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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 $>0$?

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2 Answers 2

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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.

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Let me first consider the case of nonconstant holomorphic maps $X\to T^2$, from compact connected Riemann surfaces of genus $g\ge 2$ to tori (smooth elliptic curves). Every such map is determined (up to a translation of $T^2$) by the induced map of fundamental groups $G=\pi_1(X)\to \pi_1(T^2)=Z^2$. There are only countably many homomorphisms $G\to Z^2$ and the moduli space of elliptic curves is 1-dimensional. Hence, the subset of the moduli space ${\mathcal M}_g$ consisting of Riemann surfaces admitting such holomorphic maps is a countable union of complex subvarieties of dimension $\le 1$. Since ${\mathcal M}_g$ is an irreducible variety of dimension $3g-3$, we conclude that only a meager (in the classical topology) subset of ${\mathcal M}_g$ consists of Riemann surfaces admitting non constant holomorphic maps to elliptic curves. In particular, not every Riemann surface of genus $g\ge 2$ admits such a map.

The situation is essentially the same for nonconstant holomorphic maps $X\to Y$ where $X$ and $Y$ are hyperbolic Riemann surfaces: By Severi's finiteness theorem (see my answer here), given $X$, there are only finitely many (up to a natural isomorphism) pairs $(Y, f)$ where $Y$ is a Riemann surface of genus $\ge 2$ and $f: X\to Y$ is a nonconstant holomorphic map.

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