TITLE: Existence of non-constant holomorphic map between two given compact Riemann surfaces
QUESTION [5 upvotes]: 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 $>0$?

REPLY [5 votes]: 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.