nLab complex structure



homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Manifolds and cobordisms

Complex geometry

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



Complex structure on vector spaces


Given a ground field 𝕂\mathbb{K}, a complex structure on a 𝕂\mathbb{K}-vector space VV is a 𝕂\mathbb{K}-linear automorphism

J:VV J \,\colon\, V \longrightarrow V

that squares to minus the identity:

JJ=Id V. J \circ J = - Id_V \,.

The idea here is that JJ acts by multiplication with the would-be imaginary unit i\mathrm{i}, and indeed: A real vector space (𝕂\mathbb{K} the real numbers) equipped with a complex structure JJ as in Def. becomes a complex vector space by declaring that i\mathrm{i} acts via JJ:

×V V (x+iy,v) xv+yJ(v) \array{ \mathbb{C} \times V &\longrightarrow& V \\ \big( x + \mathrm{i} \cdot y ,\; v \big) &\mapsto& x \cdot v + y \cdot J(v) }

Here a real-linear map ϕ:VV\phi \colon V \to V is complex-linear iff it commutes with the complex structure in that:

ϕJ=Jϕ. \phi \circ J \,=\, J \circ \phi \,.

Complex structure on manifolds

More generally, an almost complex structure on a smooth manifold is a smoothly varying fiberwise complex structure in the sense of Def. on its tangent spaces (which a priori are real vector 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. , 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}) } \,.

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}) } \,.

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).


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


Every Riemannian metric on an oriented 2-dimensional manifold induces an almost complex structure given by forming orthogonal tangent vectors.


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.


Complex structure on vector spaces

Textbook accounts:

See also:

Fiberwise complex structure on real vector bundles:

Complex structure on manifolds

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)

Last revised on February 27, 2024 at 06:50:56. See the history of this page for a list of all contributions to it.