# Existence of non-constant holomorphic map between two given compact Riemann surfaces

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 $$>0$$?

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.

• But if the genus of $X$ is $>1$ then is that true? – Danny Apr 18 at 17:32
• I'm not sure, I have to go to a meeting right now, but I will think about it. – Michael Albanese Apr 18 at 17:51

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.