A 'Darboux theorem' for derived schemes with shifted symplectic structure

Author: 

Brav, C
Bussi, V
Joyce, D

Publication Date: 

1 October 2018

Journal: 

Journal of the American Mathematical Society

Last Updated: 

2021-04-27T18:58:00.643+01:00

Volume: 

32

DOI: 

10.1090/jams/910

page: 

399-443

abstract: 

We prove a 'Darboux theorem' for derived schemes with symplectic forms of
degree $k<0$, in the sense of Pantev, Toen, Vaquie and Vezzosi arXiv:1111.3209.
More precisely, we show that a derived scheme $X$ with symplectic form $\omega$
of degree $k$ is locally equivalent to (Spec $A,\omega'$) for Spec $A$ an
affine derived scheme whose cdga $A$ has Darboux-like coordinates in which the
symplectic form $\omega'$ is standard, and the differential in $A$ is given by
Poisson bracket with a Hamiltonian function $H$ in $A$ of degree $k+1$.
When $k=-1$, this implies that a $-1$-shifted symplectic derived scheme
$(X,\omega)$ is Zariski locally equivalent to the derived critical locus
Crit$(H)$ of a regular function $H:U\to{\mathbb A}^1$ on a smooth scheme $U$.
We use this to show that the underlying classical scheme of $X$ has the
structure of an 'algebraic d-critical locus', in the sense of Joyce
arXiv:1304.4508.
In the sequels arXiv:1211.3259, arXiv:1305.6428, arXiv:1312.0090,
arXiv:1504.00690, 1506.04024 we will discuss applications of these results 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, and we will extend the results of this paper and
arXiv:1211.3259, arXiv:1305.6428 from (derived) schemes to (derived) Artin
stacks, and to give local descriptions of Lagrangians in $k$-shifted symplectic
derived schemes.
Bouaziz and Grojnowski arXiv:1309.2197 independently prove a similar 'Darboux
Theorem'.

Symplectic id: 

401038

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Submitted to ORA: 

Submitted

Publication Type: 

Journal Article