Characteristic Cohomology I: Singularities of Given Type

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.