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A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 5, pp. 831-908.

Pantev, Toën, Vaquié et Vezzosi [19] ont défini des schémas et des champs dérivés symplectiques k-décalés X pour k, et des Lagrangiens f:LX en eux. Ils ont des applications importantes pour la géomètrie Calabi–Yau et la quantification. Bussi, Brav et Joyce [7] et Bouaziz et Grojnowski [5] ont prouvé des « théorèmes de Darboux » donnant des modàles locaux précis Zariski ou étale pour les schémas dérivés symplectiques k-décalés X pour k<0, les présentant comme des fibrés cotangent décalés tordus.

Nous prouvons un « théorème de voisinage Lagrangien » donnant des modèles locaux précis Zariski ou étale pour les Lagrangiens f:LX dans les schémas dérivés symplectiques k-décalés X pour k<0, par rapport à la « forme Darboux » de Bussi–Brav–Joyce pour X. C’est-à-dire, localement, ces Lagrangiens peuvent être présentés sous forme de fibrés conormaux décalés tordus. Nous donnons aussi un résultat partiel lorsque k=0.

Nous espérons que nos résultats auront de futures applications à la géométrie de Poisson k-décalée de [12], à la définition de « catégories de Fukaya » de variétés symplectiques complexes ou algébriques, à la catégorification de la théorie de Donaldson–Thomas des variétés de Calabi–Yau de dimension 3, et au « Algèbres de Hall Cohomologiques ».

Pantev, Toën, Vaquié and Vezzosi [19] defined k-shifted symplectic derived schemes and stacks X for k, and Lagrangians f:LX in them. They have important applications to Calabi–Yau geometry and quantization. Bussi, Brav and Joyce [7] and Bouaziz and Grojnowski [5] proved “Darboux Theorems” giving explicit Zariski or étale local models for k-shifted symplectic derived schemes X for k<0 presenting them as twisted shifted cotangent bundles.

We prove a “Lagrangian Neighbourhood Theorem” which gives explicit Zariski or étale local models for Lagrangians f:LX in k-shifted symplectic derived schemes X for k<0, relative to the “Darboux form” local models of [7] for 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 shifted Poisson geometry [12], and to defining “Fukaya categories” of complex or algebraic symplectic manifolds, and to the categorification of Donaldson–Thomas theory of Calabi–Yau 3-folds and “Cohomological Hall Algebras”.

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DOI : https://doi.org/10.5802/afst.1616
@article{AFST_2019_6_28_5_831_0,
     author = {Dominic Joyce and Pavel Safronov},
     title = {A {Lagrangian} {Neighbourhood} {Theorem} for shifted symplectic derived schemes},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {831--908},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 28},
     number = {5},
     year = {2019},
     doi = {10.5802/afst.1616},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1616/}
}
Dominic Joyce; Pavel Safronov. A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 5, pp. 831-908. doi : 10.5802/afst.1616. https://afst.centre-mersenne.org/articles/10.5802/afst.1616/

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