This is the fifth in a series of papers [15], [4], [3], [5] on the 'k-shifted symplectic derived algebraic geometry' of Pantev, Toën, Vaquié and Vezzosi [30]. This paper extends the results of [4], [3], [5] from (derived) schemes to (derived) Artin stacks. We prove four main results:(a) If (X, ωX ) is a k-shifted symplectic derived Artin stack for k < 0 in the sense of [30], then near each x ∈ X we can find a 'minimal' smooth t * M F X,s = L n/2

Let $U$ be a smooth $\mathbb C$-scheme, $f:U\to\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\mathbb C$-subscheme of $U$. Then one can define the "perverse sheaf of vanishing cycles" $PV_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $\Phi:U\to U$ is an isomorphism with $f\circ\Phi=f$ and $\Phi\vert_X=$id$_X$. Then $\Phi$ induces an isomorphism $\Phi_*:PV_{U,f}\to PV_{U,f}$. We show that $\Phi_*$ is multiplication by det$(d\Phi\vert_X)=1$ or $-1$. (b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$, for $X^{(3)}$ the third-order thickening of $X$ in $U$, and $f^{(3)}=f\vert_{X^{(3)}}:X^{(3)}\to\mathbb A^1$. (c) If $U,V$ are smooth $\mathbb C$-schemes, $f:U\to\mathbb A^1$, $g:V\to\mathbb A^1$ are regular, $X=$Crit$(f)$, $Y=$Crit$(g)$, and $\Phi:U\to V$ is an embedding with $f=g\circ\Phi$ and $\Phi\vert_X:X\to Y$ an isomorphism, there is a natural isomorphism $\Theta_\Phi:PV_{U,f}\to\Phi\vert_X^*(PV_{V,g})\otimes_{\mathbb Z_2}P_\Phi$, for $P_\Phi$ a natural principal $\mathbb Z_2$-bundle on $X$. (d) If $(X,s)$ is an oriented d-critical locus in the sense of Joyce arXiv:1304.4508, there is a natural perverse sheaf $P_{X,s}$ on $X$, such that if $(X,s)$ is locally modelled on Crit$(f:U\to\mathbb A^1)$ then $P_{X,s}$ is locally modelled on $PV_{U,f}$. We also generalize our results to replace $U,X$ by complex analytic spaces, and $PV_{U,f}$ by $\mathcal D$-modules, or mixed Hodge modules. We discuss applications of (d) to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to defining a 'Fukaya category' of Lagrangians in a complex symplectic manifold using perverse sheaves. This is the third in a series of papers arXiv:1304.4508, arXiv:1305.6302, arXiv:1305.6428, arXiv:1312.0090, arXiv:1403.2403, arXiv:1404.1329, arXiv:1504.00690.Comment: 77 pages, LaTeX. (v4) corrections, new Appendix by Joerg Schuerman

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We prove a Darboux theorem for derived schemes with symplectic forms of degree k < 0, in the sense of Pantev, Toën, Vaquié and Vezzosi [26]. More precisely, we show that a derived scheme X with symplectic formω of degree k is locally equivalent to (Spec A, ω) for Spec A an affine derived scheme in which the cdga A has Darboux-like coordinates with respect to which the symplectic form ω is standard, and in which the differential in A is given by a Poisson bracket with a Hamiltonian function Φ of degree k + 1.When k = −1, this implies that a −1-shifted symplectic derived scheme (X,ω) is Zariski locally equivalent to the derived critical locus Crit(Φ) of a regular function Φ : U → A 1 on a smooth scheme U . We use this to show that the classical scheme X = t0(X) has the structure of an algebraic d-critical locus, in the sense of Joyce [18].In the sequels [3,5, 7,9,19] we 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.