Complex manifold

Remarks

Example: the Riemann sphere $\hat\C = \C \cup \set{\infty}$ with the charts $\varphi_0(z) = z$ on $\C$ and $\varphi_\infty(z) = 1/z$ on $\hat\C \setminus \set{0}$, with $\varphi_\infty(\infty) = 0$. On the overlap, $\varphi_0(\C \setminus \set{0}) = \C \setminus \set{0}$, and the transition map $w \mapsto 1/w$ is holomorphic there; so $\hat\C$ is a complex manifold of dimension $1$.

Non-example: on $\C$, the two charts $\varphi_1(z) = z$ and $\varphi_2(z) = \bar z$ do not form a holomorphic atlas. Each is a homeomorphism onto $\C$, but the transition map $w \mapsto \bar w$ is not holomorphic, so clause 2 fails.

Every complex manifold of dimension $n$ is a smooth manifold of dimension $2n$, since holomorphic maps are smooth. It is canonically oriented, because the real Jacobian determinant of a holomorphic map is $\abs{\det J_\C}^2 \ge 0$, where $J_\C$ is its complex Jacobian matrix. Further examples are open subsets of $\C^n$, complex projective space, complex tori and Hopf manifolds. A complex structure induces an almost complex structure; by the Newlander–Nirenberg theorem, the almost complex structures that arise this way are exactly the integrable ones.