I am reading the book Riemann surface by Donaldson. I want to understand the following Lemma (p.74).
Lemma. Let $X$ be a Riemann surface. There is a unique way to define a complex structure on $TX_p$ such that the derivative of any holomorphic function, defined on a neighborhood of $p$ in $X$, is complex linear.
By definition, a complex structure is an $\mathbb{R}$-linear map $J: TX_p \to TX_p$ such taht $J^2=-1$.
First, I don't understand how the derivative of a holomorphic function is related to $TX_p$.
Second, it is written that the proof follows from the definition of a holomorphic function but I don't see it.
Could you explain these for me?
I can't read Donaldson's mind, but if I had to guess, this is what he meant.
If we view $X$ as a real surface, then by picking coordinates $z=x+iy$ at $p$ we get a basis $\displaystyle \left\{\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right\}$ for the real tangent space $T_p X$. Let us define a complex structure by
$$\displaystyle J\left(\frac{\partial}{\partial x}\right)=\frac{\partial}{\partial y}$$
and
$$\displaystyle J\left(\frac{\partial}{\partial y}\right)=-\frac{\partial}{\partial x}$$
I claim that with this complex structure, given any holomorphic map $f:X\to Y$ between Riemann surfaces, the push-forward $f_{\ast,p}:T_p X\to T_{f(p)}Y$ (from real geometry) is actually complex linear.
Indeed, we need only check that $f_{\ast,p}$ commutes with $J$. But, let's check what $f_{\ast,p}$ does to the basis vectors of the tangent spaces. Let's pick coordinates $(x_1,y_1)$ at $p$ and $(x_2,y_2)$ at $f(p)$. Then,
$$\begin{aligned}f_{\ast,p}\left(\frac{\partial}{\partial x_1}\right)(x_2) &=\frac{\partial}{\partial x_1}(x_2\circ f)\\ &=\frac{\partial}{\partial x}(x_2\circ f\circ x_1^{-1})\\ & =\frac{\partial (f\circ x_1^{-1})}{\partial x}\frac{\partial}{\partial x_2}\end{aligned}$$
and similarly
$$f_{\ast,p}\left(\frac{\partial}{\partial y_1}\right)=\frac{\partial (f\circ y_1^{-1})}{\partial y}\frac{\partial}{\partial y_2}$$
So, for example the equality
$$f_{\ast,p}\left(J\left(\frac{\partial}{\partial x_1}\right)\right)=J\left(f_{\ast,p}\left(\frac{\partial}{\partial x_1}\right)\right)$$
amounts to the statement
$$\frac{\partial (f\circ x_1^{-1})}{\partial x}=-\frac{\partial (f\circ y_1^{-1})}{\partial y}$$
Do the other, and you'll get exactly what you might suspect: the Cauchy-Riemann equations.
Thus, with this "standard" complex structure (rotation by $i$ in the tangent space) we see that the pushforwards of holomorphic maps are actually $\mathbb{C}$-linear. Conversely, you can trace this argument back to see that this is precisely the complex structure that does this.
Please let me know if you would like me to explain any more!