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

Complex surfaces with mutually non-biholomorphic universal covers