A Darboux theorem for derived schemes with shifted symplectic structure

Abstract:

We prove a Darboux theorem for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived scheme $\mathbfit{X}$ with symplectic form $\tilde {\omega }$ of degree $k$ is locally equivalent to $(\operatorname{Spec} A,\omega)$ for $\operatorname{Spec} A$ an affine derived scheme in which the cdga $A$ has Darboux-like coordinates with respect to which the symplectic form $\omega$ is standard, and in which the differential in $A$ is given by a Poisson bracket with a Hamiltonian function $\Phi$ of degree $k+1$.

When $k=-1$, this implies that a $-1$-shifted symplectic derived scheme $(\mathbfit{X},\tilde {\omega })$ is Zariski locally equivalent to the derived critical locus $\operatorname{Crit}(\Phi )$ of a regular function $\Phi :U\rightarrow {\mathbb A}^1$ on a smooth scheme $U$. We use this to show that the classical scheme $X=t_0(\mathbfit{X})$ 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.