Local monodromy, Behrend function and Hilbert scheme of points on ${\mathbb{C}}^3$

We prove a criteria for the Behrend function values on the Hilbert scheme of $n$-points on ${\text{C}}^3$ from local monodromy and the Denef–Loeser’s Lefschetz trace formula of jet spaces. If the power two, three, or four of the monodromy is unipotent, it reduces respectively to a Hessian parity formula, a cubic-complement formula, or a quartic formula. This gives a Milnor fiber and monodromy based proof of nonconstancy of the Behrend function on ${\text{Hilb}}^n({\text{C}}^3)$ after Jelisiejew-Kool-Schmiermann. We also calculate two ideals in ${\text{Hilb}}^9({\text{C}}^3)$ and ${\text{Hilb}}^{12}({\text{C}}^3)$ so that the Behrend function is not $\pm 1$, answering a question of Ricolfi.