Riemann Existence Theorem for Real Curve

2010-12-03T12:34:14Z

By real curve, we mean a Riemann surface $X$ together with an anti-holomorphic involution $\sigma : X\rightarrow X$. Let $S$ be a finite subset of $X$. For each point $x\in S$, we associate a positive integer $m_x\geq 2$. Then by Riemann existence theorem, there exists a universal covering $\pi : Y\rightarrow X$ such that $S$ is the branch locus of $\pi$ and $m_x$ is ramification index of $\pi$ over $x\in S$.

Question:- Is it true that the involution $\sigma$ can be lift to $Y$ making it real curve?

Answer by Francesco Polizzi for Riemann Existence Theorem for Real Curve

2010-12-03T13:12:41Z

In general, the property of being real is not preserved by finite coverings, not even by Galois ones.

For instance, take $X= \mathbb{P}^1$, which is a real curve with the anti-holomorphic involution $\sigma(z) = \bar{z}$.

Now every elliptic curve $Y$ is a double cover of $\mathbb{P}^1$ branched in four points, but not all elliptic curves have a real structure.

More precisely, $\sigma$ can be lifted to $Y$ if and only if the double cover $Y \to \mathbb{P}^1$ admits an affine equation of the form

$w^2=(z-a)(z-\bar{a})(z-b)(z-\bar{b}) \quad a,b \in \mathbb{C}$.

In this case there are exactly two liftings, namely

$(z,w) \to (\bar{z}, \bar{w}) \quad $ and $ \quad (z,w) \to (\bar{z}, -\bar{w})$.

Riemann Existence Theorem for Real Curve - MathOverflow

Riemann Existence Theorem for Real Curve

Answer by Francesco Polizzi for Riemann Existence Theorem for Real Curve