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

Abstract

We prove a criteria for the Behrend function values on the Hilbert scheme of $n$-points on $\C^3$ 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 $\Hilb^n(\C^3)$ after Jelisiejew-Kool-Schmiermann. We also calculate two ideals in $\Hilb^9(\C^3)$ and $\Hilb^{12}(\C^3)$ so that the Behrend function is not $\pm 1$, answering a question of Ricolfi.

arXiv comment20 pages, comments are welcome, calculated two Behrend function values which are 2 and 3, references updated