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.