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Using his deep and beautiful idea of cutting with a Hyperplane, Lefschetzexplained how the homology groups of a projective smooth variety could beconstructed from basic pieces, that he called primitive homology. This idea canbe applied every time we have a vector space with a biliner form (i.e. homologywith cup product) and a $\pm$-symmetric nilpotent operator (i.e. cutting with ahyperplane). We will illustrate this in the context of Singularity Theory: Agerm of an isolated singular point of a hypersurface defined by $f=0$. We beginwith the algebraic setting in the Jacobian (or Milnor) Algebra of thesingularity, with Grothendieck pairing as bilinear form and multiplication by$f$ as a symmetric nilpotent operator. We continue in the topological settingof vanishing cohomology with bilinear form induced from cup product and asnilpotent map the logarithm of the unipotent map of the monodromy as ananti-symmetric operator. We then show how these 2 very different settings aretied up using the Brieskorn lattice as a D-module, on using results ofBrieskorn (1970), A. Varchenko (1980s), M. Saito (1989), and C. Hertling (1999,2004, 2005), inducing a Polarized Mixed Hodge structure at the singularity(Steenbrink, 1976) and bringing the spectrum of the singularity as a deeperinvariant than the eigenvalues of the Algebraic Monodromy. In particular, weshow how an $f$-Jordan chain is obtained from several $N$-Jordan chains bygluing them in the Brieskorn lattice.

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