We prove a 'Darboux theorem' for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toen, Vaquie and Vezzosi arXiv:1111.3209. More precisely, we show that a derived scheme $X$ with symplectic form $\omega$ of degree $k$ is locally equivalent to (Spec $A,\omega'$) for Spec $A$ an affine derived scheme whose cdga $A$ has Darboux-like coordinates in which the symplectic form $\omega'$ is standard, and the differential in $A$ is given by Poisson bracket with a Hamiltonian function $H$ in $A$ of degree $k+1$. When $k=-1$, this implies that a $-1$-shifted symplectic derived scheme $(X,\omega)$ is Zariski locally equivalent to the derived critical locus Crit$(H)$ of a regular function $H:U\to{\mathbb A}^1$ on a smooth scheme $U$. We use this to show that the underlying classical scheme of $X$ has the structure of an 'algebraic d-critical locus', in the sense of Joyce arXiv:1304.4508. In the sequels arXiv:1211.3259, arXiv:1305.6428, arXiv:1312.0090, arXiv:1504.00690, 1506.04024 we will discuss applications of these results to categorified and motivic Donaldson-Thomas theory of Calabi-Yau 3-folds, and to defining new Donaldson-Thomas type invariants of Calabi-Yau 4-folds, and to defining 'Fukaya categories' of Lagrangians in algebraic symplectic manifolds using perverse sheaves, and we will extend the results of this paper and arXiv:1211.3259, arXiv:1305.6428 from (derived) schemes to (derived) Artin stacks, and to give local descriptions of Lagrangians in $k$-shifted symplectic derived schemes. Bouaziz and Grojnowski arXiv:1309.2197 independently prove a similar 'Darboux Theorem'.

中文翻译：

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具有移位辛结构的导出方案的 Darboux 定理

在 Pantev、Toen、Vaquie 和 Vezzosi arXiv:1111.3209 的意义上，我们证明了具有阶数 $k<0$ 的辛形式的派生方案的“Darboux 定理”。更准确地说，我们证明了具有辛形式 $\omega$ 的度数 $k$ 的派生方案 $X$ 在局部等效于 (Spec $A,\omega'$) 对于 Spec $A$ 是仿射派生方案，其 cdga $ A$ 具有类达布坐标，其中辛形式 $\omega'$ 是标准形式，$A$ 中的微分由泊松括号给出，带有哈密顿函数 $H$ 在 $A$ 的度数 $k+1$ . 当 $k=-1$ 时，这意味着 $-1$-shifted 辛派生方案 $(X,\omega)$ 是 Zariski 局部等价于正则函数 $H 的派生临界轨迹 Crit$(H)$ :U\to{\mathbb A}^1$ 在平滑方案 $U$ 上。我们用它来表明 $X$ 的基本经典方案具有“代数 d 临界轨迹”的结构，在 Joyce arXiv:1304.4508 的意义上。在续集 arXiv:1211.3259、arXiv:1305.6428、arXiv:1312.0090、arXiv:1504.00690、1506.04024 中，我们将讨论这些结果在分类和动机理论中的应用Calabi-Yau 4-folds 的类型不变量，并使用逆滑轮在代数辛流形中定义拉格朗日的“Fukaya 类别”，我们将把本文和 arXiv:1211.3259、arXiv:1305.6428 的结果从（导出）方案扩展到（导出）Artin 堆栈，并在 $k$-shifted 辛导出方案中给出拉格朗日量的局部描述。Bouaziz 和 Grojnowski arXiv:1309。