§5. Polarizations 25

Assume now that CP

∞(M)

is an abelian Lie subalgebra in

C∞(M).

According to formula (32) from Appendix II.3.1 we have {f1, f2} =

σ(s-gradf1, s-gradf2). We see that skew-gradients of f ∈ CP

∞(M)

span an

isotropic subspace at every point m ∈ M. But this subspace has dimension

1

2

dim M, hence must be a maximal isotropic subspace in Tm(M).

There is a remarkable complex analog of real polarizations.

Definition 4. A complex polarization of (M, σ) is an integrable sub-

bundle P of the complexified tangent bundle T

CM

such that each fiber

P (m) is a maximal isotropic subspace in the symplectic complex vector

space (TmM,

C σC(m)).

Here the integrability is defined formally by the equivalent conditions b)

and c) in the Frobenius Criterion above.

The space CP

∞(M),

as before, is a subalgebra in the complexification of

C∞(M).

A simple description of this subalgebra can be given in a special

case.

Let P be an integrable complex subbundle of T

CM.

Then its complex

conjugate P and the intersection D := P ∩ P are also integrable (this is an

easy exercise in application of the Frobenius Criterion). On the contrary,

the subbundle E := P + P in general is not integrable.

Note that both D and E are invariant under complex conjugation, hence

can be viewed as complexifications of real subbundles D0 = D ∩ TM and

E0 = E ∩ TM, respectively.

Proposition 5. Assume that the subbundle E0 is integrable. Then in a

neighborhood of every point of M there exists a local coordinate system

{u1, . . . , uk; x1, . . . , xl; y1, . . . , yl; v1, . . . , vm} with the following prop-

erties:

(i) D0 is generated by

∂

∂vi

, 1 ≤ i ≤ k;

(ii) E0 is generated by

∂

∂vi

, 1 ≤ i ≤ k,

∂

∂xj

, 1 ≤ j ≤ l, and

∂

∂yj

, 1 ≤

j ≤ l;

(iii) P is generated by

∂

∂vi

, 1 ≤ i ≤ k, and

∂

∂xj

+ i

∂

∂yj

, 1 ≤ j ≤ l.

The crucial case is D0 = 0, E0 = TM. In this case Proposition 5

is exactly the Nirenberg-Newlander theorem on integrability of an almost

complex structure.

Let us introduce the notation zj = xj + iyj, 1 ≤ j ≤ l. Then we can

say that the algebra CP

∞(M)

consists of functions which do not depend on

coordinates vi, 1 ≤ i ≤ k, and are holomorphic in coordinates zj, 1 ≤ j ≤ l.