A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Dominic Joyce; Pavel Safronov

doi:10.5802/afst.1616

Dominic Joyce ¹; Pavel Safronov ²

¹ The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
² Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 5, pp. 831-908.

Abstract
Abstract

Pantev, Toën, Vaquié and Vezzosi [19] defined $k$ -shifted symplectic derived schemes and stacks $X$ for $k \in ℤ$ , and Lagrangians $f : L \to X$ 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 $f : L \to X$ 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”.

Pantev, Toën, Vaquié et Vezzosi [19] ont défini des schémas et des champs dérivés symplectiques $k$ -décalés $X$ pour $k \in ℤ$ , et des Lagrangiens $f : L \to X$ 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 $f : L \to X$ 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 ».

Received: 2017-08-03
Accepted: 2017-08-29
Published online: 2020-04-23

DOI: 10.5802/afst.1616

Author's affiliations:

Dominic Joyce ¹; Pavel Safronov ²

¹ The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
² Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     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},
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Dominic Joyce; Pavel Safronov. A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 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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