TITLE: Generalizations of de Franchis and function field Mordell
QUESTION [6 upvotes]: The classical de Franchis theorem, as generalized by S. Kobayashi and T. Ochiai ("Meromorphic mappings onto compact complex spaces of general type," Inventiones, 1975), states that if $X$ is a complex projective varity of general type, then for any fixed complex projective variety $Y$, there are only finitely many surjective mappings $Y \to X$. (A variant of this for Kobayashi-hyperbolic complex projective manifolds $X$ was conjectured by Lang and proved by C. Horst; those manifolds are expected to be of general type anyway, but if I understand correctly, this expectation has not been proved in full generality.)
I would like to ask here about the generalization of this problem whereby we drop the surjectivity assumption, and assume instead that $Y \to X$ is generically finite onto its image. To keep things simple, consider $\dim{Y} = 1$ and $\dim{X} = 2$, which is the first interesting case:
Question. Let $X$ be a surface of general type and $Y$ a fixed curve. Assume that there is no curve $Z$ such that $X$ is rationally dominated by $Y \times Z$. Does $X$ contain only finitely many images of non-constant morphisms $Y \to X$? [Is it plausible that the manifestly necessary assumption on $X$ should be sufficient?] 
This would be a generalization of the function field Mordell conjecture. [To see this, let $X \to B$ be a fibred surface of genus $> 1$, and choose any branched covering $Y \to B$ of genus $> 1$. Then $X \times_B Y$ is a surface of general type, and if it is domianted by a product $Y \times Z$, then $X/B$ is isotrivial. ] It would likewise generalize the theorem of Miyaoka and Lu (itself a very particular case of Lang's geometric conjecture), according to which a surface of general type contains only finitely many smooth rational curves.
I am aware of a 1985 paper by Y. Imayoshi ("Holomorphic maps of compact Riemann surfaces into $2$-dimensional compact $C$-hyperbolic manifolds," Math. Ann. 270), which solves the analogous question for a compact complex surface $X$ which admits a covering $\Omega \to X$ such that the bounded holomorphic functions on $\Omega$ separate points (i.e., the Caratheodory pseudo-distance is a distance on $\Omega$; such $X$ are a fortiori Kobayashi-hyperbolic.) In that case the conclusion is in fact the finiteness of the set of non-constant holomorhic maps from a fixed compact hyperbolic Riemann surface, and it is enough to assume that $X$ does not have a finite etale cover by the product of two compact hyperbolic Riemann surfaces.
But presumably, $C$-hyperbolicity is a very restrictive constraint on a surface of general type. Have the above algebraic statement or similar strengthenings of function field Mordell been considered, or at least explicitly stated anywhere?

REPLY [5 votes]: One of the major results in this direction, when $Y$ is either a rational or an elliptic curve, is due to Bogomolov, see 
F. Bogomolov, Families of curves on a surface of general type, Doklady AN SSSR 236 (1977),
no. 5, 1041-1044, in Russian; English translation: Soviet Math. Dokl. 18 (1977), 1294-1277.
The statement of Bogomolov's result is as follows:

Theorem (Bogomolov). Let $X$ be a surface of general type with $c_1^2(X) > c_2(X)$. Then, for any $g$, the curves of geometric genus $g$ on $X$ form a bounded family (roughly speaking, a family having only a finite number of irreducible components).

Since $X$ is of general type, it is not covered by rational or elliptic curves, i.e. curves of geometric genus $0$ and $1$ cannot deform. Then Bogomolov's theorem implies that there are only a finite number of rational or elliptic curves on $X$, that is we obtain the following

Corollary. Let $X$ be a surface of general type with $c_1^2(X) > c_2(X)$, and let $Y$ be either a rational curve or an elliptic curve. Then there are only finitely many images of non-constant morphisms $ Y \to X$. 

The condition $c_1^2(X) > c_2(X)$ is rather restrictive, for instance it is never satisfied for a hypersurface $X \subset \mathbb{P}^3$. It is expected that in any case neither rational nor elliptic curves can be dense on a variety of general type $X$ (I think this is a conjecture due to Kobayashi), but in principle one could have surfaces  of general type with $c_1^2(X) \leq c_2(X)$ and containing countably many rational or elliptic curves. However, I'm not aware of any example of this type.