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Let M be an ^-dimensional complex manifold with a real analytic kahler metric. Throughout this paper, a kahler manifold is assumed to have a real analytic kahler metric without mentioning it. A relatively compact domain D in M is called a pseudoconvex (resp. strongly pseudoconvex) domain if there exist a neighborhood U of p and a pseudoconvex (resp. strongly pseudoconvex) function (p on U satisfying Df] U= {(p<$\ for each boundary point p^dD. We write simply s-pseudoconvex domains (resp. functions) for strongly pseudoconvex domains (resp. functions). Note that pseudoconvex domains are not always Stein manifolds. The purpose of the present paper is to show the following theorem:

Pseudoconvex domains on a Kähler manifold with positive holomorphic bisectional curvature