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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. Complex manifolds can be regarded as a special case of differentiable manifolds. For example, a 1-dimensional complex manifold is geometrically a surface, known as a Riemann surface. The requirement that the transition functions be holomorphic means that unlike in the general differential case, there is no distinction between different ''C''''k''-structures for different k, since holomorphic functions are analytic, and thus any holomorphic structure is also a Ck structure, for any k ≥1.
Implications of complex structure Since complex analytic functions are much more rigid than smooth functions, the theory of complex manifolds has a very different flavor, compared to real manifolds, For example, by the Whitney embedding theorem, every real manifold can be embedded as a submanifold of Rn, while it is rare for a complex manifold to be a (complex) submanifold of Cn. Consider for example any compact complex manifold M: any entire function on it must be locally constant, by the extension to several complex variables of Liouville's theorem. This means that M cannot be embedded in Cn unless it has dimension 0. Complex manifolds which can be embedded in Cn (which are necessarily not compact, or just some points) are known as Stein manifolds. One can define an analogue of a Riemannian metric for complex manifolds, called a Kähler metric. Again, unlike the case of real manifolds, which always have Riemannian metrics, it is unusual for a complex manifold to have any Kähler metric. Examples of complex manifolds 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 that squares to −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 and . 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. It is a beautiful result that this analytic condition boils down to the algebraic requirement that the Nijenhuis tensor of the almost complex structure vanishes. The Nijenhuis tensor is defined on pairs of vector fields, by . Kähler and Calabi-Yau manifolds A Kähler manifold is a manifold which simultaneously carries a Riemannian structure, a symplectic structure, and a complex structure which are all compatible in some suitable sense. A Calabi-Yau manifold is a compact Ricci-flat Kähler manifold. In string theory the extra dimensions are curled up into a Calabi-Yau manifold. | ||||||||
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