Complex manifold - Knowlpedia

Definition

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.

“Holomorphic manifold” is a synonym, not a separate kind of object. The adjective “complex” describes the coordinate structure; it does not mean merely that the underlying topological space or tangent spaces have been complexified.

Underlying smooth and almost-complex structures

Holomorphic transition maps are smooth as maps of real variables. Forgetting the complex coordinates therefore gives an underlying smooth manifold of real dimension $2n$ . The charts also determine an almost-complex structure $J$ on its tangent bundle.

Conversely, an almost-complex structure comes from complex charts exactly when it is integrable. Thus one may equivalently describe a complex manifold as a smooth even-dimensional manifold equipped with an integrable almost-complex structure. The equivalence is supplied by the Newlander–Nirenberg theorem in the smooth category.

Morphisms

The natural morphisms between complex manifolds are holomorphic maps. In terms of the underlying almost-complex structures, a smooth map $f:X\to Y$ is holomorphic exactly when

df\circ J_X=J_Y\circ df.

Complex manifolds and holomorphic maps form the category of complex manifolds. Its isomorphisms are biholomorphisms, and the automorphisms of one object form its biholomorphism group.

Examples and distinctions

Open subsets of $\mathbb C^n$ , complex projective space, complex tori, and the Riemann sphere are complex manifolds. A real smooth manifold may support several inequivalent complex structures or none at all. A Hermitian manifold adds a compatible Riemannian metric, while a Kähler manifold imposes an additional closedness condition on the associated two-form.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: Chapters 1–2, complex manifolds, almost-complex structures, and holomorphic maps.
August Newlander and Louis Nirenberg, “Complex Analytic Coordinates in Almost Complex Manifolds,” Annals of Mathematics 65 (1957), 391–404. DOI record.