Anti-holomorphic involution induced by complex conjugation

Question

Let $X$ be a Riemann surface. What is the antiholomorphic involution $\sigma: X\to X$ induced by the complex conjugation?

I don't understand how can we define $\sigma$. For example take $x\in X$ send it with a local chart $\phi_U$, to $\phi_U(x)\in\mathbb C$, the apply the conjugation $\overline {\phi_U(x)}\in\mathbb C$. How do we come back to $X$ in order to define $\sigma(x)?$

Answer 1

Let $Y$ be a two dimensional real manifold, if it exists a Riemann surface structure $X$ on $Y$ is given by some charts $\phi_j : U_j\subset \Bbb{C}\to Y$, such that $f : X\to \Bbb{C}$ is meromorphic iff $\forall j, f\circ \phi_j$ is meromorphic on its domain, and the chart compatibility condition is that $Y$ is covered by $\bigcup_j \phi_j(U_j)$ and each $\phi_j^{-1}\phi_i$ is analytic on its domain.

There is another Riemann surface structure $X^*$ obtained from the charts $$\phi_j^* : \overline{U_j}\to Y,\qquad \phi_j^*(z)=\phi_j(\overline{z})$$ It satisfies the chart compatibility condition and the identity on the underlying real manifold is an anti-holomorphic bijection $X\to X^*$, and $p\mapsto f(p)$ is meromorphic $X\to \Bbb{C}$ iff $p\mapsto \overline{f(p)}$ is meromorphic $X^*\to \Bbb{C}$.

When $X$ is connected compact its field of meromorphic functions is isomorphic to $\Bbb{C}(x)[y]/(q(x,y))$ with $q(x,y)=\sum_{n,m} a_{n,m}x^n y^m\in \Bbb{C}(x)[y]$ monic irreducible and the function field of $X^*$ is $\Bbb{C}(z)[w]/(q^*(z,w))$ where $q^*(z,w)=\sum_{n,m} \overline{a_{n,m}}z^n w^m$. The anti-holomoprhic map above is $(x,y) \to (z,w), c\in\Bbb{C}\to \overline{c}$.

The uniformization theorem (embedding the universal cover into $\Bbb{C}\cup \infty$) is another way to visualize $X^*$.

Answer 2

This only makes sense when $X$ has additional structure. In particular, if $X$ is a curve in $\mathbb{P}^n$ which is defined by a collection of polynomials with real coefficients, then the map $\mathbb{P}^n\to\mathbb{P}^n$ given by complex conjugation on each homogeneous coordinate restricts to an anti-holomorphic involution of $X$.