arXiv 1506.040242015-06math.AG

移位辛导出格式的拉格朗日邻域定理

A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Dominic Joyce, Pavel Safronov

备注：: 68 pages
发表信息：: Ann. Fac. Sci. Toulouse Math. 28 (2019), pp. 831-908
DOI：: 10.5802/afst.1616

摘要：

Pantev，Toen，Vaquié和Vezzosi arXiv：1111.3209定义了$k$ -移位辛导出格式，并将$k\in\mathbb Z$和拉格朗日${\bf f}:{\bf L}\to{\bf X}$叠加在一起.它们在卡-丘几何和量子化中有重要的应用。Bussi、Brav和Joyce arXiv：1305.6302证明了一个“达布定理”，给出了$k$的显式Zariski或étale局部模型-用于$k<0$的移位辛推导方案${\bf X}$，将它们表示为扭曲移位余切丛。我们证明了一个“拉格朗日邻域定理”给出明确的Zerki或etale局部模型拉格朗日${\bf f}:{\bf L}\to{\bf X}$在$k$ -移位辛衍生计划${\bf X}$的$k<0$，相对于Bussi-Brav-Joyce的“达布形式”局部模型${\bf X}$。也就是说，局部的拉格朗日量可以表示为扭曲移位的余法丛。当$k=0$ .我们希望我们的结果将有未来的应用$k$ -移位泊松几何（见arXiv：1506.03699），以定义“类”的复杂或代数辛流形，并归类唐纳森-托马斯理论的Calabi-Yau 3-folds和“上同调霍尔代数”。

英文摘要：

Pantev, Toen, Vaquié and Vezzosi arXiv:1111.3209 defined $k$-shifted symplectic derived schemes and stacks ${\bf X}$ for $k\in\mathbb Z$, and Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in them. They have important applications to Calabi-Yau geometry and quantization. Bussi, Brav and Joyce arXiv:1305.6302 proved a 'Darboux Theorem' giving explicit Zariski or étale local models for $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$ presenting them as twisted shifted cotangent bundles. We prove a 'Lagrangian Neighbourhood Theorem' giving explicit Zariski or etale local models for Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$, relative to the Bussi-Brav-Joyce 'Darboux form' local models for ${\bf X}$. That is, locally such Lagrangians can be presented as twisted shifted conormal bundles. We also give a partial result when $k=0$. We expect our results will have future applications to $k$-shifted Poisson geometry (see arXiv:1506.03699), to defining 'Fukaya categories' of complex or algebraic symplectic manifolds, and to categorifying Donaldson-Thomas theory of Calabi-Yau 3-folds and 'Cohomological Hall algebras'.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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