Almost complex manifold
In
The concept is due to Charles Ehresmann and Heinz Hopf in the 1940s.[1]
Formal definition
Let M be a smooth manifold. An almost complex structure J on M is a linear complex structure (that is, a
If M admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose M is n-dimensional, and let J : TM → TM be an almost complex structure. If J2 = −1 then (det J)2 = (−1)n. But if M is a real manifold, then det J is a real number – thus n must be even if M has an almost complex structure. One can show that it must be
An easy exercise in
Examples
For every integer n, the flat space R2n admits an almost complex structure. An example for such an almost complex structure is (1 ≤ i, j ≤ 2n): for odd i, for even i.
The only spheres which admit almost complex structures are S2 and S6 (Borel & Serre (1953)). In particular, S4 cannot be given an almost complex structure (Ehresmann and Hopf). In the case of S2, the almost complex structure comes from an honest complex structure on the Riemann sphere. The 6-sphere, S6, when considered as the set of unit norm imaginary octonions, inherits an almost complex structure from the octonion multiplication; the question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.[2]
Differential topology of almost complex manifolds
Just as a complex structure on a vector space V allows a decomposition of VC into V+ and V− (the
Just as we build
In other words, each Ωr(M)C admits a decomposition into a sum of Ω(p, q)(M), with r = p + q.
As with any
so that is a map which increases the holomorphic part of the type by one (takes forms of type (p, q) to forms of type (p+1, q)), and is a map which increases the antiholomorphic part of the type by one. These operators are called the
Since the sum of all the projections must be the identity map, we note that the exterior derivative can be written
Integrable almost complex structures
Every complex manifold is itself an almost complex manifold. In local holomorphic coordinates one can define the maps
(just like a counterclockwise rotation of π/2) or
One easily checks that this map defines an almost complex structure. Thus any complex structure on a manifold yields an almost complex structure, which is said to be 'induced' by the complex structure, and the complex structure is said to be 'compatible with' the almost complex structure.
The converse question, whether the almost complex structure implies the existence of a complex structure is much less trivial, and not true in general. On an arbitrary almost complex manifold one can always find coordinates for which the almost complex structure takes the above canonical form at any given point p. In general, however, it is not possible to find coordinates so that J takes the canonical form on an entire
Given any linear map A on each tangent space of M; i.e., A is a tensor field of rank (1, 1), then the Nijenhuis tensor is a tensor field of rank (1,2) given by
or, for the usual case of an almost complex structure A=J such that ,
The individual expressions on the right depend on the choice of the smooth vector fields X and Y, but the left side actually depends only on the pointwise values of X and Y, which is why NA is a tensor. This is also clear from the component formula
In terms of the Frölicher–Nijenhuis bracket, which generalizes the Lie bracket of vector fields, the Nijenhuis tensor NA is just one-half of [A, A].
The Newlander–Nirenberg theorem states that an almost complex structure J is integrable if and only if NJ = 0. The compatible complex structure is unique, as discussed above. Since the existence of an integrable almost complex structure is equivalent to the existence of a complex structure, this is sometimes taken as the definition of a complex structure.
There are several other criteria which are equivalent to the vanishing of the Nijenhuis tensor, and which therefore furnish methods for checking the integrability of an almost complex structure (and in fact each of these can be found in the literature):
- The Lie bracket of any two (1, 0)-vector fields is again of type (1, 0)
Any of these conditions implies the existence of a unique compatible complex structure.
The existence of an almost complex structure is a topological question and is relatively easy to answer, as discussed above. The existence of an integrable almost complex structure, on the other hand, is a much more difficult analytic question. For example, it is still not known whether S6 admits an integrable almost complex structure, despite a long history of ultimately unverified claims. Smoothness issues are important. For
Compatible triples
Suppose M is equipped with a
- g(u, v) = ω(u, Jv)
- ω(u, v) = g(Ju, v)
- J(u) = (φg)−1(φω(u)).
In each of these equations, the two structures on the right hand side are called compatible when the corresponding construction yields a structure of the type specified. For example, ω and J are compatible if and only if ω(•, J•) is a Riemannian metric. The bundle on M whose sections are the almost complex structures compatible to ω has contractible fibres: the complex structures on the tangent fibres compatible with the restriction to the symplectic forms.
Using elementary properties of the symplectic form ω, one can show that a compatible almost complex structure J is an
These triples are related to the 2 out of 3 property of the unitary group.
Generalized almost complex structure
An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the
See also
- Almost quaternionic manifold– Concept in geometry
- Chern class – Characteristic classes of vector bundles
- Frölicher–Nijenhuis bracket
- Kähler manifold – Manifold with Riemannian, complex and symplectic structure
- Poisson manifold – Mathematical structure in differential geometry
- Rizza manifold
- Symplectic manifold – Type of manifold in differential geometry
References
- Newlander, August; MR 0088770.
- ISBN 3-540-42195-5. Information on compatible triples, Kähler and Hermitian manifolds, etc.
- ISBN 0-387-90419-0. Short section which introduces standard basic material.
- Rubei, Elena (2014). Algebraic Geometry, a concise dictionary. Berlin/Boston: Walter De Gruyter. ISBN 978-3-11-031622-3.
- MR 0058213.