  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 biliner form (i.e. homology
with cup product) and a $\pm$-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 2 very different settings are
tied up using the Brieskorn lattice as a D-module, on using results of
Brieskorn (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 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.
