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A (linear) complex structure on a vector space VV is an automorphism J:VVJ : V \to V that squares to minus the identity: JJ=IdJ \circ J = - Id.

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


An almost complex structure on a smooth manifold XX (of even dimension) is a rank (1,1)(1,1)-tensor field JJ, hence a smooth section JΓ(TXT *X)J \in \Gamma(T X \otimes T^* X), such that, over each point xXx \in X, JJ is a linear complex structure, def. 1, on that tangent space T xXT_x X under the canonical identification EndT xXT xXT x *XEnd T_x X \simeq T_x X\otimes T_x^* X.

Equivalently, stated more intrinsically:


An almost complex structure on a smooth manifold XX of dimension 2n2 n is a reduction of the structure group of the tangent bundle to the complex general linear group along GL(n,)GL(2n,)GL(n,\mathbb{C}) \hookrightarrow GL(2n,\mathbb{R}).


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:

BGL(n,) alm.compl.str. X τtang.bund. BGL(2n,). \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

BU(n) herm.alm.compl.str. X τtang.bund. BGL(2n,). \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}) } \,.

A complex structure on a smooth manifold XX is the structure of a complex manifold on XX. Every such defines an almost complex structure and almost complex structures arising this way are called integrable (see at integrability of G-structures).


Characterizations of integrability

The Newlander-Nierenberg theorem states that an almost complex structure JJ on a smooth manifold is integrable (see also at integrability of G-structures) precisely if its Nijenhuis tensor? vanishes, N J=0N_J = 0.

Relation to Spin cSpin^c-structures

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

BU(n) ϕ BSpin c BU(1) BSO(2n) w 2 B 2 2. \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

complex structure+ Riemannian structure+ symplectic structure
complex structureHermitian structureKähler structure


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 March 14, 2014 05:28:06 by Urs Schreiber (