geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A complex manifold is a manifold modeled on $\mathbb{C}^n$ (the complex $n$-dimensional complex line):
a smooth manifold locally isomorphic to $\mathbb{C}^n$ whose transition functions are holomorphic functions.
equivalently: a smooth manifold equipped with an integrable almost complex structure
Every complex manifold admits a good open cover in $Disk_{cmpl}$.
For instance (Maddock, lemma 3.2.8).
A complex manifold of complex dimension 1 is called a Riemann surface.
A complex manifold whose canonical bundle is trivializable is a Calabi-Yau manifold. In complex dimension 2 this is a K3 surface.
For instance
Stefan Vandoren, Lectures on Riemannian Geometry, Part II: Complex Manifolds (pdf)
Zachary Maddock, Dobeault cohomology (pdf)