§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.
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