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




  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\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)




          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:


          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.


          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 December 21, 2017 at 06:08:31. See the history of this page for a list of all contributions to it.