A finiteness property of representations of \(\pi_ 1\) of algebraic surfaces (Q2365239)

Let \(X\) be a compact complex algebraic smooth surface and \(\pi_1(X)\) be the fundamental group of \(X\). In this article we shall prove the following: Theorem. Suppose that \( X\) contains a numerically nonzero divisor \(D\) of rational curves (possibly singular) and the self intersection \(D^2=0\). Then any \(n\)-dimensional reductive representation \(\rho\) of \(\pi_1 (X)\) is either finite, or factors through a surjective morphism \(f:X \to C\) onto an algebraic curve \(C\) of connected fibers after passing to some finite étale covering and blowing of \(X\).