LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
10.
Moment maps in algebraic setting
10.1.
Symplectic algebraic varieties.
An aﬃne algebraic variety
X
is said to be
Poisson
if
C
[
X
] is equipped with a Poisson bracket.
Exercise 10.1.
Let
A
be a commutative algebra and
B
be a localization of
A
.
Let
A
be
equipped with a bracket.
Show that there is a unique bracket on
B
such that the natural
homomorphism
A
→
B
respects the bracket.
Thanks to this exercise, the
sheaf
O
X
of regular functions on
X
acquires a bracket (i.e., we
have brackets on all algebras of sections and the restriction homomorphisms are compatible
with the bracket). We say that an arbitrary (=not necessarily aﬃne) variety
X
is Poisson if
the sheaf
O
X
comes equipped with a Poisson bracket.
Recall that on a variety
X
such that
O
X
is equipped with a bracket we have a bivector
(=a bivector field)
P
∈
Γ(
X
reg
,
∧
2
TX
reg
). This gives rise to a map
v
x
:
T
∗
x
X
→
T
x
X
for
x
∈
X
reg
, α
7→
P
x
(
α,
·
). We say that
P
is nondegenerate in
x
if this map is an isomorphism.
In this case, we can use this map to get a 2-form
ω
x
∈
∧
2
T
∗
x
X
:
ω
x
(
v
x
(
α
)
, v
x
(
β
)) =
P
x
(
α, β
) =
⟨
α, v
x
(
β
)
⟩
=
−⟨
v
x
(
α
)
, β
⟩
.
Now suppose
X
is smooth.
Suppose that
P
is non-degenerate (=non-degenerate at all
points).
So we have a non-degenerate form
ω
on
X
.
The condition that
P
is Poisson is
equivalent to
dω
= 0. A non-degenerate closed form
ω
is called
symplectic
(and
X
is called
a
symplectic variety
).
The most important for us class of symplectic varieties is cotangent bundles. Let
X
0
be
a smooth algebraic variety, set
X
:=
T
∗
X
0
.
A symplectic form
ω
on
X
is introduced as
follows. First, let us introduce a canonical 1-form
α
. We need to say how
α
x
pairs with a
tangent vector for any
x
∈
X
. A point
X
can be thought as a pair (
x
0
, β
), where
x
0
∈
X
0
and
β
∈
T
∗
x
0
X
0
. Consider the projection
π
:
X
X
0
(defined by
π
(
x
) =
x
0
). For
x
= (
x
0
, β
)
we define
α
x
by
⟨
α
x
, v
⟩
=
⟨
β, d
x
π
(
v
)
⟩
.
We can write
α
in “coordinates”. If we worked in the
C
∞
- or analytic setting, we could
use the usual coordinates. However, we cannot do this because we want to show that
α
is an
algebraic form. So we will use an algebro-geometric substitute for coordinate charts: ´
etale
coordinates. Namely, we can introduce ´
etale coordinates in a neighborhood of each point
x
0
∈
X
0
. Let us choose functions
x
1
, . . . , x
n
with a property that
d
x
0
x
1
, . . . d
x
0
x
n
form a basis
in
T
∗
x
0
X
0
. Then
dx
1
, . . . , dx
n
are linearly independent at any point from some neighborhood
X
0
0
of
x
0
. So the map
φ
:
X
0
0
→
C
n
given by (
x
1
, . . . , x
n
) is ´
etale and we call
x
1
, . . . , x
n
´
etale
coordinates. Then we can get ´
etale coordinates
y
1
, . . . , y
n
on
T
∗
X
0
0
as follows: by definition
y
i
(
x
0
, β
) is the coeﬃcient of
d
x
0
x
i
in
β
, i.e.,
β
=
∑
n
i
=1
y
i
(
x
0
, β
)
d
x
0
x
i
(and we view
x
1
, . . . , x
n
as functions on
T
∗
X
0
0
via pull-back). Then, on
T
∗
X
0
0
,
α
is given by
∑
n
i
=1
y
i
dx
i
.