Linear Shafarevich Conjecture in positive characteristic, Hyperbolicity and Applications

Given a complex quasi-projective normal variety $X$ and a linear representation $\varrho:\pi_1(X)\to {\rm GL}_{N}(K)$ with $K$ any field of positive characteristic, we mainly establish the following results: 1. the construction of the Shafarevich morphism ${\rm sh}_\varrho:X\to {\rm Sh}_\varrho(X)$ associated with $\varrho$. 2. In cases where $X$ is projective, $\varrho$ is faithful and the $\Gamma$-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 $\varrho(\pi_1(X))$ 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 $\varrho(\pi_1(X))$ is virtually abelian. We also prove Claudon-H\"oring-Koll\'ar's conjecture for complex projective manifolds with linear fundamental groups of any characteristic.