Page 208 - Curvature and Homology

190 V. COMPLEX MANIFOLDS
3. Show that a holomorphic function on a (connected) complex manifold M
which vanishes on some non-empty open subset must vanish everywhere on M.
4. Let a be a holomorphic 1-form on the Riemann sphere SB. Then, in C,-the
complex plane, a = f(2)d.z where f(x) is an entire function. By employing the
map given by 112 at oo show that f(l/z)l/xa has a pole at the origin unless
f(a) = 0. In this way, we obtain a direct proof of the fact that SB is of genus 0.

B. Almost complex manifolds [SO]
1. Let X and Y be any two vector fields of type (0,l) on the almost complex
manifold M. Then, in order that M be complex it is necessary that [X, Y] be of
type (0,l). Denote by T1-O and TO-' the spaces of tangent vector fields of types
(1,O) and (0,1), respectively, on M.
2. On an almost complex manifold M the following conditions are equivalent:
(a) [TO*l, TO**] C PI;
(b) d Aq*C c A Q+lsr @ A Qer+' for every q and r ;
(c) h(X, Y) = [X, Y] + J[JX, Y] + J[X, JY] - [ JX, JYI = 0 for any vector
fields X and Y where J is the almost complex structure operator of M.
Hence, in order that M be complex it is necessary that h(X,Y) = 0, for
any X and Y. Show that the condition (c) is equivalent to (5.2.18).
3. h(X,Y) is a tensor of type (1,2) with the properties:
(i) h(X + Y,Z) = h(X,Z) + h(Y,Z),
(ii) h(X, Y) = - h(Y,X),
(iii) h(X, f Y) = f h(X, Y)
for any X,Y,Z E T and f E F.

4. If dim M = 2, M is complex.
Hint: h(X, JX) = 0 for all X.
5. Let G be a 2n-dimensional Lie group, L the Lie algebra of left invariant
vector fields on G and J an almost complex structure on G. If the tensor field
J of type (1,l) on G is left invariant, that is, if J is a left invmiattt almost complex
structure, then JL = L. The integrability condition may consequently be
expressed as h(X, Y)=O for any X, Y E L. Since every bi-invariant (that is, both
left and right invariant) tensor field on a Lie group is analytic it follows that
every left invariant almost complex structure on an abelian Lie group defines a
complex structure on the underlying manifo!d. (It is known that a bi-invariant
almost complex structure on any Lie group is integrable.)
6. Show that any two complex structures on a differentiable manifold which
define the same almost complex structure coincide.