I have suspected that the following is well-known:

If $P$ is a homogeneous polynomial of degree $d$ in $n$ variables (for example, Fermat quintic $x_1^5 + \cdots + x_5^5$). The Milnor ring is ${\mathbb C}[x_1, \ldots, x_n]$ modulo the Jacobian ideal of $P$. On the other hand, $P$ defines a hypersurface $S$ in ${\mathbb P}^{n-1}$. The middle dimensional cohomology of $S$ (or maybe the primitive part) should correspond to the ${\mathbb Z}_d$-invariant subspace of the Milnor ring.

For example, for the Fermat quintic, the Milnor ring has dimension 1024 and the $h_3$ of the Calabi-Yau 3-fold defined by $P$ is 204 which is equal to the dimension of the ${\mathbb Z}_5$-invariant subspace of the Milnor ring.

But I don't know if there is any elementary literature which discusses this correspondence. And I would like to see some geometric treatment using vanishing cycles/Lefschetz pencils/thimbles to see this correspondence. For example, for Fermat quintic, there are 1024 Lefschetz thimbles if we perturb $P$ by adding a small linear term; but why only the ${\mathbb Z}_5$-invariant part survives at "infinity"?

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    $\begingroup$ A very beautiful and natural way to think about this is as a decategorified (taking Hochschild homology + Hodge data) version of the Calabi-Yau Landau-Ginzburg correspondence. See Segal's www2.imperial.ac.uk/~epsegal/papers/… and generalizations that came thereafter. I should of course make clear that this interpretation should recover Griffiths' formula for all degrees not just in degree $d$. $\endgroup$
    – user36931
    Feb 3 '14 at 13:04

This is implied by Griffiths' celebrated presentation of the Hodge filtration of a hypersurface in terms of the Jacobian ring. He gives an isomorphism of the primitive part of $H^{p,q}$ of a smooth hypersurface of degree $d$ in $\mathbb{P}^{n+1}$ with the degree $(p+1)d-n-2$ part of the Jacobian ring. You can find an excellent presentation in Voisin's book Hodge Theory and Complex Algebraic Geometry.

  • $\begingroup$ There is also a nice presentation of the Griffiths residue calculus in the article by Carlsson and Toledo, "Discriminant Complements and Kernels of Monodromy Representations". $\endgroup$ Feb 3 '14 at 14:32
  • $\begingroup$ I guess my question wasn't asked properly. I had a look at Voisin's book and it has a very geometric approach to Lefschetz theorem. But she only answers: vanishing cohomology of a smooth member of a Lefschetz pencil is "generated" by vanishing spheres, simply by some Morse-theoretic argument, some n-disks are glued along the vanishing spheres. However, she didn't say if there are (linear) relations among the vanishing spheres. It seems to me that the number of vanishing spheres are much more than the rank of cohomology. $\endgroup$
    – UVIR
    Feb 3 '14 at 18:51

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