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It is a widely known (perhaps slightly folkloric) fact that compact complex manifolds, understood as first-order structures with a predicate for each analytic subset, are definable in an expansion of the field of reals with restricted analytic functions.

What is known about other classes of complex manifolds? Are there other results of the form "complex manifolds belonging to the class $\mathscr C$ are definable in the o-minimal expansion of the real field $\mathcal R = (\mathbf R, +, \cdot, \ldots)$" where $\mathscr C, \mathcal R$ are replaced with something concrete?

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