\left(V\right)$. In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for $V$ a hypersurface), and complement and link (for the general case). It captures the cohomology of $V_0$ inherited from $V$ and is given by subalgebras of the cohomology for $V_0$ for the Milnor fiber and complements, and is a subgroup for the cohomology of the link. We showed these cohomologies are functorial and invariant under diffeomorphism groups of equivalences $K_H$ for Milnor fibers and $K_V$ for complements and links. We also gave geometric criteria for detecting the nonvanishing of the characteristic cohomology.

In this paper, we apply these methods in the case $V$ denotes any of the varieties of singular $m\times m$ complex matrices, which may be either general, symmetric, or skew-symmetric (with $m$ even). For these varieties, we have shown in another paper that their Milnor fibers and complements have compact “model submanifolds”  for their homotopy types, which are classical symmetric spaces in the sense of Cartan. As a result, we first give the structure of the characteristic cohomology subalgebras for the Milnor fibers and complements as images of exterior algebras (or in one case a module on two generators over an exterior algebra). For links, the characteristic cohomology group is the image of a shifted upper truncated exterior algebra. In addition, we extend these results for the complement and link to the case of general $m\times p$ complex matrices.

Second, we then apply the geometric detection methods introduced in Part I to detect when specific characteristic cohomology classes for the Milnor fiber or complement are nonzero. We identify an exterior subalgebra on a specific set of generators and for the link that it contains an appropriate shifted upper truncated exterior subalgebra. The detection criterion involves a special type of “kite map germ of size $\ell$” based on a given flag of subspaces. The general criterion that detects such nonvanishing characteristic cohomology is then given in terms of the defining germ $f_0$ containing such a kite map germ of size $\ell$. Furthermore, we use a restricted form of kite spaces to give a cohomological relation between the cohomology of local links and the global link for the varieties of singular matrices.