geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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:
Notice that by further reduction along the maximal compact subgroup inclusion of the unitary group this yields explicitly a unitary/hermitean vector bundle 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 at integrability of G-structures).
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, $N_J = 0$.
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.
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