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Given a ground field $\mathbb{K}$, a complex structure on a $\mathbb{K}$-vector space $V$ is a $\mathbb{K}$-linear automorphism
that squares to minus the identity:
The idea here is that $J$ acts by multiplication with the would-be imaginary unit $\mathrm{i}$, and indeed: A real vector space ($\mathbb{K}$ the real numbers) equipped with a complex structure $J$ as in Def. becomes a complex vector space by declaring that $\mathrm{i}$ acts via $J$:
Here a real-linear map $\phi \colon V \to V$ is complex-linear iff it commutes with the complex structure in that:
More generally, an almost complex structure on a smooth manifold is a smoothly varying fiberwise complex structure in the sense of Def. on its tangent spaces (which a priori are real vector 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. , 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.
Textbook accounts:
See also:
Fiberwise complex structure on real vector bundles:
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
Last revised on February 27, 2024 at 06:50:56. See the history of this page for a list of all contributions to it.