# Relation between Milnor ring and middle dimensional homology of hypersurface

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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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 $d$. – 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.