For questions about or involving complex manifolds.

An $n$-dimensional complex manifold is a topological manifold equipped with an open cover $\\{U_{\alpha}\\}$ and homeomorphisms $\varphi_{\alpha} : U_{\alpha} \to B$ where $B$ is an open ball in $\mathbb{C}^n$ such that the transition maps $\varphi_{\alpha}\circ\varphi_{\beta}^{-1}$ are holomorphic.

Every complex manifold is smooth, but not every smooth manifold is complex. An intermediate notion is that of an almost complex manifold which is a smooth manifold $M$ equipped with a bundle endomorphism $J : TM \to TM$ such that $J\circ J = -\operatorname{id}_{TM}$. Such a $J$ is called an almost complex structure.