D-critical loci for local toric Calabi-Yau 3-folds

TL;DR

This paper establishes a canonical d-critical locus structure on the Hilbert scheme of zero-dimensional subschemes on local toric Calabi-Yau 3-folds, linking derived symplectic geometry with classical degeneracy loci.

Contribution

It demonstrates the existence and compatibility of the d-critical locus structure with known descriptions for local toric Calabi-Yau 3-folds, connecting derived and classical geometric frameworks.

Findings

Canonical d-critical locus structure on Hilbert schemes established

Compatibility with degeneracy locus descriptions confirmed

Isomorphism with known structures for local $P^2$ and $F_n$

Abstract

The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [BJM19]. There is a canonical d-critical locus structure on the Hilbert scheme of dimension zero subschemes on local toric Calabi-Yau 3-folds. This is obtained by truncating the $-1$-shifted symplectic structure on the derived moduli stack [BBBBJ15]. In this paper we show the canonical d-critical locus structure has critical charts consistent with the description of Hilbert scheme as a degeneracy locus [BBS13]. In particular, the canonical d-critical locus structure is isomorphic to the one constructed in [KS12] for local $\mathbb{P}^2$ and local $\mathbb{F}_n$.