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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
DOI: https://doi.org/10.1090/jams/910
Published electronically: October 1, 2018
MathSciNet review: 3904157
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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 $\mathbfit{X}$ with symplectic form $\tilde {\omega }$ of degree $k$ is locally equivalent to $(\operatorname{Spec} A,\omega)$ for $\operatorname{Spec} A$ 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 $(\mathbfit{X},\tilde {\omega })$ is Zariski locally equivalent to the derived critical locus $\operatorname{Crit}(\Phi )$ of a regular function $\Phi :U\rightarrow {\mathbb A}^1$ on a smooth scheme $U$. We use this to show that the classical scheme $X=t_0(\mathbfit{X})$ 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
MR Author ID: 867687
Email: chris.i.brav@gmail.com

Vittoria Bussi
Affiliation: ICTP, Strada Costiera 11, Trieste, Italy
MR Author ID: 1093359
Email: vbussi@ictp.it

Dominic Joyce
Affiliation: The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
MR Author ID: 306920
Email: joyce@maths.ox.ac.uk

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