geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A complex manifold is a manifold holomorphically modeled on polydiscs $D$ in $\mathbb{C}^n$ (complexified $n$-dimensional cartesian space):
a smooth manifold locally isomorphic to $D \hookrightarrow \mathbb{C}^n$ whose transition functions are holomorphic functions;
equivalently: a smooth manifold equipped with an integrable almost complex structure;
equivalently a smooth complex analytic space.
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.
Textbook accounts include
Lectures notes inclide
Stefan Vandoren, Lectures on Riemannian Geometry, Part II: Complex Manifolds (pdf)
Zachary Maddock, Dobeault cohomology (pdf)