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)