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\begin{document}

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\section*{complex structure}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

[[!include homotopy - contents]]

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{characterizations_of_integrability}{Characterizations of integrability}\dotfill \pageref*{characterizations_of_integrability} \linebreak
\noindent\hyperlink{on_2dimensional_manifolds}{On 2-dimensional manifolds}\dotfill \pageref*{on_2dimensional_manifolds} \linebreak
\noindent\hyperlink{relation_to_structures}{Relation to $Spin^c$-structures}\dotfill \pageref*{relation_to_structures} \linebreak
\noindent\hyperlink{relation_to_hermitian_and_khler_structure}{Relation to Hermitian and K\"a{}hler structure}\dotfill \pageref*{relation_to_hermitian_and_khler_structure} \linebreak
\noindent\hyperlink{moduli_stacks_of_complex_structures}{Moduli stacks of complex structures}\dotfill \pageref*{moduli_stacks_of_complex_structures} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{LinearComplexStructure}\hypertarget{LinearComplexStructure}{}
A \emph{(linear) complex structure} on a [[vector space]] $V$ is an [[automorphism]] $J : V \to V$ that squares to minus the [[identity]]: $J \circ J = - Id$.

\end{defn}
More generally, an almost complex structure on a [[smooth manifold]] is a smoothly varying [[fiber]]wise complex structure on its [[tangent spaces]]:

\begin{defn}
\label{}\hypertarget{}{}
An \emph{almost complex structure} on a [[smooth manifold]] $X$ (of even [[dimension]]) is a rank $(1,1)$-[[tensor field]] $J$, hence a smooth [[section]] $J \in \Gamma(T X \otimes T^* X)$, such that, over each point $x \in X$, $J$ is a linear complex structure, def. \ref{LinearComplexStructure}, on that [[tangent space]] $T_x X$ under the canonical identification $End T_x X \simeq T_x X\otimes T_x^* X$.

\end{defn}
Equivalently, stated more intrinsically:

\begin{defn}
\label{}\hypertarget{}{}
An \emph{almost complex structure} on a [[smooth manifold]] $X$ of [[dimension]] $2 n$ is a [[reduction of the structure group]] of the [[tangent bundle]] to the [[complex numbers|complex]] [[general linear group]] along $GL(n,\mathbb{C}) \hookrightarrow GL(2n,\mathbb{R})$.

\end{defn}
\begin{remark}
\label{InTermsOfSmoothStacks}\hypertarget{InTermsOfSmoothStacks}{}
In terms of modulating maps of bundles into their [[smooth infinity-groupoid|smooth]] [[moduli stacks]], this means that an almost complex structure is a lift in the following [[diagram]] in [[Smooth∞Grpd]]:

\begin{displaymath}
\itexarray{
    && \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})
  }
  \,.
\end{displaymath}
By further [[reduction of the structure group|reduction]] along the [[maximal compact subgroup]] inclusion of the [[unitary group]] this yields an [[almost Hermitian structure]]

\begin{displaymath}
\itexarray{
    && \mathbf{B} U(n)
    \\
    & {}^{\mathllap{herm.alm.compl.str.}}\nearrow & \downarrow^{\mathrlap{}}
    \\
    X
    &\underoverset{\tau}{tang.\,bund.}{\to}&
    \mathbf{B} GL(2n,\mathbb{R})
  }
  \,.
\end{displaymath}
\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A \emph{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 \emph{integrable} (see also at \emph{[[integrability of G-structures]]} the section \emph{\href{integrability+of+G-structures#ExampleComplexStructure}{Examples -- Complex structure}}).

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{characterizations_of_integrability}{}\subsubsection*{{Characterizations of integrability}}\label{characterizations_of_integrability}

The \emph{[[Newlander-Nirenberg theorem]]} states that an almost complex structure $J$ on a smooth manifold is integrable (see also at \emph{[[integrability of G-structures]]}) precisely if its [[Nijenhuis tensor]] vanishes, $N_J = 0$.

See also at \emph{[[integrability of G-structures]]} the section \emph{\href{integrability+of+G-structures#ExampleComplexStructure}{Examples -- Complex structure}}.

\hypertarget{on_2dimensional_manifolds}{}\subsubsection*{{On 2-dimensional manifolds}}\label{on_2dimensional_manifolds}

\begin{prop}
\label{}\hypertarget{}{}
Every [[Riemannian metric]] on an [[orientation|oriented]] 2-[[dimension|dimensional]] [[manifold]] induces an almost complex structure given by forming orthogonal tangent vectors.

\end{prop}
\begin{prop}
\label{In2dAnyAlmostComplexStructureIsIntegrable}\hypertarget{In2dAnyAlmostComplexStructureIsIntegrable}{}
Every almost complex structure on a 2-[[dimension|dimensional]] [[manifold]] is integrable, hence is a complex structure.

\end{prop}
In the special case of [[real analytic manifolds]] this fact was known to [[Carl Friedrich Gauss]]. For the general case see for instance \hyperlink{Audin}{Audin, remark 3 on p. 47}.

\hypertarget{relation_to_structures}{}\subsubsection*{{Relation to $Spin^c$-structures}}\label{relation_to_structures}

Every almost complex structure canonically induces a [[spin{\tt \symbol{94}}c-structure]] by postcomposition with the universal [[characteristic map]] $\phi$ in the [[diagram]]

\begin{displaymath}
\itexarray{
    \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
  }
  \,.
\end{displaymath}
See at \emph{[[spin{\tt \symbol{94}}c-structure]]} for more.

\hypertarget{relation_to_hermitian_and_khler_structure}{}\subsubsection*{{Relation to Hermitian and K\"a{}hler structure}}\label{relation_to_hermitian_and_khler_structure}

\begin{itemize}%
\item An almost complex structure equipped with a compatible [[Riemannian metric]] is a \emph{[[Hermitian structure]]}.


\item An almost complex structure equipped with a compatible [[Riemannian structure]] and [[symplectic structure]] is a \emph{[[Kähler structure]]}.



\end{itemize}
\begin{tabular}{l|l|l}
[[complex structure]]&+ [[Riemannian structure]]&+ [[symplectic structure]]\\
\hline 
[[complex structure]]&[[Hermitian structure]]&[[Kähler structure]]\\
\end{tabular}

\hypertarget{moduli_stacks_of_complex_structures}{}\subsubsection*{{Moduli stacks of complex structures}}\label{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 \href{moduli+space+of+curves#OverTheComplexNumbers}{here}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[stable complex structure]]


\item [[real structure]]


\item [[quaternionic structure]]


\item [[holomorphic vector bundle]], [[pseudoholomorphic vector bundle]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Lecture notes include

\begin{itemize}%
\item Mich\`e{}le Audin, \emph{Symplectic and almost complex manifolds} (\href{http://www-irma.u-strasbg.fr/~maudin/HolomorphicChapII.pdf}{pdf})

\end{itemize}
Discussion from the point of view of [[integrable G-structures]] includes

\begin{itemize}%
\item [[Robert Bryant]], \emph{Remarks on the geometry of almost complex 6-manifolds}, The Asian Journal of Mathematics, vol. 10 no. 3 (September, 2006), pp. 561--606. (\href{http://arxiv.org/abs/math/0508428}{arXiv:math/0508428})

\end{itemize}
A discussion of deformations of complex structures is in

\begin{itemize}%
\item [[Domenico Fiorenza]], \emph{[[domenicofiorenza:The periods map of a complex projective manifold. An oo-category perspective]]}

\end{itemize}
The \emph{[[moduli space of complex structures]]} on a manifold is discussed for instance from page 175 on of

\begin{itemize}%
\item [[Yongbin Ruan]], \emph{Symplectic topology and complex surfaces} in \emph{Geometry and analysis on complex manifolds} (1994)

\end{itemize}
and in

\begin{itemize}%
\item Yurii M. Burman, \emph{Relative moduli spaces of complex structures: an example} (\href{http://arxiv.org/abs/math/9903029}{arXiv:math/9903029})

\end{itemize}
[[!redirects complex structures]]

[[!redirects almost complex structure]] [[!redirects almost complex structures]]

[[!redirects almost complex manifold]] [[!redirects almost complex manifolds]]

[[!redirects linear complex structure]] [[!redirects linear complex structures]]



\end{document}