A 'Darboux Theorem' for shifted symplectic structures on derived Artin
stacks, with applications

This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302,
arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived
algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This
paper extends the previous three from (derived) schemes to (derived) Artin
stacks. We prove four main results:
(a) If $(X,\omega)$ is a $k$-shifted symplectic derived Artin stack for $k<0$
in the sense of arXiv:1111.3209, then near each $x\in X$ we can find a
'minimal' smooth atlas $\varphi:U\to X$ with $U$ an affine derived scheme, such
that $(U,\varphi^*(\omega))$ may be written explicitly in coordinates in a
standard 'Darboux form'.
(b) If $(X,\omega)$ is a $-1$-shifted symplectic derived Artin stack and $X'$
the underlying classical Artin stack, then $X'$ extends naturally to a
'd-critical stack' $(X',s)$ in the sense of arXiv:1304.4508.
(c) If $(X,s)$ is an oriented d-critical stack, we can define a natural
perverse sheaf $P^\bullet_{X,s}$ on $X$, such that whenever $T$ is a scheme and
$t:T\to X$ is smooth of relative dimension $n$, then $T$ is locally modelled on
a critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and
$t^*(P^\bullet_{X,s})[n]$ is locally modelled on the perverse sheaf of
vanishing cycles $PV_{U,f}^\bullet$ of $f$.
(d) If $(X,s)$ is a finite type oriented d-critical stack, we can define a
natural motive $MF_{X,s}$ in a ring of motives $\bar{\mathcal
M}^{st,\hat\mu}_X$ on $X$, such that whenever $T$ is a finite type scheme and
$t:T\to X$ is smooth of dimension $n$, then $T$ is locally modelled on a
critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and ${\mathbb
L}^{-n/2}\odot t^*(MF_{X,s})$ is locally modelled on the motivic vanishing
cycle $MF^{mot,\phi}_{U,f}$ of $f$ in $\bar{\mathcal M}^{st,\hat\mu}_T$.
Our results have applications to categorified and motivic extensions of
Donaldson-Thomas theory of Calabi-Yau 3-folds