Reductive Shafarevich Conjecture

Author:

Ya Deng, Katsutoshi Yamanoi, Ludmil Katzarkov

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Complex Variables (math.CV)

journal:

date:

2023-06-04 16:00:00

Abstract

In this paper, we present a more accessible proof of Eyssidieux's proof of the reductive Shafarevich conjecture in 2004, along with several generalizations. In a nutshell, we prove the holomorphic convexity of the covering of a projective normal variety $X$, which corresponds to the intersection of kernels of reductive representations $\varrho:\pi_1(X)\to {\rm GL}_{N}(\mathbb{C})$. Our approach avoids the necessity of using the reduction mod $p$ method employed in Eyssidieux's original proof. Moreover, we extend the theorems to singular normal varieties under a weaker condition of absolutely constructible subsets, thereby answering a question by Eyssidieux, Katzarkov, Pantev, and Ramachandran. Additionally, we construct the Shafarevich morphism for reductive representations over quasi-projective varieties unconditionally, and proving its algebraic nature at the function field level.

PDF: Reductive Shafarevich Conjecture.pdf