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-Kollár's conjecture for complex projective manifolds with linear fundamental groups of any characteristic.