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"?