A Darboux theorem for derived schemes with shifted symplectic structure

Abstract

We prove a Darboux theorem for derived schemes with symplectic forms of degree k > 0 k>0 , in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived scheme X \mathbfit {X} with symplectic form ω ~ \tilde {\omega } of degree k k is locally equivalent to ( Spec ⁡ A , ω ) (\operatorname {Spec} A,\omega ) for Spec ⁡ A \operatorname {Spec} A an affine derived scheme in which the cdga A A has Darboux-like coordinates with respect to which the symplectic form ω \omega is standard, and in which the differential in A A is given by a Poisson bracket with a Hamiltonian function Φ \Phi of degree k + 1 k+1 . When k = − 1 k=-1 , this implies that a − 1 -1 -shifted symplectic derived scheme ( X , ω ~ ) (\mathbfit {X}, \tilde {\omega }) is Zariski locally equivalent to the derived critical locus Crit ⁡ ( Φ ) \operatorname {Crit}(\Phi ) of a regular function Φ : U → A 1 \Phi :U\rightarrow \mathbb {A}^1 on a smooth scheme U U . We use this to show that the classical scheme X = t 0 ( X ) X=t_0(\mathbfit {X}) has the structure of an algebraic d-critical locus, in the sense of Joyce. In a series of works, the authors and their collaborators 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.