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$(-1)$-Shifted Darboux theorem of derived schemes in characteristic $p>2$

Jiaqi Fu

TL;DR

The paper develops a characteristic-$p$ version of the Brav–Bussi–Joyce Darboux theorem for $(-1)$-shifted symplectic derived schemes by introducing and leveraging derived infinitesimal cohomology and jet–Frobenius residues. It shows that, locally, $(-1)$-shifted infinitesimal symplectic forms admit Darboux charts modulo a residue, and that perturbations by higher $p$-powers can be controlled, yielding a conceptual, characteristic-free perspective on shifted Darboux theory. A key advancement is constructing a de Rham $2$-shifted symplectic form on the derived moduli stack of perfect complexes over Calabi–Yau $3$-folds in characteristic $p>2$, along with a conjectural infinitesimal structure for this form. The framework unifies derived foliations, square-zero extensions, and formal quotients to articulate local Darboux charts and demonstrates potential avenues to define Donaldson–Thomas-type invariants in positive characteristic. The paper also provides explicit examples via AKSZ-type formalisms and Lagrangian intersections, illustrating the practical reach of the theory in arithmetic-geometric settings.

Abstract

The derived geometry approach to Donaldson--Thomas theory (over $\mathbb{C}$) is built on Pantev--Toën--Vezzosi--Vaquié's existence theorem of $(-1)$-shifted symplectic forms \cite{pantev2013shifted} and Brav--Bussi--Joyce's shifted Darboux theorem \cite{brav2019darboux}. In this paper, we prove a Darboux theorem in characteristic $p>2$ for the $(-1)$-shifted symplectic forms endowed with an \textit{infinitesimal structure}. A key ingredient is Antieau's derived infinitesimal cohomology \cite{antieau2025filtrations}, which enjoys a Poincaré-type lemma. Our argument is in fact characteristic-free and provides a conceptual understanding of the Brav--Bussi--Joyce theorem. Moreover, we extend the existence theorem of Pantev--Toën--Vaquié--Vezzosi by constructing a de Rham $(-1)$-shifted symplectic form on $\operatorname{Map}_k(X,\underline{\operatorname{Perf}})$, where $X$ is a Calabi--Yau $3$-fold over a field $k$ in characteristic $p>2$. We conjecture that this $(-1)$-shifted symplectic form admits an infinitesimal structure.

$(-1)$-Shifted Darboux theorem of derived schemes in characteristic $p>2$

TL;DR

The paper develops a characteristic- version of the Brav–Bussi–Joyce Darboux theorem for -shifted symplectic derived schemes by introducing and leveraging derived infinitesimal cohomology and jet–Frobenius residues. It shows that, locally, -shifted infinitesimal symplectic forms admit Darboux charts modulo a residue, and that perturbations by higher -powers can be controlled, yielding a conceptual, characteristic-free perspective on shifted Darboux theory. A key advancement is constructing a de Rham -shifted symplectic form on the derived moduli stack of perfect complexes over Calabi–Yau -folds in characteristic , along with a conjectural infinitesimal structure for this form. The framework unifies derived foliations, square-zero extensions, and formal quotients to articulate local Darboux charts and demonstrates potential avenues to define Donaldson–Thomas-type invariants in positive characteristic. The paper also provides explicit examples via AKSZ-type formalisms and Lagrangian intersections, illustrating the practical reach of the theory in arithmetic-geometric settings.

Abstract

The derived geometry approach to Donaldson--Thomas theory (over ) is built on Pantev--Toën--Vezzosi--Vaquié's existence theorem of -shifted symplectic forms \cite{pantev2013shifted} and Brav--Bussi--Joyce's shifted Darboux theorem \cite{brav2019darboux}. In this paper, we prove a Darboux theorem in characteristic for the -shifted symplectic forms endowed with an \textit{infinitesimal structure}. A key ingredient is Antieau's derived infinitesimal cohomology \cite{antieau2025filtrations}, which enjoys a Poincaré-type lemma. Our argument is in fact characteristic-free and provides a conceptual understanding of the Brav--Bussi--Joyce theorem. Moreover, we extend the existence theorem of Pantev--Toën--Vaquié--Vezzosi by constructing a de Rham -shifted symplectic form on , where is a Calabi--Yau -fold over a field in characteristic . We conjecture that this -shifted symplectic form admits an infinitesimal structure.

Paper Structure

This paper contains 27 sections, 42 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Poincaré-type lemma.
  3. Jet-Frobenius residue.
  4. Infinitesimal structures and Darboux charts.
  5. Sheaf moduli.
  6. Higher Lie algebroids and Calabi--Yau $4$-folds.
  7. Arrangement of this paper.
  8. Acknowledgement.
  9. Derived geometry away from $\mathbb{Q}$
  10. Filtrations and derived rings
  11. Frobenius
  12. Derived algebraic geometry
  13. Formal derived schemes
  14. Shifted symplectic structures via infinitesimal cohomology
  15. Infinitesimal cohomology and foliations
  16. ...and 12 more sections

Key Result

Theorem 1.1

Assume that $B$ is a simplicial commutative ring which is almost of finite presentation over $k$. Then there is a natural equivalence of derived commutative rings where $B^{(n)}$ denotes the Frobenius twist relative to $k$. In particular, $\pi_{i}\widehat{\mathbbb{\Pi}}\mathstrut_{B/k}=0$ for $i\ne0,-1$.

Theorems & Definitions (121)

  • Theorem 1.1
  • Remark 1.2
  • Example 1.3
  • Remark 1.5
  • Theorem 1.6
  • Definition 2.2
  • Remark 2.4
  • Remark 2.6
  • Definition 2.8
  • Definition 2.9
  • ...and 111 more