6 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
REMARK. If I is holomorphic, a(u, v)/a(x, y) = Idet(al.,jaz.)I
Z ~ o.
Proof We write it out for n = 2 and leave the general case to the reader.
We use the Cauchy-Riemann equations
and set
a.A = aUA/aX. = aVA/ay.,
bVA = aVA/aX. = -au}../ay •. Then
au, av, oUz avz
all bll al2 bl2
ax, ax, ax, ax,
av,
=
-b'l all -bJ2 al2
aUI aU2 avz
ay,
aYI ay, aYI
a21 b21 a22 b22
-b21 a2'
-b22 a22
We perform the following sequence of operations: Multiply column 2 by i and
add it to column I ; do the same with columns 4 and 3. Then multiply row 1
by i and subtract it from row 2; do the same with rows 3 and 4. Making use
of the fact that B.A = aIA/aZ. = a.A + ib.A, we get
gil gl2
* *
a(u, v) gZI g22
* * = Idet(g.A)1
2 --=
0 0 gl2 o(x, y) gil
0 0 gz, gZ2
by interchanging columns 2 and 3 and rows 2 and 3. Q.E.D.
THEOREM 1.3. (Inverse Mapping Theorem) Let/: V -+ en be a holomor
phic map.
If
det(oJ,./oz.)lz= .. :F-0, then for a sufficiently small neighborhood N
of a,Jis a bijective map N -+ I(N);J(N) is open and/-'I/(N) is holomorphic
on/(N).
Proof The remark gives o(u, L,)/a(X, y) :F-0 at a. We then use the inverse
mapping theorem for differentiable (real variable) functions
to conclude that
I(N) is open,
I is bijective, and I-I is differentiable on I(N). Set qJ(w) =
/-I(W); then z" = cp,,[J(z)]. Computing,
o = a~1l = ± aCPIl a~A + a~" a~A
az. A=I aw}.az. awAaz.
But det(alA/az.) = det(a/A/az.) :F-O. So by linear algebra, aqJ,,/aWA = 0 and
qJ =/-1 is holomorphic. Q.E.D.