# Complex structure

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