COMPLEX GEOMETRY Course notes

Chapter 3
COMPLEX MANIFOLDS
3.1 Complex manifolds and forms
Recall that for a smooth R-manifold M, there is an ideal I(x) for each x ∈ M, given by
I(x) = {f ∈ C ∞ (M) / f(x) = 0} ↩→ C ∞ (M)
The cotangent plane at x ∈ M can be defined as the quotient
T ∨ x (M) := I(x)/I(x) 2
and the tangent plane at x ∈ M is simply the dual space of the cotangent plane T ∨ x (X), i.e.,
T x (M) := (T ∨ x (M)) ∨
Definition 3.1.1. An almost complex structure on an R-differentiable manifold X of dim R = 2n is an
epimorphism J of T X such that J 2 = −1. Or equivalently, it is the structure of a complex vector bundle on
T X.
A complex structure on X induces an almost complex structure on X by setting J = i = √ −1. We obtain a
map J : T X,R −→ T X,R with √ −1 acting on the domain and J acting in the codomain. We have
∂
∂z = 1 ( ∂
− i ∂ ) ( ) ∂ ∂
↦→ ,
2 ∂x i ∂y i ∂x i ∂y i
Locally, J is defined by
( ) ( ∂ ∂ ∂
, ↦→ , − ∂ )
∂x i ∂y i ∂y i ∂x i
corresponding to
the eigen-value i, an an eigen-value T 0,1
X
corresponding to the eigen-value −i, for the operator J. Note that
is naturally isomorphic to T X,R by taking the real part, and this isomorphism identifies i with J. Hence
is generated by vectors of the form u − iJu, with u ∈ T X,R.
Let (X, J) be an almost complex manifold. Then T X,R ⊗ C contains an eigen-bundle T 1,0
X
T 1,0
X
T 1,0
X
Theorem 3.1.1. A complex manifold has a complex structure J on T X,R and its associated subbundle
⊆ T X,R ⊗ C is naturally the same as T X (by taking the real part).
T 1,0
X
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