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Let $(M,J)$ be an almost-complex structure. By definition, if $M$ admits local holomorphic coordinates for $J$ around every point and these patch together to form a holomorphic atlas (complex structure) for $M$, then $J$ is said to be integrable.

I was able to verify that a complex manifold $M$ can admit many almost-complex structures.

But I couldn't verify this: given different integrable almost-complex structures $J_1,J_2$, can we say that they integrate to different complex structures?

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    $\begingroup$ certainly, right? If the complex structures were identical then the collection of charts would be identical, and the derivative of multiplication-by-$i$ in a chart is how you define the $J_i$. $\endgroup$ – user98602 Nov 26 '18 at 20:12
  • $\begingroup$ @MikeMiller What I mean by "complex structures $\{(U_i,\varphi_i)\}$ and $\{(V_j,\psi_j)\}$ being the same" is that $\varphi_i\circ\psi_j^{-1}$ is holomorphic for every $i,j$. I don't understand what you said about multiplication by $i$ defining $J_i$, since $J_1,J_2$ are already given. $\endgroup$ – rmdmc89 Nov 26 '18 at 20:28
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    $\begingroup$ You may as well pass to a maximal atlas, which means that both atlases contain exactly the same charts. For your charts to integrate your given almost complex structure means that, on each chart, the map $J$ is the derivative of multiplication-by-$i$. $\endgroup$ – user98602 Nov 26 '18 at 20:38

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