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

Sheldon Katz, Yun Shi

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].

D-critical locus structure on the Hilbert schemes of some 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].

Paper Structure

This paper contains 9 sections, 10 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Notations
  4. Background
  5. Introduction to motivic DT invariants
  6. Construction of $\operatorname{Hilb}^n(\mathbb{C}^3)$
  7. D-critical Locus Structure on $\operatorname{Hilb}^n(\omega_S)$
  8. Orientation on $(\operatorname{Hilb}^n(\omega_S), s)$
  9. Motivic DT invariants for ${\textcolor{black}{\operatorname{Hilb}}}^n(\omega_S)$

Key Result

Theorem 2.2

(Joy15, Theorem 2.1) There exists a sheaf $\mathcal{S}_Y$ of $\mathbb{C}$ vector spaces, uniquely characterized by two properties. (i) Suppose $R\subset Y$ is a Zariski open subset of $Y$, and $i:R\hookrightarrow U$ a closed embedding in some smooth scheme $U$. Define the sheaf of ideals $I_{R, U}$ Then there is an exact sequence of sheaves of vector spaces on $R$: where $\iota_{R, U}$ is a morp

Theorems & Definitions (20)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 10 more