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.