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Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?

The following sub question is rewritten thanks to the comment of Robert Bryant:

Is it true that if $(M,J)$ admits a vector field that preserves $J$ then there is $J'$ on $M$ homotopic to $J$ that is preserved by an $S^1$-action on $M$?

After three years I don't think indeed that the following is such a reasonable motivation

POSSIBLE MOTIVATION. Claire Voisin gave a construction of the Hilbert scheme of points for every almost complex 4-fold by an analogy with the Hilbert scheme of points of a complex surface. The first calculation of the Euler charactericstics of Hilbert scheme of points of complex surfaces was done via localisation techniques, for $CP^2$. Now, if we have a "holomorphic" vector field on an almost complex manifold, this could potentially help to reduce the calculation of the Euler charachteristics of its Hilbert scheme to the study of fixed points of the manifold. So the question is how flexible this notion is... But in its nature this question seems to be more a question (maybe not a hard one) on dynamical systems.

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This sounds really cool. Is this recent work of Voisin? – Kevin H. Lin Dec 22 '09 at 9:09
Thanks :) Maybe this sound cooler than it should be. He is the reference, it was published in 2002 – Dmitri Dec 22 '09 at 10:45
It's an intriguing question, but it seems to me a surprisingly delicate way to calculate the Euler characteristic of the Hilbert scheme, which one expects to be a universal function of the Betti numbers of the 4-manifold, as in the integrable case. Perhaps one can prove this using the Cech spectral sequence coming from the open cover of Hilb arising from a good cover of the 4-manifold? [BTW, I know you're quoting Voisin, but the following rule is worth insisting on: a 4-fold is an algebraic or analytic variety of dimension 4. A 4-manifold has real dimension 4.] – Tim Perutz Dec 22 '09 at 11:55
Tim, thanks for the comment! In fact the approach I propose is exactly to give a justification of your words:)) How do we calculate the Euler charecteristics of a manifold - count the number of zeros of a vector field. I want to say, that for a Hilbert scheme this should be a result of the same caclulation. I think it would be cool to have a "simple" calculation of the Euler charateristics. By the way this speculation can also be applied to DT invariants -- when you count 0-dim subschemes on 3 dimensional CY manifolds. This calculation was done in 2005 and was one of MNOP conjectures. – Dmitri Dec 22 '09 at 12:27
When you say the vector field preserves the almost-complex structure, are you referring to the flow associated to the vector field? And do you want the vector field to be everywhere nonzero? Nonzero at all but finitely many points? – Ryan Budney Jan 16 '10 at 22:51

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