Let $s: \mathbb C^n \to \mathbb C$ be a homogeneous degree-$d$ polynomial which is nonsingular (in the sense that the hypersurface it defines in $\mathbb{CP}^{n-1}$ is smooth; equivalently the discriminant does not vanish). Write $\Omega$ for the canonical holomorphic $n$-form $\mathrm d z_1\cdots\mathrm d z_n$ on $\mathbb C^n$. I am interested in integrals against the measure $\exp(s)\Omega$. Specifically, one can ask about integrals of the form $\int_\gamma f\exp(s)\Omega$ for some contour $\gamma$, where $f$ is a polynomial.

By a "contour" I mean a boundary-free (but not compact) real-$n$-dimensional submanifold of $\mathbb C^n$. For the integral to converge, you'd like exponential decay at the ends of $\gamma$; i.e. at the ends of $\gamma$ we should have $\Re(s)\to -\infty$. By Stokes' Theorem, small perturbations of $\gamma$ don't matter, and neither do compact cycles. So all that matters in $\gamma$ is the class it represents in the relative homology group $\mathrm H_n(\mathbb C^n,\lbrace \Re(s)\lt 0\rbrace)$. If $n\gt 1$, by the long exact sequence for relative homology this group is equal to $\mathrm H_{n-1}(\lbrace \Re(s)\lt 0\rbrace)$. (When $n=1$ it is the quotient of this by $\mathrm H_0(\mathbb C^n)$ which is one-dimensional.)

Stoke's theorem also implies that integrals of total derivatives (of functions that vanish at infinity) vanish. Namely (provided $\gamma$ is as above), if $f$ is of the form $\frac{\partial g}{\partial z_i} + \frac{\partial s}{\partial z_i}g$ for some $i=1,\dots,n$ and some polynomial $g$, then $\int_\gamma f\exp(s)\Omega = 0$.

Thus the integral determines a pairing of the form $$ \textstyle \mathrm H_n\bigl(\mathbb C^n,\lbrace \Re(s)\lt 0\rbrace\bigr) \otimes \bigl(\mathbb C[z_1,\dots,z_n] / \sum_i (\text{image of } \frac{\partial}{\partial z_i} + \frac{\partial s}{\partial z_i})\bigr) \to \mathbb C $$ My question is whether this pairing is perfect. If it matters, I am primarily interested in the case when $s$ is generic.

The answer is (somewhat trivially) "yes" when $n=1$ or when $d=2$. When $n=2$ I convinced myself that the answer is yes for "diagonal" $s(z_1,z_2) = z_1^d + z_2^d$; at least, we calculated the dimension for $d=3$, and a similar argument should work for higher $d$. At least some of the people I have asked have given the prediction that the answer is "no" in general.

For generic $s$ I have complete control over the "algebraic side": I can give an explicit basis for the quotient and for this basis and the monomial basis explicit formulas for the map. An easy part of this is to see that this "algebraic" piece is $(d-1)^n$-dimensional.

On the other hand, I'm not very good at algebraic topology and algebraic geometry, and the homology group is definitely inaccessible to pure algebra and is rather a piece of real algebraic geometry. After talking with a number of folks at Berkeley, we've been unable to calculate even the dimension of the relative homology group, but maybe there were tricks we didn't think of. Or maybe there's an a priori reason why the pairing is perfect, and then I would have a calculation of this "real topology" side.