# nLab complex structure

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

A (linear) complex structure on a vector space $V$ is an automorphism $J:V\to V$ that squares to minus the identity: $J\circ J=-\mathrm{Id}$.

An almost complex structure on a smooth manifold $X$ is a smooth section $J\in \Gamma \left(TX\otimes {T}^{*}X\right)$ such that, over each point $x\in X$, $J$ is a linear complex structure on that tangent space ${T}_{x}X$ under the canonical identification $\mathrm{End}{T}_{x}X\simeq {T}_{x}X\otimes {T}_{x}^{*}X$. Equivalently this is a reduction of the structure group of the tangent bundle to the complex general linear group along $\mathrm{GL}\left(n,ℂ\right)↪\mathrm{GL}\left(2n,ℝ\right)$.

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 .

## Properties

### Characterizations of integrability

The Newlander-Nierenberg theorem states that an almost complex structure $J$ on a smooth manifold is integrable precisely if its Nijenhuis tensor? vanishes, ${N}_{J}=0$.

### Relation to ${\mathrm{Spin}}^{c}$-structures

Every alomost complex structure canonically induces a spin^c-structure. See there for more.

## References

A discussion of deformations of complex structures is in

Revised on November 7, 2012 18:17:08 by Urs Schreiber (82.169.65.155)