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

This is the fifth in a series of papers on the 'k-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi. We extend our earlier results from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If (X, omega(X)) is a k-shifted symplectic derived Artin stack for k < 0, then near each x is an element of X we can find a 'minimal' smooth atlas phi: U -> X, such that (U, phi*(omega(X))) may be written explicitly in coordinates in a standard 'Darboux form'. (b) If (X, omega(X)) is a (-1)-shifted symplectic derived Artin stack and X = t(0)(X) the classical Artin stack, then X extends to a 'd-critical stack' (X, s), as by Joyce. (c) If (X, s) is an oriented d-critical stack, we define a natural perverse sheaf (sic)(X,s)(center dot) on X, such that whenever T is a scheme and t : T -> X is smooth of relative dimension n, T is locally modelled on a critical locus Crit (f: U -> A(1)), and t*((sic)(X, s)* )[n] is modelled on the perverse sheaf of vanishing cycles P nu(center dot)(U,f) of f. (d) If (X, s) is a finite-type oriented d-critical stack, we can define a natural motive MFX,s in a ring of motives (M) over bar (st,(mu) over cap)(X) on X, such that if T is a scheme and t: T -> X is smooth of dimension n, then T is modelled on a critical locus Crit (f: U -> A(1)), and L-n/2 circle dot t* (MFX,s) is modelled on the motivic vanishing cycle MFU,Fmot,phi of f. Our results have applications to categorified and motivic extensions of DonaldsonThomas theory of Calabi-Yau 3-folds.