2019/11/05 by Damon, James
#14M12 #20G05 #32S25 #55R80 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary: 11S90 #Secondary: 57T15
paper · doi:10.48550/arxiv.1911.02092
For a germ of a variety V, 0 ⊂ \mathbb CN, 0, a singularity V0 of type V, is given by a germ f0 : \mathbb Cn, 0 → \mathbb CN, 0 which is transverse to V in an appropriate sense so that V0 = f0-1(V). For these singularities, we introduce "characteristic cohomology" to capture the contribution of the topology of V to that of V0, for the Milnor fiber (for V, 0 a hypersurface), and complement and link of V0 (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 V, they are functorial over the category of singularities of type V. In addition, for the link of V0 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 KV-equivalence of defining germs f0, resp. for the link invariant under the \mathcal KH-equivalence of f0 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.