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I understand a deformation of complex manifold $X$ to be a pair of complex manifolds $\mathcal{X}$ and $S$, with holomorphic map of complex manifolds $\pi: \mathcal{X} \rightarrow S$ where for some point "$0$" in $S$, the fiber $\pi^{-1}(0)$ is biholomorphic to $X$. This is the definition used, for example, in the Kodaira-Spencer theorem.

Is $\mathcal{X}$ the moduli space? Or is $S$? Or is the moduli space a completely different concept?

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    $\begingroup$ $\mathcal X$ is some larger complex manifold, and $S$ is some "parameterizing family", often the complex unit disc. $\endgroup$ – user98602 May 8 '17 at 20:51
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The moduli space $\mathcal{M}$ is the isomorphism classes of complex structures on the smooth manifold $X$ and is topologized so that 'nearby' complex structures are deformations of one another. Note, the moduli space need not be a manifold.

There is a map $S \to \mathcal{M}$ given by $s \mapsto \pi^{-1}(s)$, but this need not be injective (consider $\mathcal{X} = X\times S$). If you could find some family $\mathcal{X} \to S$ where $S$ is connected and the map $S \to \mathcal{M}$ is surjective (in particular, $\mathcal{M}$ is surjective), then all the complex structures on $X$ would be deformation equivalent.

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  • $\begingroup$ Thanks! Can you recommend a reference for this? Particularly from the analytic perspective? $\endgroup$ – orlandpm May 8 '17 at 21:16
  • $\begingroup$ I don't know of a reference actually. If you find one, let me know. $\endgroup$ – Michael Albanese May 8 '17 at 21:30

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