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

TL;DR

This paper proves a Darboux theorem for $(-1)$-shifted symplectic forms with infinitesimal structure in characteristic p>2, extending derived geometry tools to positive characteristic and providing new insights into shifted symplectic forms.

Contribution

It establishes a Darboux theorem in characteristic p>2 for $(-1)$-shifted symplectic forms with infinitesimal structure and extends the existence theorem to new geometric contexts.

Findings

Proved a Darboux theorem in characteristic p>2 for $(-1)$-shifted symplectic forms.

Constructed a de Rham $(-1)$-shifted symplectic form on mapping stacks involving Calabi-Yau 3-folds.

Provided a characteristic-free conceptual understanding of existing shifted Darboux theorems.

Abstract

The derived geometry approach to Donaldson--Thomas theory (over $\mathbb{C}$) is built on Pantev--To\"en--Vezzosi--Vaqui\'e'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 Poincar\'e-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--To\"en--Vaqui\'e--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.