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A complex structure on a real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c0241701.png" /> is the structure of a complex vector space on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c0241702.png" /> that is compatible with the original real structure. The complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c0241703.png" /> is completely determined by the operator of multiplication by the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c0241704.png" />, the role of which can be taken by an arbitrary linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c0241705.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c0241706.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c0241707.png" /> is the identity. Therefore, a transformation of this type is often called a complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c0241708.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c0241709.png" /> is endowed with a complex structure and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417010.png" /> is a basis of this space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417012.png" /> forms a basis of it over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417013.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417015.png" /> is a complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417016.png" /> then the complexification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417018.png" /> decomposes into a direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417020.png" /> are the eigen spaces of the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417021.png" /> extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417022.png" /> corresponding to the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417024.png" />. Conversely, each complex subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417026.png" /> determines a complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417027.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417028.png" />.
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Any two complex structures on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417029.png" />-dimensional real space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417030.png" /> can be mapped into each other by some automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417031.png" />. The set of all complex structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417032.png" /> is thus a homogeneous space of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417033.png" /> and is identified with the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417035.png" /> is the subgroup of non-singular matrices of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417036.png" /></td> </tr></table>
+
A complex structure on a real vector space  $  V $
 +
is the structure of a complex vector space on  $  V $
 +
that is compatible with the original real structure. The complex structure on  $  V $
 +
is completely determined by the operator of multiplication by the number  $  i $,
 +
the role of which can be taken by an arbitrary linear transformation  $  I :  V \rightarrow V $
 +
satisfying  $  I  ^ {2} = - E $,
 +
where  $  E $
 +
is the identity. Therefore, a transformation of this type is often called a complex structure on  $  V $.
 +
If  $  V $
 +
is endowed with a complex structure and  $  v _ {1} \dots v _ {n} $
 +
is a basis of this space over  $  \mathbf C $,
 +
then  $  v _ {1} \dots v _ {n} , I v _ {1} \dots I v _ {n} $
 +
forms a basis of it over  $  \mathbf R $,
 +
so that  $  \mathop{\rm dim} _ {\mathbf R }  V = 2  \mathop{\rm dim} _ {\mathbf C }  V $.
 +
If  $  I $
 +
is a complex structure on  $  V $
 +
then the complexification  $  V ^ {\mathbf C } $
 +
of  $  V $
 +
decomposes into a direct sum  $  V ^ {\mathbf C } = V _ {+} \dot{+} V _ {-} $,
 +
where  $  V _  \pm  $
 +
are the eigen spaces of the transformation  $  I $
 +
extended to  $  V ^ {\mathbf C } $
 +
corresponding to the eigen values  $  \pm  i $,
 +
and  $  V _ {-} = \overline{ {V _ {+} }}\; $.
 +
Conversely, each complex subspace  $  S \subset  V ^ {\mathbf C } $
 +
such that  $  V ^ {\mathbf C } = S \dot{+} \overline{S}\; $
 +
determines a complex structure on  $  V $
 +
for which  $  V _ {+} = S $.
  
A complex structure on a differentiable manifold is the structure of a complex-analytic manifold (cf. [[Analytic manifold|Analytic manifold]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417037.png" /> is a differentiable manifold, then a complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417038.png" /> is a complex-analytic atlas on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417039.png" /> that is compatible with the real differentiable atlas defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417040.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417041.png" />. A complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417042.png" /> induces a complex structure on each tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417043.png" />, and therefore induces on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024170/c02417044.png" /> an [[Almost-complex structure|almost-complex structure]] which completely determines it.
+
Any two complex structures on a  $  2n $-
 +
dimensional real space  $  V $
 +
can be mapped into each other by some automorphism of  $  V $.
 +
The set of all complex structures on  $  V $
 +
is thus a homogeneous space of the group  $  \mathop{\rm GL} ( 2n , \mathbf R ) $
 +
and is identified with the quotient space  $  \mathop{\rm GL} ( 2n , \mathbf R ) / H $,
 +
where  $  H \cong  \mathop{\rm GL} ( n , \mathbf C ) $
 +
is the subgroup of non-singular matrices of the form
 +
 
 +
$$
 +
\left \|
 +
 
 +
\begin{array}{rl}
 +
A  & B  \\
 +
- B  & A  \\
 +
\end{array}
 +
\
 +
\right \| .
 +
$$
 +
 
 +
A complex structure on a differentiable manifold is the structure of a complex-analytic manifold (cf. [[Analytic manifold|Analytic manifold]]). If $  M $
 +
is a differentiable manifold, then a complex structure on $  M $
 +
is a complex-analytic atlas on $  M $
 +
that is compatible with the real differentiable atlas defined on $  M $.  
 +
Here $  \mathop{\rm dim} _ {\mathbf R }  M = 2  \mathop{\rm dim} _ {\mathbf C }  M $.  
 +
A complex structure on $  M $
 +
induces a complex structure on each tangent space $  T _ {x} ( M) $,  
 +
and therefore induces on $  M $
 +
an [[Almost-complex structure|almost-complex structure]] which completely determines it.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Lichnerowicz,  "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.O. Wells jr.,  "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Lichnerowicz,  "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.O. Wells jr.,  "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


A complex structure on a real vector space $ V $ is the structure of a complex vector space on $ V $ that is compatible with the original real structure. The complex structure on $ V $ is completely determined by the operator of multiplication by the number $ i $, the role of which can be taken by an arbitrary linear transformation $ I : V \rightarrow V $ satisfying $ I ^ {2} = - E $, where $ E $ is the identity. Therefore, a transformation of this type is often called a complex structure on $ V $. If $ V $ is endowed with a complex structure and $ v _ {1} \dots v _ {n} $ is a basis of this space over $ \mathbf C $, then $ v _ {1} \dots v _ {n} , I v _ {1} \dots I v _ {n} $ forms a basis of it over $ \mathbf R $, so that $ \mathop{\rm dim} _ {\mathbf R } V = 2 \mathop{\rm dim} _ {\mathbf C } V $. If $ I $ is a complex structure on $ V $ then the complexification $ V ^ {\mathbf C } $ of $ V $ decomposes into a direct sum $ V ^ {\mathbf C } = V _ {+} \dot{+} V _ {-} $, where $ V _ \pm $ are the eigen spaces of the transformation $ I $ extended to $ V ^ {\mathbf C } $ corresponding to the eigen values $ \pm i $, and $ V _ {-} = \overline{ {V _ {+} }}\; $. Conversely, each complex subspace $ S \subset V ^ {\mathbf C } $ such that $ V ^ {\mathbf C } = S \dot{+} \overline{S}\; $ determines a complex structure on $ V $ for which $ V _ {+} = S $.

Any two complex structures on a $ 2n $- dimensional real space $ V $ can be mapped into each other by some automorphism of $ V $. The set of all complex structures on $ V $ is thus a homogeneous space of the group $ \mathop{\rm GL} ( 2n , \mathbf R ) $ and is identified with the quotient space $ \mathop{\rm GL} ( 2n , \mathbf R ) / H $, where $ H \cong \mathop{\rm GL} ( n , \mathbf C ) $ is the subgroup of non-singular matrices of the form

$$ \left \| \begin{array}{rl} A & B \\ - B & A \\ \end{array} \ \right \| . $$

A complex structure on a differentiable manifold is the structure of a complex-analytic manifold (cf. Analytic manifold). If $ M $ is a differentiable manifold, then a complex structure on $ M $ is a complex-analytic atlas on $ M $ that is compatible with the real differentiable atlas defined on $ M $. Here $ \mathop{\rm dim} _ {\mathbf R } M = 2 \mathop{\rm dim} _ {\mathbf C } M $. A complex structure on $ M $ induces a complex structure on each tangent space $ T _ {x} ( M) $, and therefore induces on $ M $ an almost-complex structure which completely determines it.

References

[1] A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[2] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
How to Cite This Entry:
Complex structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complex_structure&oldid=46432
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article