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Let $(M, \omega, J)$ be a compact symplectic manifold with a compatible almost complex structure $J$, such that the symplectic form determines an integer cohomology class, ie $$ [\omega] \in H^2(M, \mathbb{Z}).$$ Does there exist an "almost homolorphic line bundle" $L \rightarrow M $ such that its first Chern class is $[\omega]$?

By "almost holomorphic line bundle", I mean a complex line bundle, whose transition data is pseudo holomorphic (ie differential commutes with J).

Note that, if M was a Kahler manifold (ie J is integrable), the answer is yes.

$\textbf{Modified Question:} $ It seems the answer to the above question is no. Here is a weaker question: Suppose $(M,\omega)$ is a compatc symplectic manifold, with integral symplectic class. Does there exist an $\omega$-compatible almost complex structure $J$, such that the answer to the above question is yes (ie there will exist an "almost homolorphic" line bunde (wrt to the J) whose first Chern class is $\omega$)?

It seems there are examples of $(M, \omega)$ such that for a generic compatible $J$ the answer is no. But still there could be some compatible $J$ for which the answer is yes.

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I would like to make sure I'm understanding the Kahler case correctly, so let me know if this makes sense: what I see is that in our scenario we can always find a line bundle $L\to M$ and unitary connection $A$ with $F_A=2\pi\omega$ (giving $c_1(L)=[\omega]$), and then an integrable $J$ making $\omega$ Kahler would impose a holomorphic structure on $L$ via $\partial^{0,1}_A$? –  Chris Gerig Jul 27 '13 at 19:12
    
@Chris: It's actually around the other way. The OP wants to fix an $\omega$-compatible $J$ in advance (and not assume that $J$ is integrable, for that would make $(M,\omega,J)$ be a Kähler structure, a special case that is already known), and require that the bundle $L$ be constructible by specifying local trivializations on the elements of an open cover for which the transition functions are $J$-pseudo-holomorphic. –  Robert Bryant Jul 27 '13 at 19:32
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1 Answer

The answer is already 'no' in dimension $4$. The generic almost complex structure compatible with a symplectic structure in dimension $4$ does not admit any pseudoholomorphic functions (in your sense) other than the constants, so, for such data $(M,\omega,J)$, you are asking whether there is a line bundle $L$ that can be described by constant transition functions such that its first Chern class is $[\omega]$.

To see that this is not always possible, consider the special case in which $M$ is simply-connected, and choose $J$ generically compatible with $\omega$ such that the sheaf of $J$-pseudoholomorphic functions is the constant sheaf. Then any $L$ that can be built with constant transition functions will inherit from that construction a flat connection $\nabla$ with holonomy in $\mathbb{C}^\ast$. Since $M$ is simply connected and nabla is flat, the holonomy will be trivial, so $L$ will have a nonvanishing parallel section and hence be the trivial line bundle, so its first Chern class will be zero.

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Thank you for your answer. I have modified the question now. I am now asking if there exists some compatible $J$ for which the answer to my question is yes (as opposed to fixing some $J$ before hand). –  Ritwik Jul 28 '13 at 8:35
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