Non-Abelian Hodge Theories and the Reductive Shafarevich Conjecture

Speaker

My research subjects contain complex algebraic-analytic geometry, complex hyperbolicity, non-abelian Hodge theories in both Archimedean and non-Archimedean settings, Nevanlinna theory, and the interplay among them. My current interest is harmonic mapping to Euclidean buildings, linear Shafarevich conjecture, hyperbolicity of algebraic varieties via representation of fundamental groups.

Course Description

In his book “Basic Algebraic Geometry II”, Shafarevich asked the question whether the universal covering of a complex projective variety is holomorphically convex. This problem, now known as the Shafarevich conjecture, has been extensively studied with the introduction of non-abelian Hodge theories by Simpson, Eyssidieux, Katzarkov, Pantev, Ramachandran, the speaker, etc. In this lecture, I will present a comprehensive proof of the reductive Shafarevich conjecture, which establishes that the universal covering of a complex projective normal variety is holomorphically convex if their fundamental groups can be faithfully represented as Zariski-dense subgroups of complex reductive algebraic groups. If times allows, I will explain some extension of the reductive Shafarevich conjecture to the quasi-projective cases.

The mini-course is structured as follows:

1. Non-Abelian Hodge Theories in the Archimedean Setting by Simpson: Simpson correspondence, C* -action Character varieties, Ubiquity theorem, etc.

2. Non-Abelian Hodge Theories in the Non-Archimedean Setting: Harmonic mapping to Euclidean buildings, Spectral coverings, Reduction theorem for representation into non-archimedean local fields, etc.

3. Construction of the Shafarevich Morphism.

4. Proof of the Reductive Shafarevich Conjecture.