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 Kahler manifolds ( X , omega ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,\omega )$$\end{document} , where X is the complement of a divisor in a compact Kahler manifold and we impose some conditions on the cohomology class and the asymptotic behaviour of the Kahler form omega \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} . 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.