View source for Complex structure

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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|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====
<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>