# 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.