I think I am confused about some definitions involving equivalence of almost complex structures.

Is it true that given an almost complex structure on a manifold, there is at most one complex structure that realizes it? This seems to be true but I am not completely sure.

Also, I know that there is a whole moduli space of complex structures on a surface. If I have two inequivalent complex structures on a surface, does this imply that the almost complex structures are also not equivalent? In particular, is there a moduli space of almost complex structures? Somehow I thought all almost complex structures were equivalent since almost complex structures are a topological as opposed to geometric notion.

See my related question: Space of smooth structures


1 Answer 1


Hint: Suppose that $M, M'$ are complex manifolds; let $J, J'$ denote their respective almost complex structures. Let $f: M\to M'$ be a diffeomorphism such that $f_*(J)=J'$. Show that $f$ is complex-differentiable and, hence, biholomorphic. If it helps, start with the case of complex dimension 1. If it does not help, see this wikipedia article.

Edit. See e.g. here for the discussion of deformations of almost complex structures. Unlike smooth or symplectic structures, almost complex structures in general admit nontrivial deformations.

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    $\begingroup$ Great. Does this mean that there is a continuous moduli of almost complex structures? For example, it is known that on a closed manifold, any two symplectic structures that can be deformed to each other and are cohomologous are symplectomorphism; so the moduli space is discrete. Every symplectic structure has a contractible choice of compatible almost complex structures. $\endgroup$
    – user39598
    Feb 11, 2017 at 0:06

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