Paths and cylinders
More generally, an almost complex structure on a smooth manifold is essentially a fiber-wise 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-Nierenberg 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, .
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
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)
Revised on October 6, 2013 21:23:35
by Urs Schreiber