#### Volume 19, issue 3 (2015)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications

### Oren Ben-Bassat, Christopher Brav, Vittoria Bussi and Dominic Joyce

Geometry & Topology 19 (2015) 1287–1359
##### Abstract

This is the fifth in a series of papers on the ‘$k$–shifted symplectic derived algebraic geometry’ of Pantev, Toën, Vaquié and Vezzosi. We extend our earlier results from (derived) schemes to (derived) Artin stacks. We prove four main results:

(a) If $\left(X,{\omega }_{X}\right)$ is a $k$–shifted symplectic derived Artin stack for $k<0$, then near each $x\in X$ we can find a ‘minimal’ smooth atlas $\phi :U\to X$, such that $\left(U,{\phi }^{\ast }\left({\omega }_{X}\right)\right)$ may be written explicitly in coordinates in a standard ‘Darboux form’.

(b) If $\left(X,{\omega }_{X}\right)$ is a $\left(-1\right)$–shifted symplectic derived Artin stack and $X={t}_{0}\left(X\right)$ the classical Artin stack, then $X$ extends to a ‘d–critical stack’ $\left(X,s\right)$, as by Joyce.

(c) If $\left(X,s\right)$ is an oriented d–critical stack, we define a natural perverse sheaf ${\stackrel{̌}{P}}_{X,s}^{\bullet }$ on $X$, such that whenever $T$ is a scheme and $t:T\to X$ is smooth of relative dimension $n$, $T$ is locally modelled on a critical locus $Crit\left(f:U\to {\mathbb{A}}^{1}\right)$, and ${t}^{\ast }\left({\stackrel{̌}{P}}_{X,s}^{\bullet }\right)\left[n\right]$ is modelled on the perverse sheaf of vanishing cycles ${\mathsc{P}\mathsc{V}}_{U,f}^{\bullet }$ of $f$.

(d) If $\left(X,s\right)$ is a finite-type oriented d–critical stack, we can define a natural motive ${MF}_{X,s}$ in a ring of motives ${\overline{\mathsc{ℳ}}}_{X}^{st,\stackrel{̂}{\mu }}$ on $X$, such that if $T$ is a scheme and $t:T\to X$ is smooth of dimension $n$, then $T$ is modelled on a critical locus $Crit\left(f:U\to {\mathbb{A}}^{1}\right)$, and ${\mathbb{L}}^{-n∕2}\odot {t}^{\ast }\left({MF}_{X,s}\right)$ is modelled on the motivic vanishing cycle ${MF}_{U,f}^{mot,\varphi }$ of $f$.

Our results have applications to categorified and motivic extensions of Donaldson–Thomas theory of Calabi–Yau $3$–folds.

##### Keywords
derived algebraic geometry, derived stack, shifted symplectic structure, perverse sheaf, vanishing cycles, motivic invariant, Calabi–Yau manifold, Donaldson–Thomas theory
##### Mathematical Subject Classification 2010
Primary: 14A20
Secondary: 14F05, 14D23, 14N35, 32S30