From: Kenneth Simpson <"ken"@VirtualMachines.$nospam$.COM>
Subject: Re: finding complex structures
Date: Sun, 23 Sep 2001 07:56:14 GMT
Newsgroups: sci.physics.research
Summary: Complex structures on Kaehler manifolds

Urs Schreiber wrote:
 
> The thing is, I have a metric g and am looking for any *almost*
> (sorry...) complex structure J such that g(Ju,v) is a Kaehler form.


The Kaehler form is

	f(u,v) = g(u,Jv)

which implies the manifold is complex. Since you have an 
almost complex structure, then the manifold is complex iff 
the almost complex structure is integrable. 

There are lots of ways to determine if the almost complex 
structure is integrable. jeff.c1123 mentioned one, namely, 
that the Nijenhuis torsion tensor N(u,v) of J vanish where 

	N(u,v) = 2([Ju,Jv]-[u,v]-J[u,Jv]-J[Ju,v])

(which is the obstruction to the integration of J.)

If the Kaehler form is closed, then the manifold is Kaehler
and the Kaehler form is real and harmonic 

	dK=0
	gK=0

or 

	(dg+gd)K = 0

where g is the codifferential (using the Hermitian metric)
and (dg+gd) is the Laplacian. 

And since the manifold is Kaehler, using dK=0 you can write 
the Hermitian metric in terms of a Kaehler potential F(u,v)

	h(u,v) = d/du d/dv F(u,v)   

(where the derivatives are the usual complex partials.)

The potential completely determines the Kaehler manifold.

-- Ken


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