# nLab complex structure

Contents

### Context

#### Manifolds and cobordisms

Definitions

Genera and invariants

Classification

Theorems

complex geometry

### Examples

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.

### 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.

## References

### Complex structure on vector spaces

Textbook accounts:

Fiberwise complex structure on real vector bundles:

### 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

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)

Last revised on February 27, 2024 at 06:50:56. See the history of this page for a list of all contributions to it.