Let $f: X \to Y$ be a proper morphism between complex varieties (the varieties as well as the map may be non-smooth) and let $\phi \in \text{Perv}(X)$ be a perverse sheaf on $X$. Given this data, it is natural to form the following Laurent polynomial:
$$F(y) = \sum_{k \in \mathbb{Z}} y^{k} \, \chi\big({}^{\mathfrak{p}}\mathcal{H}^{k}(Rf_{*}\phi)\big)$$
Pushing forward, $Rf_{*}\phi$ is generally no longer perverse on $Y$, so we apply the perverse cohomology sheaves. Finally, $\chi(\cdot)$ is the Euler characteristic extracting a number from a complex. By Verdier duality, $F(y)$ is actually a symmetric Laurent polynomial, i.e. invariant under $y \leftrightarrow y^{-1}$.
This setup arises in curve-counting on Calabi-Yau threefolds. In this case, $f:X \to Y$ is the Hilbert-Chow morphism from a certain moduli space of sheaves to the Chow variety of curves. And $\phi \in \text{Perv}(X)$ is the natural perverse sheaf of vanishing cycles. For details, see (https://arxiv.org/pdf/1610.07303.pdf). In this context, $F(y)$ computes the Gopakumar-Vafa invariants, which are the closest thing we have to honest counts of curves.
Question: There is some condition on the data $(f,X,Y)$ such that the coefficient of the largest power of $y$ in $F(y)$ is given by the Euler characteristic of the base space $Y$, up to a sign. Maybe assume $\phi$ is the natural perverse sheaf of vanishing cycles. In the curve counting context, you would say that the maximal genus Gopakumar-Vafa invariant is given by the (weighted) Euler characteristic of Chow. Can someone explain what these conditions are? I don't think they're too strong. Like, you don't have to assume the map is smooth or something.
I get the feeling that this is some sort of folklore result that all the perverse sheaf experts know. I'm sure it's simply a straightforward computation, but I've heard it in a number of contexts, but never written down anywhere.
