Paths and cylinders
Manifolds and cobordisms
A (linear) complex structure on a vector space is an automorphism that squares to minus the identity?: .
More generally, an almost complex structure on a smooth manifold is a smoothly varying fiberwise complex structure on its tangent spaces:
An almost complex structure on a smooth manifold (of even dimension) is a rank -tensor field , hence a smooth section , such that, over each point , is a linear complex structure, def. 1, on that tangent space under the canonical identification .
Equivalently, stated more intrinsically:
Characterizations of integrability
The Newlander-Nirenberg theorem states that an almost complex structure on a smooth manifold is integrable (see also at integrability of G-structures) precisely if its Nijenhuis tensor vanishes, .
On 2-dimensional manifolds
Every almost complex structure on a 2-dimensional manifold is integrable, hence is a complex structure.
In the special case of real analytic manifolds this fact was known to Carl Friedrich Gauss. For the general case see for instance Audin, remark 3 on p. 47.
Relation to -structures
Every almost complex structure canonically induces a spin^c-structure by postcomposition with the universal characteristic map in the diagram
See at spin^c-structure for more.
Relation to Hermitian and Kähler structure
Moduli stacks of complex structures
One may consider the moduli stack of complex structures on a given manifold. For 2-dimensional manifolds these are famous as the Riemann moduli stacks of complex curves. They may also be expressed as moduli stacks of almost complex structures, see here.
- Michèle Audin, Symplectic and almost complex manifolds (pdf)
A discussion of deformations of complex structures is in
The moduli space of complex structures on a manifold is discussed for instance from page 175 on of
- Yongbin Ruan, Symplectic topology and complex surfaces in Geometry and analysis on complex manifolds (1994)
- Yurii M. Burman, Relative moduli spaces of complex structures: an example (arXiv:math/9903029)