Nonlinear Equations - UFRJ

46 [CH. 4: DIFFERENTIAL FORMS
Lemma 4.3. If A is an n × n matrix, then
√ n∧ n∑ −1
| det(A)| 2 dV =
2 A kiĀkj dz i ∧ d¯z j
Proof. As in exercise 4.2,
and
k=1 i,j=1
det(A) dz 1 ∧ · · · ∧ dz n =
det(A) d¯z 1 ∧ · · · ∧ d¯z n =
n∧
k=1 i=1
n∧
k=1 j=1
n∑
A ki dz i
n∑
Ā kj d¯z j .
The Lemma is proved by wedging the two expressions above and
multiplying by ( √ −1/2) n .
If U is an open subset of C n , then C ∞ (U, C) is the complex space
of all smooth complex valued functions of U. Here, smooth means
of class C ∞ and real derivatives are assumed. The holomorphic and
anti-holomorphic derivatives are defined as
and
∂f
∂z i
= 1 2
( ∂f
∂x i
− √ −1 ∂f
∂y i
)
∂f
= 1 ( ∂f
+ √ −1 ∂f )
∂¯z i 2 ∂x i ∂y i
The Cauchy-Riemann equations for a function f to be holomorphic
are just
∂f
∂¯z i
= 0.
We denote by ∂ : A kl (U) → A k+1,l (U) the holomorphic differential,
and by ¯∂ : A kl (U) → A k,l+1 (U) the anti-holomorphic differential.
If
∑
α(z) =
α i1,...,j l
(z) dz i1 ∧ · · · ∧ d¯z jl ,
1≤i 1