Characteristic Cohomology I: Singularities of Given Type

TL;DR

This paper introduces characteristic cohomology to analyze the topology of singularities of a given type, providing invariants for Milnor fibers, complements, and links, with functorial and invariance properties.

Contribution

It defines characteristic cohomology for singularities of a fixed type, establishing its functoriality, invariance under equivalences, and geometric criteria for detecting nonvanishing subalgebras.

Findings

Characteristic cohomology captures the topology of singularities.

Invariance under specific equivalence relations.

Geometric criteria for nonvanishing subalgebras.

Abstract

For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of type $\mathcal{V}$, is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense so that $\mathcal{V}_0 = f_0^{-1}(\mathcal{V})$. For these singularities, we introduce "characteristic cohomology" to capture the contribution of the topology of $\mathcal{V}$ to that of $\mathcal{V}_0$, for the Milnor fiber (for $\mathcal{V}, 0$ a hypersurface), and complement and link of $\mathcal{V}_0$ (in the general case). The characteristic cohomology of both the Milnor fiber and complement are subalgebras of the cohomology of the Milnor fibers, respectively the complement. For a fixed $\mathcal{V}$, they are functorial over the category of singularities of type $\mathcal{V}$. In addition, for the link of $\mathcal{V}_0$ there is a characteristic cohomology subgroup of the cohomology of the link over a field of characteristic 0. The characteristic cohomologies for Milnor fiber and complement are shown to be invariant under the $\mathcal K_{\mathcal{V}}$-equivalence of defining germs $f_0$, resp. for the link invariant under the $\mathcal K_{H}$-equivalence of $f_0$ for $H$ the defining equation of $\mathcal V, 0$. We give a geometric criteria involving "vanishing compact models", which detect nonvanishing subalgebras of the characteristic cohomologies, resp. subgroups for the link. In part II of this paper we specialize to the case of square matrix singularities, which may be general, symmetric or skew-symmetric.