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Low degree motivic Donaldson-Thomas invariants of the three-dimensional projective space

Anna M. Viergever

Abstract

Levine has constructed motivic analogues of virtual fundamental classes, living in cohomology of Witt sheaves. We use this to define motivic Donaldson-Thomas invariants $\tilde{I}_n$ for $\mathbb{P}^3$ over $\mathbb{R}$. We show that for $n$ odd, $\tilde{I}_n = 0$ and we compute $\tilde{I}_2 = 10, \tilde{I}_4 = 25$ and $\tilde{I}_6 = -50$. We then make a conjecture about the general case, which could be a motivic analogue of a classical theorem of Maulik-Nekrasov-Okounkov-Pandharipande. The results presented here also form a chapter in the authors thesis, which was submitted on May 30'th, 2023.

Low degree motivic Donaldson-Thomas invariants of the three-dimensional projective space

Abstract

Levine has constructed motivic analogues of virtual fundamental classes, living in cohomology of Witt sheaves. We use this to define motivic Donaldson-Thomas invariants for over . We show that for odd, and we compute and . We then make a conjecture about the general case, which could be a motivic analogue of a classical theorem of Maulik-Nekrasov-Okounkov-Pandharipande. The results presented here also form a chapter in the authors thesis, which was submitted on May 30'th, 2023.
Paper Structure (14 sections, 14 theorems, 93 equations)

This paper contains 14 sections, 14 theorems, 93 equations.

Table of Contents

  1. Motivic Donaldson-Thomas invariants of a smooth projective threefold
  2. Witt cohomology of $BN_S$ and Euler classes of canonical bundles
  3. Perfect obstruction theory
  4. Orientation
  5. Motivic Donaldson-Thomas invariants of $\mathbb{P}^3$
  6. Action of $N_S$ on $\mathbb{P}^3$
  7. Motivic Donaldson-Thomas invariants for $\mathbb{P}^3$
  8. Applying virtual localization
  9. Signs coming from the orientation
  10. Computing the trace
  11. Computations of $\tilde{I}_n$ for $n\leq 6$
  12. The computation for $n=2$
  13. The computation for $n=4$
  14. The computation for $n=6$

Key Result

Theorem 1

Let $X = \mathbb{P}^3_{\mathbb{R}}$. For $n\geq 0$, let $\tilde{I}_n$ be the quadratic degree of the motivic virtual fundamental class associated to $\text{Hilb}^n(X)$. Then $\tilde{I}_2 = 10, \tilde{I}_4 = 25$ and $\tilde{I}_6 = -50$.

Theorems & Definitions (38)

  • Theorem 1
  • Conjecture 2
  • Remark 1.4
  • Proposition 1.6: LevineMEC, Proposition 5.5
  • Theorem 1.7: LevineMEC, Theorem 7.1
  • Remark 1.8
  • Remark 1.9
  • Definition 1.10
  • Remark 1.11
  • Remark 1.12
  • ...and 28 more