a $\cc $-linear space on $V$ is defined by $z\cdot v=(x+iy)\cdot v:=x\cdot v +y\cdot Jv$. In the case $(\rr^{2n},J\st )$ this gives $\cc^n$.

$2^0.$ Let $x_0\in (X,J)$. By an affine change of coordinates in some chart $U\ni x_0$ we can always suppose that $x_0=0$ and $J(0)=J\st $ in basis ${\d \over \d x_1},...,{\d \over \d x_{2n}}$. One can now put $y_j=x_{2j}$ in order to have $J(0) {\d \over \d x_j}={\d \over \d y_j}$.

*$3^0.$ Complex manifolds. By definition, a complex manifold is an almost-complex manifold $(X,J)$ such that in the neighborhood of any point $x_0\in X$ a coordinate system $x_1,y_1,...,x_n,y_n$ can be found such that $J {\d \over \d x_j}={\d \over \d y_j}$ everywhere in this neighborhood.*

Equivalently there is in a neighborhood of each point a $C^1$-diffeomorphism $\phi $ into $(\rr^{2n},J\st )$ such that its differential is $(J,J\st )$-linear. The latter means that $d\phi \scirc J=J\st \scirc d\phi $ as mappings $TX\to \rr^{2n}$.

Definition 1.1.2. [*Such an almost-complex structure is called complex.* ]{}

*$4^0.$ Riemann surfaces. Riemann surfaces are oriented, connected two-dimensional manifolds. Fix some Riemannian metric $g$ on such $S$. For $v\in TS$ define $Jv=\{$ rotation of $v$ in positive direction onto $90^\circ \} $. $J$ is an 