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Chapter 1
Basic symplectic geometry
This chapter is an introduction on symplectic geometry. Symplectic geometry has its origin in
classical mechanics. Many important geometry problems can be naturally formulated in the
context of symplectic geometry, thus it is also a widely useful language in mathematic physics,
representation theory etc. Since 1970’s, after Kostant and Souriau introduced the geometric
quantization, symplectic geometry became an independent mathematic subject which is an ex-
tension of complex geometry. Complex geometry is a classical and still very active area, and
K¨ahler manifolds in complex geometry are naturally symplectic manifolds which belong to a
large class of manifolds: Poisson manifolds. These three classes of manifolds are basic objects of
this chapter.
We start in Section 1.1 the definition on the symplectic vector spaces and show the space
of its compatible complex structures is contractible. More precisely, we construct a smooth
surjective map from the space of metrics on the vector space to the space of its compatible
complex structures, this allows us to extend it easily to the symplectic vector bundles case.
In Section 1.2, after recall basic facts on differential manifolds, we explain the Moser’s trick
which is very useful to treat the problems on the existence of certain diffeomorphisms and as
applications, we establish the Darboux theorem which explain locally, any symplectic manifold
is same as a symplectic vector space, thus any possible symplectic invariant should be of a global
nature. In Section 1.3, we explain the Poisson structure on a symplectic manifold and give a brief
introduction on Poisson manifolds. In Section 1.4, we recall the definition of a K¨ahler manifold.
3
CHAPTER1. BASICSYMPLECTICGEOMETRY 4
1.1 Linear symplectic geometry
This section s a continuation of linear algebra. We explain basic facts on symplectic vector
spaces, compatible complex structures and symplectic groups.
1.1.1 Symplectic vector spaces
Let V be a real vector space of dimension m. We will denote by V ∗ its dual space, and for k ∈ N,
k ∗
let Λ V be the space of antisymmetric (i.e., alternating) multilinear mappings from V × ··· × V
| {z }
to R. Certainly, for k > m, we have k times
k ∗
Λ V =0. (1.1.1)
Weget easily that
0 ∗ 1 ∗ ∗ m ∗
Λ V =R, Λ V =V , dimΛ V =1. (1.1.2)
Anonvanishing element of ΛmV∗ defines an orientation of V. The antisymmetric multiplication
k ∗ r ∗
for α ∈ Λ V , β ∈ Λ V is defined by: for v1,...,vk+r ∈ V,
1 X |σ|
(α∧β)(v ,...,v ) := (−1) α(v , . . . , v )β(v , . . . , v ), (1.1.3)
1 k+r k!r! σ(1) σ(k) σ(k+1) σ(k+r)
σ∈Sk+r
• ∗ m k ∗
and |σ| is the sign of σ ∈ Sk+r, the (k + r)-th permutation group. Then Λ V = ⊕ Λ V
k=0
• ∗
becomes an algebra with its Z-grading induced by its degree, and Λ V is called the exterior
∗ j m ∗ k ∗
algebra of V . Any basis {e }j=1 of V induces a following basis of Λ V :
I i1 ik
e :=e ∧···∧e for I ={1≤i <··· 0 if u 6= 0. (1.1.5)
Definition 1.1.1. We say (V,ω) is a symplectic vector space if V is a finite dimensional real
vector space, and ω : V × V → R is a nondegenerate antisymmetric bilinear form. In this case,
we call ω a symplectic form on V.
Definition 1.1.2. Let (V ,ω ), (V ,ω ) be two symplectic vector spaces. A linear map φ : V →
1 1 2 2 1
V2 is called symplectic, if
ω =φ∗ω :=ω (φ·,φ·). (1.1.6)
1 2 2
If the linear map φ : V → V is symplectic, then as ω is nondegenerate, φ is injective. If φ
1 2 1
is also an isomorphism, we call that φ is a symplectic isomorphism.
Proposition 1.1.3. If (V,ω) is a symplectic vector space of dimension m, then m is even and
m/2 m ∗
ω ∈Λ V is nonvanishing which defines an orientation of V. Moreover, the map
v ∈ V → ω(v,·) ∈ V∗ (1.1.7)
is an isomorphism.
CHAPTER1. BASICSYMPLECTICGEOMETRY 5
Proof. Let h·,·i be a scalar product on V . Then there exists an antisymmetric invertible endo-
morphism A ∈ End(V) such that
ω(·,·) = h·,A·i. (1.1.8)
As
t m
detA=det(A )=(−1) detA, (1.1.9)
thus m is even.
If h·, ·i′ is another scalar product on V , and A′ is the corresponding antisymmetric invertible
endomorphism. Then there is P ∈ GL(V) such that PAPt = A′. Thus detA and detA′ have
the same signature. This means V has a canonical orientation. In fact, this is equivalent to
m/2 m ∗ m/2
ω ∈Λ V andω 6=0. ∗ ∗
As ω is nondegenerate, the map v ∈ V → ω(v,·) ∈ V is injective. As dim V = dim V ,
R R
(1.1.7) is an isomorphism. The proof of Proposition 1.1.3 is completed.
The basic example is the following. In fact, as we will see in Theorem 1.1.15, it is the only
symplectic vector space.
Example 1.1.4. Let L be a vector space. Then L ⊕ L∗ is a symplectic vector space with a
∗
symplectic form ωL⊕L defined by: for (l ,l∗), (l ,l∗) ∈ L⊕L∗,
1 1 2 2
∗
ωL⊕L ((l ,l∗),(l ,l∗)) = (l ,l∗) − (l ,l∗), (1.1.10)
1 1 2 2 1 2 2 1
∗ ∗ n n∗
here we denote by (l ,l ) := l (l ). In particular, if we identify R with R by the canonical
1 2 2 1
n t t n
scalar product of R defined by: for x = (x1,...,xn) , y = (y1,...,yn) ∈ R ,
n
hx,yi = Xx y . (1.1.11)
i i
i=1
n n∗
We call R2n,ω := (Rn ⊕ Rn∗,ωR ⊕R ) the standard symplectic space. Sometimes, we also
0
denote by ωst the canonical symplectic form ω0.
From now on, let (V,ω) be a symplectic vector space. For W ⊂ V a linear subspace, let
⊥
W ω ={v∈V :ω(v,w)=0, for all w ∈W}, (1.1.12)
be the ω-orthogonal complement of W. We denote by v⊥ u for u,v ∈ V if ω(u,v) = 0. In the
ω
same way, u⊥ωW for W ⊂ V if ω(u,v) = 0 for any v ∈ W.
Definition 1.1.5. For W a linear subspace of a symplectic vector space (V,ω), we call
⊥
1. W is symplectic if W ∩W ω = 0;
⊥
2. W is isotropic if W ⊂ W ω;
⊥
3. W is coisotropic if W ω ⊂ W;
⊥
4. W is Lagrangian if W = W ω.
Proposition 1.1.6. For W a linear subspace of (V,ω), we have
⊥
⊥ ⊥ ω
dimW +dimW ω =dimV, W ω =W. (1.1.13)
⊥
If W is symplectic, then W ω is also symplectic and we have the direct decomposition of sym-
plectic vector spaces
⊥
(V,ω) = (W,ω| )⊕(W ω,ω| ⊥ ). (1.1.14)
W W ω
CHAPTER1. BASICSYMPLECTICGEOMETRY 6
⊥ ⊥
Proof. Let h, i be a scalar product on V . Let A ∈ End(V) as in (1.1.8). Then W ω = (AW) .
Hence,
⊥ ⊥
dimW ω =dim(AW) =dimV −dim(AW). (1.1.15)
As A is invertible, by (1.1.15), we get the first equation of (1.1.13), in particular, we have
⊥ ⊥
⊥ ω ⊥ ω
dimW = dim W ω . But by (1.1.12), we have W ⊂ W ω . This means the second
equation of (1.1.13) holds.
⊥
⊥ ⊥ ω ⊥ ⊥
If W is symplectic, then W ω ∩ W ω =W ω∩W={0},thusW ω issymplectic. Now
we get (1.1.14) by the first equation of (1.1.13).
The proof of Proposition 1.1.6 is completed.
Proposition 1.1.7. If (V,ω) is a linear symplectic space of dimension 2n. Then there exists
e ,f , ..., e , f a basis of V , such that, for 1 6 i,j 6 n,
1 1 n n
ω(e ,f ) = δ , ω(e ,e ) = 0, ω(f ,f ) = 0. (1.1.16)
i j ij i j i j
This basis will be called a symplectic basis of V .
Proof. We shall prove this proposition by induction on dimV/2. If dimV = 0, certainly it holds.
Wesuppose dimV > 2 and the proposition is true for the symplectic vector space of dimension
smaller than dimV −2.
Let e1 ∈ V\{0}. As ω is nondegenerate, there is f1 ∈ V such that
ω(e ,f ) = 1. (1.1.17)
1 1
Set W = Re ⊕Rf . By Proposition 1.1.6, we have a ω-orthogonal decomposition
1 1
⊥
V =W⊕W ω. (1.1.18)
⊥
By the induction hypotheses, we have a symplectic basis e ,f ,...,e ,f on W ω. Hence, {e ,
2 2 n n 1
f1, ..., en, fn} is a symplectic basis of V .
The proof of Proposition 1.1.7 is completed.
Wegive two applications of the symplectic basis.
Corollary 1.1.8. Let ω,ω′ be two symplectic forms on V. Then there exists A ∈ GL(V) such
that
ω′(A·,A·) = ω(·,·). (1.1.19)
′ ′ ′
Proof. Let {e ,f } (resp. {e ,f }) be a symplectic basis of (V,ω) (resp. (V,ω )). Let A ∈ End(V )
i j i j
be defined by
Ae =e′, Af =f′. (1.1.20)
i i j j
As (e ,f ),(e′,f′) are bases of V , A is invertible. Moreover,
i j i j
′ ′ ′ ′ ′ ′ ′ ′
ω (Ae ,Ae ) = ω (e ,e ) = 0, ω (Af ,Af ) = ω (f ,f ) = 0, (1.1.21)
i j i j i j i j
ω′(Ae ,Af ) = ω′(e′,f′) = δ .
i j i j ij
This means (1.1.19) holds. The proof of Corollary 1.1.8 is completed.
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