Bilinear forms in the Jacobian module and binding of N-spectral chains of a hypersurface with an isolated singularity

Abstract

Using his deep and beautiful idea of cutting with a Hyperplane, Lefschetz explained how the homology groups of a projective smooth variety could be constructed from basic pieces, that he called primitive homology. This idea can be applied every time we have a vector space with a bilinear form (i.e., homology with cup product) and a ±-symmetric nilpotent operator (i.e., cutting with a hyperplane). We will illustrate this in the context of Singularity Theory: A germ of an isolated singular point of a hypersurface defined by $f=0$. We begin with the algebraic setting in the Jacobian (or Milnor) Algebra of the singularity, with Grothendieck pairing as bilinear form and multiplication by f as a symmetric nilpotent operator. We continue in the topological setting of vanishing cohomology with bilinear form induced from cup product and as nilpotent map the logarithm of the unipotent map of the monodromy as an anti-symmetric operator. We then show how these two very different settings are tied up using the Brieskorn lattice as a D-module, on using results of Brieskorn (Manuscripta Math, 2:103–161, 1970), Varchenko (Math USSR Izvestiya, 26:31, 1986), Saito (Ann Inst Fourier Grenoble, 39:27–72, 1989), and Hertling (Compositio Mathematica, 116:1–37, 1999, Camb Tracts Math, 151, 2004, in: Barlet (ed) Singularités, Institut Élie Cartan Nancy, Villers-lés-Nancy, 2005), inducing a Polarized Mixed Hodge structure at the singularity (Steenbrink in Mixed Hodge structure on the vanishing cohomology, Nordic Summer School, Simposium in Mathematics, Oslo, 1976) and bringing the spectrum of the singularity as a deeper invariant than the eigenvalues of the algebraic monodromy. In particular, we show how an f-Jordan chain is obtained from several N-Jordan chains by gluing them in the Brieskorn lattice.