Finding Anti-commuting Complex structures

[GcSanchez05]

Say we complex structure J on a space with even dimension. Is there a trick to finding another complex stucture I that anti-commutes with J?

Moreover I'm that these be integrable complex structres. Any ideas?

 

[Ben Niehoff]

Suppose I, J are complex structures and

IJ = -JI

i.e., they anticommute. Then

$$(IJ)^2 = (IJ)(IJ) = -(IJ)(JI) = + I^2 = -1$$
and hence (IJ) is also a complex structure. Call it K. It is not too hard to verify that these 3 complex structures satisfy the quaternion algebra,

$$IJ = K, \qquad JK = I, \qquad KI = J, \qquad I^2 = J^2 = K^2 = IJK = -1$$
Such a manifold is called hyper-Kaehler and must be of dimension 4n. A hyper-Kaehler manifold has holonomy Sp(n). In 4 real dimensions (i.e., n = 1), we have Sp(1) = SU(2). In general, Sp(n) is a subgroup of SU(2n). Hence all hyper-Kaehler manifolds are Calabi-Yau.

So, in order for such a triplet of complex structures to exist, it is at least necessary that the Ricci form vanish. In 4 real dimensions, this condition is also sufficient.

Suppose we're only interested in 4 dimensions. Define the orientation such that the Kaehler form J is anti-self-dual. Then I and K are also anti-self-dual, and orthogonal to J. That is, I, J, and K span the space $\Lambda^2_- (M)$. So if you are given one of the complex structures, you can work out the others algebraically.

 

[GcSanchez05]

Well, I understood up to hyperkahler manifolds..

I'm participating in an REU and I think I'm venturing too far from my knowledge. Either way thank you for your explanations!

 

[Ben Niehoff]

OK, well the general result is:

If two complex structures I, J exist and they anticommute, this implies that a third complex structure K = IJ exists, such that I, J, K are a representation of the unit quaternions. A manifold that has this structure is called "hyper-Kahler".

Only certain manifolds have this property, since it requires, among other things, that the Ricci tensor vanish. So you won't be able to find two anticommuting complex structures in general.