Complex manifold

A complex manifold of complex dimension $n$ is a Hausdorff, second-countable space covered by coordinate charts $\varphi_i:U_i\to V_i\subseteq\mathbb C^n$ such that each transition map $\varphi_j\circ\varphi_i^{-1}$ is holomorphic wherever it is defined.

Forgetting complex coordinates gives an underlying smooth manifold of real dimension $2n$. The complex charts determine an almost-complex structure on its tangent bundle; the existence of holomorphic charts is the additional integrability condition.