A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Journal article

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

Authors: Joyce, D; Safronov, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Université Paul Sabatier
• Journal: Annales de la Faculte des Sciences de Toulouse
• Volume: 28
• Issue: 5
• Pages: 831-908
• Publication date: 2020-04-23
• Acceptance date: 2017-08-30
• DOI: 10.5802/afst.1616
• Language: English
• Keywords: math.AG