I became again interested in finding preferred extremals of Kähler action, which would have 4-D CP_{2} and perhaps also M^{4} projections. This would correspond to Maxwell phase that I conjectured long time ago. Deformations of CP_{2} type vacuum extremals would correspond also to these extremals. The signature of the induced metric might be also Minkowskian. It however turns out that the
solution ansatz requires Euclidian signature and that M^{4} projection is 3-D so that original hope is not realized.

I proceed by the following arguments to the ansatz.

- Effective 3-dimensionality for action (holography) requires that action decomposes to vanishing j
^{α}A_{α} term + total divergence giving 3-D "boundary" terms. The first term certainly vanishes (giving effective 3-dimensionality and therefore holography) for
D_{β}J^{αβ}=j^{α}=0 .

Empty space Maxwell equations, something extremely natural. Also for the proposed GRT limit these equations are true.

- How to obtain empty space Maxwell equations j
^{&}alpha;=0? Answer is simple: assume self duality or its slight modification:
J=*J

holding for CP_{2} and CP_{2} type vacuum extremals or a more general condition

J=k*J ,

k some constant not far from unity. * is Hodge dual involving 4-D permutation symbol.k=constant requires that the determinant of the induced metric is apart from constant equal to that of CP_{2} metric. It does not require that the induced metric is proportional to the CP_{2} metric, which is not possible since M^{4} contribution to metric has Minkowskian signature and cannot be therefore proportional to CP_{2} metric.

- Field equations reduce with these assumptions to equations differing from minimal surfaces equations only in that metric g is replaced by Maxwellian energy momentum tensor T. Schematically:
Tr(TH^{k})=0 ,

where T is Maxwellian energy momentum tensor and H^{k} is the second fundamental form - asymmetric 2-tensor defined by covariant derivative of gradients of imbedding space coordinates.

- It would be nice to have minimal surface equations since they are the non-linear generalization of massless wave equations. This is achieved if one has
T= Λ g .

Maxwell energy momentum tensor would be proportional to the metric! One would have dynamically generated cosmological constant! This begins to look really interesting since it appeared also at the proposed GRT limit of TGD.

- Very skematically and forgetting indices and being sloppy with signs, the expression for T reads as
T= JJ -g/4 Tr(JJ) .

Note that the product of tensors is obtained by generalizing matrix product. This should be proportional to metric.

Self duality implies that Tr(JJ) is just the instanton density and does not depend on metric and is constant.

For CP_{2} type vacuum extremals one obtains

T= -g+g=0 .

Cosmological constant would vanish in this case.

- Could it happen that for deformations a small value of cosmological constant is generated? The condition would reduce to
JJ= (Λ-1)g .

Λ must relate to the value of parameter k appearing in the generalized self-duality condition. This would generalize the defining condition for Kähler form

JJ=-g (i^{2}=-1 geometrically)

stating that the square of Kähler form is the negative of metric. The only modification would be that index raising is carried out by using the induced metric containing also M^{4} contribution rather than CP_{2} metric.

- Explicitly:
J_{αμ} J^{μ}_{β} = (Λ-1)g_{αβ} .

Cosmological constant would measure the breaking of Kähler structure.

One could try to develop ansatz to a more detailed form. The most obvious guess is that the induced metric is apart from constant conformal factor the metric of CP