nLab
complex structure

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 $J\in \Gamma (TX\otimes T^{*}X)$ 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 $\mathrm{GL}(n,ℂ)\hookrightarrow \mathrm{GL}(2n,ℝ)$.

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

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