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, Toën, Vaquié and Vezzosi. We extend our earlier results from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If (X, ω_X) is a k –shifted symplectic derived Artin stack for k < 0, then near each x ∈ X we can find a ‘minimal’ smooth atlas φ: U → X, such that (U, φ*(ωX)) may be written explicitly in coordinates in a standard ‘Darboux form’. (b) If (X, ω_X) is a (-1)-shifted symplectic derived Artin stack and X = t₀(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 P^• _X,s on X, such that whenever T is a scheme and t W T → X is smooth of relative dimension n, T is locally modelled on a critical locus Crit(f : U → A¹), and t*(P^• _X,s)[n] is modelled on the perverse sheaf of vanishing cycles PV ^• _U,f 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 M^st,μ _X on X , such that if T is a scheme and t W T → X is smooth of dimension n, then T is modelled on a critical locus Crit(f : U → 𝔸¹), and 𝕃^-n/2 ⊙ t*(MF_X,s) is modelled on the motivic vanishing cycle MF^mot,Φ _U,f of f. Our results have applications to categorified and motivic extensions of Donaldson– Thomas theory of Calabi–Yau 3–folds.