  In this paper, we prove the holomorphic convexity of the covering of a
complex projective {normal} variety $X$, which corresponds to the intersection
of kernels of reductive representations $\rho:\pi_1(X)\to {\rm
GL}_{N}(\mathbb{C})$, therefore answering a question by Eyssidieux, Katzarkov,
Pantev, and Ramachandran in 2012. It is worth noting that Eyssidieux had
previously proven this result in 2004 when $X$ is smooth. While our approach
follows the general strategy employed in Eyssidieux's proof, it introduces
several improvements and simplifications. Notably, it avoids the necessity of
using the reduction mod $p$ method in Eyssidieux's original proof.
  Additionally, we construct the Shafarevich morphism for complex reductive
representations of fundamental groups of complex quasi-projective varieties
unconditionally, and proving its algebraic nature at the function field level.
