nLab complex structure

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Differential geometry

differential geometry

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Definition

Definition

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

More generally, an almost complex structure on a smooth manifold is essentially a fiber-wise complex structure on its tangent spaces:

Definition

An almost complex structure on a smooth manifold $X$ (of even dimension) is a rank $\left(1,1\right)$-tensor field $J$, hence 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, def. 1, 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, stated more intrinsically:

Definition

An almost complex structure on a smooth manifold $X$ of dimension $2n$ is a reduction of the structure group of the tangent bundle to the complex general linear group along $\mathrm{GL}\left(n,ℂ\right)↪\mathrm{GL}\left(2n,ℝ\right)$.

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:

$\begin{array}{ccc}& & B\mathrm{GL}\left(n,ℂ\right)\\ & {}^{\mathrm{alm}.\mathrm{compl}.\mathrm{str}.}↗& {↓}^{}\\ X& \underset{\tau }{\overset{\mathrm{tang}.\phantom{\rule{thinmathspace}{0ex}}\mathrm{bund}.}{\to }}& B\mathrm{GL}\left(2n,ℝ\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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}) } \,.

Notice that by further reduction along the maximal compact subgroup inclusion of the unitary group this yields explicitly a unitary/hermitean vector bundle structure

$\begin{array}{ccc}& & BU\left(n\right)\\ & {}^{\mathrm{herm}.\mathrm{alm}.\mathrm{compl}.\mathrm{str}.}↗& {↓}^{}\\ X& \underset{\tau }{\overset{\mathrm{tang}.\phantom{\rule{thinmathspace}{0ex}}\mathrm{bund}.}{\to }}& B\mathrm{GL}\left(2n,ℝ\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 at integrability of G-structures).

Properties

Characterizations of integrability

The Newlander-Nierenberg 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, ${N}_{J}=0$.

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

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

$\begin{array}{ccccc}BU\left(n\right)& \stackrel{\varphi }{\to }& B{\mathrm{Spin}}^{c}& \to & BU\left(1\right)\\ & ↘& ↓& & {↓}^{}\\ & & B\mathrm{SO}\left(2n\right)& \stackrel{{w}_{2}}{\to }& {B}^{2}{ℤ}_{2}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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.

References

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)

Revised on October 6, 2013 21:23:35 by Urs Schreiber (195.37.209.182)