Complex manifold

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In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. More precisely, a complex manifold has an atlas of charts to Cⁿ, such that the change of coordinates between charts are holomorphic.

Contents 1 Implications of complex structure 2 Examples of complex manifolds 3 Almost-complex structures 4 Kähler and Calabi-Yau manifolds

[edit] Implications of complex structure

Since complex analytic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors. For example, the Whitney embedding theorem tells us that every smooth manifold can be embedded as a smooth submanifold of Rⁿ, whereas it is "rare" for a complex manifold to have a holomorphic embedding into Cⁿ. Consider for example any compact, connected complex manifold M: any holomorphic function on it is locally constant by Liouville's theorem. Now if we had a holomorphic embedding of M into Cⁿ, then the coordinate functions of Cⁿ would restrict to nonconstant holomorphic functions on M, contradicting compactness, except in the case that M is just a point. Complex manifolds that can be embedded in Cⁿ are called Stein manifolds and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties.

Since the transition maps between charts are holomorphic, complex manifolds are, in particular, smooth and canonically oriented. The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given toplogical manifold has at most finitely many smooth structures, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research.

[edit] Examples of complex manifolds

• Complex vector spaces.
• Complex projective spaces, CPⁿ. Nota Bene: This means that all complex projective spaces are orientable.
• Complex and quaterionic Grassmannians.
• Riemann surfaces.
• Complex Lie groups such as GL(n,C) or Sp(n,C).
• Smooth complex algebraic varieties.
• More generally, the inverse image of any noncritical value of a holomorphic map.
• The Cartesian product of two complex manifolds.

[edit] Almost-complex structures

An almost complex structure on any manifold (for instance, a real manifold as opposed to a complex one) is an endomorphism of the tangent bundle whose square is −Id.

Any complex manifold has an almost complex structure, but not every almost complex structure comes from a complex structure. For example, the 6 dimensional sphere 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. (It is not currently known whether or not the 6-sphere has a complex structure.) 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 and the eigenspaces form sub-bundles denoted by T^0,1M and T^1,0M. The Newlander-Niremberg 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. When this happens, we say that the almost complex structure is integrable.

The Nijenhuis tensor is defined on pairs of vector fields,

X,Y

by

N_J(X,Y) = [X,Y] + J[JX,Y] + J[X,JY] − [JX,JY].

[edit] 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. If the skew symmetric part of such a metric is symplectic, i.e. closed and nondegenerate, then the metric is called Kähler. Kähler structures are much more difficult to come by 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 theorem, it cannot be Kähler.

A Calabi-Yau manifold is a compact Ricci-flat Kähler manifold or equivalently one whose first Chern class vanishes. In string theory the "extra dimensions" are curled up into a Calabi-Yau manifold.