Topology, Hyperbolicity, and the Shafarevich Conjecture for Complex Algebraic Varieties

TL;DR

This survey reviews recent advances in the understanding of complex algebraic varieties, focusing on hyperbolicity, topology, and conjectures like Shafarevich, highlighting techniques and interrelations among these areas.

Contribution

It synthesizes recent progress on conjectures and theories connecting hyperbolicity, topology, and Hodge theory in complex algebraic varieties, emphasizing linear versions and characterizations.

Findings

Connections between hyperbolicity and fundamental group representations

Characterizations of hyperbolic varieties via local systems

Progress on conjectures related to algebraic and topological properties

Abstract

This survey presents recent developments concerning the Shafarevich conjecture, non-abelian Hodge theories, hyperbolicity, and the topology of complex algebraic varieties, as well as the interplay among these areas. More precisely, we present the main ideas and techniques involved in the linear versions of the following conjectures: the Shafarevich conjecture, the Chern-Hopf-Thurston conjecture, Koll\'ar's conjecture on the holomorphic Euler characteristic, the de Oliveira-Katzarkov-Ramachandran conjecture, and Campana's nilpotency conjecture. In addition, we discuss characterizations of the hyperbolicity of complex quasi-projective varieties via representations of their fundamental groups, together with the generalized Green-Griffiths-Lang conjecture in the presence of a big local system.