An almost complex structure on a real differentiable manifold $M$ is a tensor field $J \in \Gamma(\mbox{End}(TM))$ satisfying $J^{2}=-I$, where $I$ is the identity tensor field. The pair $(M,J)$ is called an almost complex manifold.
Question: The space $$\mathbb{C}^{n}=\{(z_1, \ldots, z_n): z_k=x_k+iy_k\} $$ carries a natural almost complex structure defined by $$J(\partial/\partial x{_k})=\partial/\partial y_{k}, \quad \quad J(\partial/\partial y_k)=-\partial/\partial x_{k}, $$ for $1 \le k \le n$.
Could someone show that $f:U \subset \mathbb{C}^{n}\rightarrow \mathbb{C}^{n}$ is holomorphic if and only if $f_{*}\circ J=J \circ f_{*}$, by showing that this condition is equivalent to the Cauchy–Riemann equation for each coordinate function? I didn't understand the generalization of Cauchy–Riemann equation for holomorphic function from $\mathbb{C}^{n}$ into $\mathbb{C}^{n}$. I know just this equations from $\mathbb{C}$ into $\mathbb{C}$.
Best Answer
Let's do it for $n=1$ first.
We have $$f(x+iy) = u(x, y) + iv(x, y),$$ and the Cauchy–Riemann equations are $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.$$
We have $$f_* = \begin{pmatrix} \partial_x u & \partial_y u\\ \partial_x v & \partial_y v \end{pmatrix}, \quad J= \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}.$$
So that $$f_*\circ J = \begin{pmatrix}\partial_y u & -\partial_x u\\ \partial_y v & -\partial_x v\end{pmatrix}, \quad J\circ f_* = \begin{pmatrix}-\partial_x v & -\partial_y v\\\partial_x u & \partial_y u\end{pmatrix}.$$ Hence, $J\circ f_*=f_*\circ J$ are just the Cauchy–Riemann equations.
In the general case you have variables $x_1, \dots, x_n, y_1, \dots, y_n$ and $$\begin{align*} f(x_1 + iy_1, \dots, x_n + iy_n) = \big( &u_1(x_1, \dots, y_n) +iv_1(x_1, \dots, y_n), \\ &\dots, \\ &u_n(x_1, \dots, y_n) +iv_n(x_1, \dots, y_n) \big).\end{align*}$$ This function is holomorphic if each component $u_k + iv_k$ is holomorphic. Cauchy–Riemann equations for this case can be found e.g. on wikipedia: $$\frac{\partial u_k}{\partial x_l} = \frac{\partial v_k}{\partial y_l},\quad \frac{\partial u_k}{\partial y_l} = -\frac{\partial v_k}{\partial x_l}.$$
The calculation with matrices is completely analogous to the case $n=1$. (Treat each entry as an $n\times n$ matrix).