D-critical locus structure on the Hilbert schemes of some local toric Calabi-Yau threefolds

TL;DR

This paper establishes a d-critical locus structure on Hilbert schemes of zero-dimensional subschemes on certain local toric Calabi-Yau threefolds, linking it to motivic Donaldson-Thomas invariants and confirming their consistency with prior definitions.

Contribution

It demonstrates the existence of a d-critical locus structure on these Hilbert schemes and shows the resulting motivic invariants align with previous formulations.

Findings

Existence of d-critical locus structure on Hilbert schemes

Alignment of motivic invariants with previous definitions

Application to local toric Calabi-Yau threefolds

Abstract

The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [BJM19]. In this paper we show that there is a d-critical locus structure on the Hilbert scheme of dimension zero subschemes on some local toric Calabi-Yau 3-folds. We also show that using this d-critical locus structure and a choice of orientation data, the resulting motivic invariants agree with the definition given by the previous work of [BBS13].