nLab complex structure

Definition

Complex structure on vector spaces

Definition

Given a ground field $\mathbb{K}$, a complex structure on a $\mathbb{K}$-vector space $V$ is a $\mathbb{K}$-linear automorphism

$$J \,\colon\, V \longrightarrow V$$

that squares to minus the identity:

$$J \circ J = - Id_V \,.$$

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 $J$:

$$\array{ \mathbb{C} \times V &\longrightarrow& V \\ \big( x + \mathrm{i} \cdot y ,\; v \big) &\mapsto& x \cdot v + y \cdot J(v) }$$

Here a real-linear map $\phi \colon V \to V$ is complex-linear iff it commutes with the complex structure in that:

$$\phi \circ J \,=\, J \circ \phi \,.$$

Complex structure on manifolds

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):

Definition

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:

Definition

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 $GL(n,\mathbb{C}) \hookrightarrow GL(2n,\mathbb{R})$.

Remark

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:

$$\array{ && \mathbf{B} GL(n,\mathbb{C}) \\ & {}^{\mathllap{alm.compl.str.}}\nearrow & \downarrow^{\mathrlap{}} \\ X &\underoverset{\tau}{tang.\,bund.}{\to}& \mathbf{B} GL(2n,\mathbb{R}) } \,.$$

By further reduction along the maximal compact subgroup inclusion of the unitary group this yields an almost Hermitian structure

$$\array{ && \mathbf{B} U(n) \\ & {}^{\mathllap{herm.alm.compl.str.}}\nearrow & \downarrow^{\mathrlap{}} \\ X &\underoverset{\tau}{tang.\,bund.}{\to}& \mathbf{B} GL(2n,\mathbb{R}) } \,.$$

Definition

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).

Properties

Characterizations of integrability

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.

On 2-dimensional manifolds

Proposition

Every Riemannian metric on an oriented 2-dimensional manifold induces an almost complex structure given by forming orthogonal tangent vectors.

Proposition

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 $Spin^c$-structures

Every almost complex structure canonically induces a spin^c-structure by postcomposition with the universal characteristic map $\phi$ in the diagram

$$\array{ \mathbf{B}U(n) &\stackrel{\phi}{\to}& \mathbf{B}Spin^c &\to& \mathbf{B}U(1) \\ &\searrow& \downarrow && \downarrow^{\mathrlap{}} \\ && \mathbf{B}SO(2n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,.$$

See at spin^c-structure for more.

Relation to Hermitian and Kähler structure

• 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

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.

Related concepts

• stable complex structure

• real structure

• quaternionic structure

• holomorphic vector bundle, pseudoholomorphic vector bundle

• para-complex structure

References

Complex structure on vector spaces

Textbook accounts:

• Shoshichi Kobayashi, Katsumi Nomizu, p. 114 in: Foundations of differential geometry, Volume 1 (1963), Volume 2 (1969) Interscience Publishers, reprinted by Wiley Classics Library (1996) [ISBN:978-0-470-55558-3, Wikipedia entry]

See also:

• Wikipedia, Linear complex structure

Fiberwise complex structure on real vector bundles:

• Emery Thomas, Complex Structures on Real Vector Bundles, American Journal of Mathematics 89 4 (1967) 887-908 [arXiv;10.2307/2373409, jstor:2373409]

Complex structure on manifolds

Lecture notes include

• Michèle Audin, Symplectic and almost complex manifolds (pdf)

Discussion from the point of view of integrable G-structures includes

• Robert Bryant, Remarks on the geometry of almost complex 6-manifolds, The Asian Journal of Mathematics, vol. 10 no. 3 (September, 2006), pp. 561–606. (arXiv:math/0508428)

A discussion of deformations of complex structures is in

• Domenico Fiorenza, The periods map of a complex projective manifold. An oo-category perspective

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)

and in

• Yurii M. Burman, Relative moduli spaces of complex structures: an example (arXiv:math/9903029)