Linear Shafarevich Conjecture in positive characteristic, Hyperbolicity and Applications

Abstract

Given a complex quasi-projective normal variety $X$ and a linear representation $\varrho :\ddot{I}{€}_1(X)\to {\mathrm{GL}}_N(K)$ with $K$ any field of positive characteristic, we mainly establish the following results: 1. the construction of the Shafarevich morphism ${\mathrm{sh}}_{\varrho }:X\to {\mathrm{Sh}}_{\varrho }(X)$ associated with $\varrho$. 2. In cases where $X$ is projective, $\varrho$ is faithful and the $\hat{I}“$-dimension of $X$ is at most two (e.g. $\dim X=2$), we prove that the Shafarevich conjecture holds for $X$. 3. In cases where $\varrho$ is big, we prove that the Green-Griffiths-Lang conjecture holds for $X$. 4. When $\varrho$ is big and the Zariski closure of $\varrho (\ddot{I}{€}_1(X))$ is a semisimple algebraic group, we prove that $X$ is pseudo Picard hyperbolic, and strongly of log general type. 5. If $X$ is special or $h$-special, then $\varrho (\ddot{I}{€}_1(X))$ is virtually abelian. We also prove Claudon-Höring-Kollár's conjecture for complex projective manifolds with linear fundamental groups of any characteristic.

Final version, 34 pages. Exposition greatly improved according to the referees' suggestions. To appear in Crelle's Journal