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If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $J$ is actually a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ of $J$ vanishes.

Assume $D\subset X$ is a J-invariant symplectic divisor (i.e. real codimension two). Under what condition on $J$, locally around points of $D$ (on some small open set around every $p\in D$), there are holomorphic functions

$$ F\colon U \to \mathbb{C}, \quad \bar\partial F =dF+i dF\circ J=0,$$ such that $D\cap U = F^{-1}(0)$.

** A weaker, and may be more feasible question is to ask for the $J$-holomorphicity only along $D$.

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The existence of a holomorphic function in a neighborhood is a strong condition. I don't think that there is any condition which may be written simply in terms of the derivatives of $J$ restricted to $D$. –  Brett Parker Jan 12 at 22:36
    
How about $J$-holomorphicity condition only along $D$? –  Mohammad F. Tehrani Jan 13 at 14:23
    
I know of two reasonable conditions on $J$ which are useful because they make holomorphic curves behave themselves well close to the divisor. One is the condition used by Eleny Ionel and Tom Parker in their construction of relative Gromov-Witten invariants, the other is my condition of a $\bar\partial \log$ compatible $J$ described in the last section of arxiv.org/abs/1108.3713 –  Brett Parker Jan 13 at 23:17

1 Answer 1

Your initial statement, that $X$ admits a complex structure if and only if $N_J=0$ is not correct. I think you mean that $J$ comes from a complex structure if and only if $N_J=0$; that is correct, but in principle $X$ could be a complex manifold and $J$ a non-integrable complex structure.

But that's just a minor mis-statement; I know it's not what you meant. So let me make a suggestion that could be more to the point (though I don't have an answer to your question).

The vanishing of $N_J$ is equivalent to saying that the associated $\bar \partial$ operator satisfies $\bar \partial^2 = 0$. This is a condition that gives the existence of a lot of holomorphic functions in a neighborhood (under sufficient regularity assumptions). When $\bar \partial ^2 \neq 0$, the conditions for finding local holomorphic functions is too overdetermined, and in general you will have no holomorphic functions at all. Constructing these holomorphic functions, when $\bar \partial ^2 =0$, is a PDE problem called the Newlander-Nirenberg theorem. Perhaps if you follow the proof of that theorem, you could tell if the vanishing of $\bar \partial ^2$ (or maybe $N_J$) along certain directions related to your divisor could allow you to construct the functions you seek.

There is yet another equivalent statement to integrability; it is that the following: The almost complex structure $J$ has eigenvalues $\pm \sqrt{-1}$ so the complexification $T_X \times \mathbb{C}$ of $T_X$ splits as a direct sum $T^{1,0}_X\oplus T^{0,1}_X$ of eigenvectors with eigenvalue $\sqrt{-1}$ and $-\sqrt{-1}$ respectively. Integrability means that $T^{1,0}_X$ is closed under (the complexification of the) Lie bracket. If you assume the underlying structure is real analytic, then this integrability condition can be used with the complex version of Frobenius' integrability condition to give you (locally) a complex manifold in the complexified tangent bundle that projects onto the base. This gives the complex structure of $X$. Perhaps this picture could help you construct the divisor from holomorphic functions in the real analytic case; then if you get a good answer in this case, you can follow the method of Newlander-Nirenberg to adapt things in the smooth case.

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