Nonlinear Equations - UFRJ

48 [CH. 4: DIFFERENTIAL FORMS
2. closed: dω z ≡ 0.
The canonical Kähler form in C n is
√ −1
ω =
2 dz 1 ∧ d¯z 1 +
√ −1
2 dz 2 ∧ d¯z 2 + · · · +
√ −1
2 dz n ∧ d¯z n .
Given a Kähler form, its volume form can be written as
dV z = 1 n! ω z ∧ ω z ∧ · · · ∧ ω
} {{ z .
}
n times
The definition above is for a Kähler structure on a subset of C n .
This definition can be extended to a complex manifold, or to a 2nmanifold
where a ‘complex multiplication’ J : T z M → T z M , J 2 =
−I, is defined.
An amazing fact about Kähler manifolds is the following.
Theorem 4.5 (Wirtinger). Wirtinger Let S be a d-dimensional complex
submanifold of a Kähler manifold M. Then it inherits its Kähler
form, and
Vol(S) = 1 ∫
ω z ∧ · · · ∧ ω z .
d! S
} {{ }
d times
Since ω is a closed form, ω∧· · ·∧ω is also closed. When S happens
to be a boundary, its volume is zero.
4.4 The co-area formula
Definition 4.6. A smooth (real, complex) fiber bundle is a tuple
(E, B, π, F ) such that
1. E is a smooth (real, complex) manifold (known as total space).
2. B is a smooth (real , complex) manifold (known as base space).
3. π : E ↦→ B is a smooth surjection (the projection).
4. F is a (real, complex) smooth manifold (the fiber).