Rational points of varieties with ample cotangent bundle over function fields (Q1659912)

The paper addresses the following variation of Bombieri-Lang conjecture over function fields. Conjecture 1. Let $K_0$ be the function field of a variety defined over an algebraically closed field $k_0$. Let $Z$ be a variety of general type over $K_0$. Suppose that $\text{Zar}(Z(K_0))=Z$, where Zar denotes the Zariski closure. Then there exists a variety $Z_0$ over $k_0$ and a rational, dominant, generically finite $K_0^{\text{sep}}$-map $g: Z_{0,K_0^{\text{sep}}} \to Z_{K_0^{\text{sep}}}$. Similarly to the original Bombieri-Lang conjecture, Conjecture 1 has been proved (with several contributions) in the case where $Z$ is embeddable in an abelian variety. Another case where Conjecture 1 has been proved (again with several contributions) is when $\dim(Z)=1$. The paper under review proves Conjecture 1.1 in the situation where $K_0$ has transcendence degree 1 over its prime field and $Z$ is a subvariety of a larger variety $Z'$, where $Z'$ is smooth and has ample cotangent bundle over $K_0$. It is worth noticing that this result allows to give examples of varieties of general type in positive characteristic which satisfy Conjecture 1 and are not embeddable in abelian varieties.