Nonlinear Equations - UFRJ

[SEC. 4.2: COMPLEX DIFFERENTIAL FORMS 45
switch to another convention: if z is a complex number, x is its real
part and y its imaginary part. This convention extends to vectors so
z = x + √ −1 y.
The sets C n and R 2n may be identified by
⎡ ⎤
x 1
y 1
z =
x 2
.
⎢ ⎥
⎣ . ⎦
y n
It is possible to define alternating k-forms in C n as complex-valued
alternating k-forms in R 2n . However, this approach misses some of
the structure related to the linearity over C and holomorphic functions.
Instead, it is usual to define A k0 as the space of complex valued
alternating k-forms in C n . A basis for A k0 is given by the expressions
dz i1 ∧ · · · ∧ dz ik , 1 ≤ i 1 < i 2 < · · · < i k ≤ n.
They are interpreted as
dz i1 ∧ · · · ∧ dz ik (u 1 , . . . , u k ) = ∑
σ∈S k
(−1) |σ| u σ(1)i1 u σ(2)i2 · · · u σ(k)ik .
Notice that dz i = dx i + √ −1 dy i . We may also define d¯z i =
dx i − √ −1 dy i . Next we define A kl as the complex vector space
spanned by all the expressions
dz i1 ∧ · · · ∧ dz ik ∧ d¯z j1 ∧ · · · ∧ d¯z jl
for 1 ≤ i 1 < i 2 < · · · < i k ≤ n, 1 ≤ j 1 < j 2 < · · · < j l ≤ n. Since
dx i ∧ dy i = −2 √ −1 dz i ∧ d¯z i ,
the standard volume form in C n is
(√ ) n
−1
dV = dx 1 ∧ dy 1 ∧ · · · ∧ dy n = dz 1 ∧ d¯z 1 ∧ · · · ∧ d¯z n .
2
The following fact is quite useful: