To what extent does Poincare duality hold on moduli stacks?

Question

Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the manifold, and $\frown$ is the cap product.

This allows us (for example) to, given an embedding $f : X \to Y$ of manifolds to push forward integration:

$$ \int_X f^*\omega = \int_Y \omega \smile (f_*[X])^\vee $$

which is a simple application of the push-pull formula: $f_*(a \frown f^*b) = (f_*a) \frown b$.

To what extent do results like this hold for stacks such as $\overline{M}_{g,n}(X,\beta)$? I know that these are not in general nice, that they can have components of a variety of dimensions, etc. They do, however, have virtual fundamental classes of pure dimension which behave in many ways like the fundamental classes that we would want.

My question is then the following:

Is there a Poincare duality for such stacks, using the virtual fundamental class? Does the push-pull formula hold in such a case?

Bonus question: What about moduli stacks with reduced obstruction theories, such as for studying Gromov-Witten invariants of K3 and Abelian surfaces? Is there any chance that it is still true there?

Answer 1

I know of one very limited case, which will almost certainly not satisfy you, where the "virtual cohomology" mentioned by Angelo exists, with a version of Poincare duality.

Presumably the generality one would like to address this question is for a moduli space (or stack) $M$ carrying a perfect obstruction theory (let me ignore the stackyness everywhere below). Let me restrict right away to virtual dimension 0, and in fact further to the case when the obstruction theory is symmetric (in the sense of Behrend, Behrend-Fantechi). Note that this already excludes Kontsevich moduli spaces $\bar M_{g,n}(X,\beta)$, even for $X$ a Calabi--Yau threefold, but we are still OK with DT- or other sheaf-theoretic moduli spaces.

In this case, we of course have Behrend's result that locally $M$ is cut out in a smooth ambient space $N$ by the zeros of an almost closed one-form. Now specialize further, and assume that in fact $M$ is cut out by the zeros of a closed one-form. Finally, specialize to the case when in fact globally $M=Z(df)\subset N$; here $N$ is a smooth variety, $f\colon N\to {\mathbb C}$ is a smooth function, and $M$ is cut out by the zeros of the exact one-form $df$. Note that there are still reasonably interesting examples, such as the Hilbert scheme of points on ${\mathbb C}^3$ or related toric or quivery examples.

In this case, it seems that the correct "virtual cohomology" to consider is $H^*(M,\phi_f)$, the cohomology of $M$ with coefficients in the perverse $\mathbb Q$-sheaf $\phi_f\in{\rm Perv}_{\mathbb Q}(M)$ of vanishing cycles of $f$ (appropriately shifted). This is called critical cohomology by Kontsevich-Soibelman. For example, this cohomology has the correct Euler characteristic, namely the integral of the Behrend function on $M$. It also carries a mixed Hodge structure, and the Euler characteristic of the weight filtration (which will differ from the degree filtration because the Hodge structure is not pure) gives what seems to be the right "quantization" (or "refinement") of the numerical invariant.

Now the point is that by standard theory, $D_M\phi_f\cong\phi_f$, where $D_M$ is the Verdier duality functor on $M$. This will give you a kind of Poincare duality $$H^n(M,\phi_f)\cong H^{-n}_c(M,\phi_f)$$ albeit between ordinary and compact-support cohomology, since $M$ is of course almost always noncompact. But at least (using the appropriate shifts) it is elegantly symmetric around zero, which is what we would expect in the virtual dimension 0 case.

I don't really know how well this generalizes; almost nothing is known (to me) about gluing, functoriality, etc.