Complex manifold

Examples of complex manifolds

• Riemann surfaces.
• Calabi–Yau manifolds.
• The Cartesian product of two complex manifolds.
• The inverse image of any noncritical value of a holomorphic map.

Smooth complex algebraic varieties

Smooth complex algebraic varieties are complex manifolds, including:

• Complex vector spaces.
• Complex projective spaces, Pⁿ(C).
• Complex Grassmannians.
• Complex Lie groups such as GL(n, C) or Sp(n, C).

Simply connected

The simply connected 1-dimensional complex manifolds are isomorphic to either:

• Δ, the unit disk in C
• C, the complex plane
• Ĉ, the Riemann sphere

Note that there are inclusions between these as Δ ⊆ C ⊆ Ĉ, but that there are no non-constant holomorphic maps in the other direction, by Liouville's theorem.

Disc vs. space vs. polydisc

The following spaces are different as complex manifolds, demonstrating the more rigid geometric character of complex manifolds (compared to smooth manifolds):

• complex space ${\displaystyle {\mathbb{C}}^n}$.
• the unit disc or open ball

${\displaystyle \left\{z\in {\mathbb{C}}^n:\Vert z\Vert <1\right\}.}$

• the polydisc

${\displaystyle \left\{z=(z_1,\dots ,z_n)\in {\mathbb{C}}^n:|z_j|<1\mathrm{\forall }j=1,\dots ,n\right\}.}$

Almost complex structures

An almost complex structure on a real 2n-manifold is a GL(n, C)-structure (in the sense of G-structures) – that is, the tangent bundle is equipped with a linear complex structure.

Concretely, this is an endomorphism of the tangent bundle whose square is −I; this endomorphism is analogous to multiplication by the imaginary number i, and is denoted J (to avoid confusion with the identity matrix I). An almost complex manifold is necessarily even-dimensional.

An almost complex structure is weaker than a complex structure: any complex manifold has an almost complex structure, but not every almost complex structure comes from a complex structure. Note that every even-dimensional real manifold has an almost complex structure defined locally from the local coordinate chart. The question is whether this almost complex structure can be defined globally. An almost complex structure that comes from a complex structure is called integrable, and when one wishes to specify a complex structure as opposed to an almost complex structure, one says an integrable complex structure. For integrable complex structures the so-called Nijenhuis tensor vanishes. This tensor is defined on pairs of vector fields, X, Y by

${\displaystyle N_J(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY].}$

For example, the 6-dimensional sphere S⁶ has a natural almost complex structure arising from the fact that it is the orthogonal complement of i in the unit sphere of the octonions, but this is not a complex structure. (The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.) Using an almost complex structure we can make sense of holomorphic maps and ask about the existence of holomorphic coordinates on the manifold. The existence of holomorphic coordinates is equivalent to saying the manifold is complex (which is what the chart definition says).

Tensoring the tangent bundle with the complex numbers we get the complexified tangent bundle, on which multiplication by complex numbers makes sense (even if we started with a real manifold). The eigenvalues of an almost complex structure are ±i and the eigenspaces form sub-bundles denoted by T^0,1M and T^1,0M. The Newlander–Nirenberg theorem shows that an almost complex structure is actually a complex structure precisely when these subbundles are involutive, i.e., closed under the Lie bracket of vector fields, and such an almost complex structure is called integrable.

Kähler and Calabi–Yau manifolds

One can define an analogue of a Riemannian metric for complex manifolds, called a Hermitian metric. Like a Riemannian metric, a Hermitian metric consists of a smoothly varying, positive definite inner product on the tangent bundle, which is Hermitian with respect to the complex structure on the tangent space at each point. As in the Riemannian case, such metrics always exist in abundance on any complex manifold, i.e. a Hermitian structure is not rigid.

A Hermitian form can be decomposed into a real part and a complex part as ${\displaystyle h(X,Y)=g(X,Y)+i\omega (X,Y)}$. The real part is a Riemannian metric, and the complex part is skew-symmetric. If the skew-symmetric part is symplectic, i.e. closed and nondegenerate, then the metric is called Kähler. Kähler structures are rare and are much more rigid.

Examples of Kähler manifolds include smooth projective varieties and more generally any complex submanifold of a Kähler manifold. The Hopf manifolds are examples of complex manifolds that are not Kähler. To construct one, take a complex vector space minus the origin and consider the action of the group of integers on this space by multiplication by exp(n). The quotient is a complex manifold whose first Betti number is one, so by the Hodge theory, it cannot be Kähler.

A Calabi–Yau manifold can be defined as a compact Ricci-flat Kähler manifold or equivalently one whose first Chern class vanishes.