Complex surfaces with mutually non-biholomorphic universal covers

TL;DR

This paper constructs a family of complex surfaces, specifically Kodaira fibrations, with uncountably many mutually non-biholomorphic universal covers, expanding understanding of universal covers in complex geometry.

Contribution

It introduces explicit examples of complex surfaces with diverse universal covers, demonstrating uncountably many non-biholomorphic cases, and determines their slopes.

Findings

Constructed explicit family of complex surfaces with diverse universal covers

Proved uncountably many non-biholomorphic universal covers exist

Determined the slopes of these surfaces

Abstract

It is known that the universal cover of compact Riemann surface is either the projective line, the complex plane or the unit disk. In this article we construct a very explicit family of complex surfaces that gives rise to uncountably many mutually non-biholomorphic universal covers. The slope of these surfaces, which are going to be total spaces of Kodaira fibrations, is also determined