Hermitian-Einstein metrics and Chern number inequalities on parabolic stable bundles over Kähler manifolds

Abstract

Let X̄ be a compact complex manifold with a smooth Kähler metric and D = ∑i=1mDi a divisor in X̄ with normal crossings. Let E be a holomorphic vector bundle over X̄ with a stable parabolic structure along D. We prove that there exists a Hermitian-Einstein metric on E′ = E|X̄\D and obtain a Chern number inequality for a stable parabolic bundle. Without the assumption that the irreducible components Di of D meet transversely, using Hironaka's theorem on the resolution of singularities, we also get a Chern number inequality for a more general stable parabolic bundle.