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complex structure

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Definition

Definition

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 a smoothly varying fiberwise complex structure on its tangent spaces:

Definition

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:

Definition

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}).

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:

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}) } \,.

By further reduction along the maximal compact subgroup inclusion of the unitary group this yields an almost Hermitian 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}) } \,.
Definition

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 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 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.

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 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

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

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)

Revised on January 22, 2015 11:12:57 by Urs Schreiber (88.100.66.95)