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?

smoothprojective Kobayashi varieties. $\endgroup$