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

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 \((X,\omega )\) $(X,\omega )$ , 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 \(\omega \) $\omega$ . 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.