On the holomorphic convexity of reductive Galois coverings over compact K\"ahler surfaces

TL;DR

This paper extends the holomorphic convexity results for reductive Galois coverings from algebraic to K"ahler surfaces, using singularity analysis of exhaustion functions instead of p-adic methods.

Contribution

It generalizes previous algebraic surface results to K"ahler surfaces by replacing p-adic techniques with singularity analysis of exhaustion functions.

Findings

Proves holomorphic convexity for certain Galois coverings over K"ahler surfaces.

Replaces p-adic factorization with analysis of singularities in exhaustion functions.

Extends known results from algebraic to K"ahler geometry.

Abstract

This article generalizes the result of Katzarkov and Ramachandran from algebraic surfaces to K\"ahler surfaces. We follow their argument to prove the holomorphic convexity of a reductive Galois covering over a compact K\"ahler surface which does not have two ends, except that we replace the $p$-adic factorization theorem by an analysis of the singularities of the continuous subanalytic plurisubharmonic exhaustion function.