Given two compact Riemann surfaces $X,Y$, can we always find a non-constant holomorphic map from $X$ to $Y$? In particular, when $Y$ is a elliptic curve, does that map exist?
Michael Albanese has given a negative answer for the case that when $X$ is $\mathbb{C}P^1$, but I still want to know if it's true for the cases that $X$ has genus $>0$?
Best Answer
No. For example, if $f : \mathbb{CP}^1 \to \mathbb{C}/\Lambda$ is holomorphic, then because $\mathbb{CP}^1$ is simply connected, it has a lift $\tilde{f} : \mathbb{CP}^1 \to \mathbb{C}$. That is, $\tilde{f}$ satisfies $f = \pi\circ\tilde{f}$ where $\pi : \mathbb{C} \to \mathbb{C}/\Lambda$ is the universal covering map (i.e. the quotient map). Every holomorphic function on a compact complex manifold is constant, so $\tilde{f}$ is constant, and therefore so is $f$.
More generally, if $f : X \to Y$ is a non-constant holomorphic map, then $g(X) \geq g(Y)$. To see this, recall that a non-constant holomorphic map between Riemann surfaces is a branched covering. Let $f : X \to Y$ be a non-constant holomorphic map, then by the Riemann-Hurwitz formula,
$$\chi(X) = d\chi(Y) - \sum_{p \in X}(e_p - 1)$$
where $e_p$ is the ramification index at $p$ and $d \in \mathbb{Z}_+$. If $g(Y) = 0$ (i.e. $Y = \mathbb{CP}^1$), there is nothing to prove, so suppose $g(Y) > 0$. Note that
$$\chi(X) = d\chi(Y) - \sum_{p \in X}(e_p - 1) \leq d\chi(Y) \leq \chi(Y)$$
where the last inequality uses the fact that $\chi(Y) \leq 0$. Therefore, $g(X) \geq g(Y)$.
As for the existence of such maps, there are non-constant holomorphic maps $f : X \to \mathbb{CP}^1$ for any Riemann surface $X$ (of any genus); this is equivalent to the existence of a non-zero meromorphic function. I'm not sure about the case where the target has positive genus, but it is related to this question.