Linear Shafarevich Conjecture in positive characteristic, Hyperbolicity and Applications
Ya Deng, Katsutoshi Yamanoi
TL;DR
This work extends the linear Shafarevich program to positive characteristic by constructing the Shafarevich morphism for representations $\rho: \pi_1(X)\to{\rm GL}_N(K)$ with $K$ of characteristic $p>0$ and establishing a suite of equivalences that tie Zariski-closed subvarieties to finiteness of monodromy. It proves holomorphic convexity of universal covers in low $\Gamma$-dimension cases, and proves a generalized Green–Griffiths–Lang framework for big representations, including pseudo Picard and Brody hyperbolicity notions as well as log general type. The paper then applies these results to Campana’s abelianity conjecture in positive characteristic, showing special or $h$-special varieties have virtually abelian fundamental groups, and derives structure theorems akin to Kollár’s conjectures. Additional consequences include a structure theorem for varieties with big fundamental groups, a factorization principle for representations through Shafarevich morphisms, and a spectral-cover approach with canonical currents that yields Steinness results for universal coverings. Overall, the methods blend Katzarkov–Eyssidieux reductions, spectral covers, and non-abelian Hodge-type techniques adapted to positive characteristic to derive hyperbolicity and structural results with broad implications for algebraic geometry in any characteristic.
Abstract
Given a complex quasi-projective normal variety $X$ and a linear representation $\varrho:π_1(X)\to {\rm GL}_{N}(K)$ 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 $Γ$-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 $\varrho(π_1(X))$ 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 $\varrho(π_1(X))$ is virtually abelian. We also prove Claudon-Höring-Kollár's conjecture for complex projective manifolds with linear fundamental groups of any characteristic.