Hermitian–Yang–Mills connections on some complete non-compact Kähler manifolds

Abstract We give an algebraic criterion for the existence of projectively Hermitian–Yang–Mills metrics on a holomorphic vector bundle E over some complete non-compact Kähler manifolds (X, ) (X, ω), where X is the complement of a divisor in a compact Kähler manifold and we impose some conditions on the cohomology class and the asymptotic behaviour of the Kähler form ω. We introduce the notion of stability with respect to a pair of (1, 1) -classes which generalizes the standard slope stability. We prove that this new stability condition is both sufficient and necessary for the existence of projectively Hermitian–Yang–Mills metrics in our setting.