Analyticity of compact complements of complete kähler manifolds

Let X be a Stein manifold, dimC X ≥ 2, K a compact subset of X, and ω an open subset of X containing K such that ω\K is connected. Suppose that ω\K carries a complete Kähler metric of bounded bisectional curvature, and locally of finite volume near K. If K admits a Stein neighborhood V, V. ω, such that V/K is connected and H² (V,R) = 0, then K is a complex analytic subvariety of X, hence reduced to a finite number of points.