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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.12584 |
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| _version_ | 1866915621305843712 |
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| author | Fu, Jiaqi |
| author_facet | Fu, Jiaqi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12584 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $(-1)$-Shifted Darboux theorem of derived schemes in characteristic $p>2$ Fu, Jiaqi Algebraic Geometry 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. |
| title | $(-1)$-Shifted Darboux theorem of derived schemes in characteristic $p>2$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2511.12584 |