Parabolic bundles and representations of the fundamental group

Abstract

Let X be a smooth complex projective variety with Neron-Severi group isomorphic to Z, and D an irreducible divisor with normal crossing singularities. Assume 1 <r ≤ 3. We prove that if π₁(X) doesn't have irreducible PU(r) representations, then π₁(X-D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth and X is a complex surface.