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A Darboux theorem for derived schemes with shifted symplectic structure

Authors: Christopher Brav, Vittoria Bussi and Dominic Joyce
Journal: J. Amer. Math. Soc. 32 (2019), 399-443
MSC (2010): Primary 14A20; Secondary 14F05, 14D23, 14N35
Published electronically: October 1, 2018
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Abstract: We prove a Darboux theorem for derived schemes with symplectic forms of degree $ k<0$, in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived scheme $ \boldsymbol {X}$ with symplectic form $ \tilde {\omega }$ of degree $ k$ is locally equivalent to $ (\mathop {\boldsymbol {\rm Spec}}\nolimits A,\omega )$ for $ \mathop {\boldsymbol {\rm Spec}}\nolimits A$ an affine derived scheme in which the cdga $ A$ has Darboux-like coordinates with respect to which the symplectic form $ \omega $ is standard, and in which the differential in $ A$ is given by a Poisson bracket with a Hamiltonian function $ \Phi $ of degree $ k+1$.

When $ k=-1$, this implies that a $ -1$-shifted symplectic derived scheme $ (\boldsymbol {X},\tilde {\omega })$ is Zariski locally equivalent to the derived critical locus $ \boldsymbol {\mathop {\rm Crit}(\Phi )}$ of a regular function $ \Phi :U\rightarrow {\mathbin {\mathbb{A}}}^1$ on a smooth scheme $ U$. We use this to show that the classical scheme $ X=t_0(\boldsymbol {X})$ has the structure of an algebraic d-critical locus, in the sense of Joyce.

In a series of works, the authors and their collaborators extend these results to (derived) Artin stacks, and discuss a Lagrangian neighbourhood theorem for shifted symplectic derived schemes, and applications 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.

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Additional Information

Christopher Brav
Affiliation: Faculty of Mathematics, Higher School of Economics, 7 Vavilova Street, Moscow, Russia

Vittoria Bussi
Affiliation: ICTP, Strada Costiera 11, Trieste, Italy

Dominic Joyce
Affiliation: The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom

Received by editor(s): December 20, 2013
Received by editor(s) in revised form: February 4, 2016, and July 13, 2018
Published electronically: October 1, 2018
Additional Notes: This research was supported by EPSRC Programme Grant EP/I033343/1.
Article copyright: © Copyright 2018 American Mathematical Society