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A (linear) complex structure on a vector space $V$ is an automorphism $J : V \to V$ that squares to minus the identity: $J \circ J = - Id$.
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 $X$ (of even dimension) is a rank $(1,1)$-tensor field $J$, hence a smooth section $J \in \Gamma(T X \otimes T^* X)$, such that, over each point $x \in X$, $J$ is a linear complex structure, def. 1, on that tangent space $T_x X$ under the canonical identification $End T_x X \simeq T_x X\otimes T_x^* X$.
Equivalently, stated more intrinsically:
An almost complex structure on a smooth manifold $X$ of dimension $2 n$ is a reduction of the structure group of the tangent bundle to the complex general linear group along $GL(n,\mathbb{C}) \hookrightarrow GL(2n,\mathbb{R})$.
In terms of modulating maps of bundles into their smooth moduli stacks, this means that an almost complex structure is a lift in the following diagram in Smooth∞Grpd:
By further reduction along the maximal compact subgroup inclusion of the unitary group this yields an almost Hermitian structure
A complex structure on a smooth manifold $X$ is the structure of a complex manifold on $X$. Every such defines an almost complex structure and almost complex structures arising this way are called integrable (see also at integrability of G-structures the section Examples – Complex structure).
The Newlander-Nirenberg theorem states that an almost complex structure $J$ on a smooth manifold is integrable (see also at integrability of G-structures) precisely if its Nijenhuis tensor vanishes, $N_J = 0$.
See also at integrability of G-structures the section Examples – Complex structure.
Every Riemannian metric on an oriented 2-dimensional manifold induces an almost complex structure given by forming orthogonal tangent vectors.
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.
Every almost complex structure canonically induces a spin^c-structure by postcomposition with the universal characteristic map $\phi$ in the diagram
See at spin^c-structure for more.
An almost complex structure equipped with a compatible Riemannian metric is a Hermitian structure.
An almost complex structure equipped with a compatible Riemannian structure and symplectic structure is a Kähler structure.
complex structure | + Riemannian structure | + symplectic structure |
---|---|---|
complex structure | Hermitian structure | Kähler structure |
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.
Lecture notes include
Discussion from the point of view of integrable G-structures includes
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
and in