Analyticity of compact complements of complete Kähler manifolds

Let $X$ be a Stein manifold, $\dim _{\mathbb{C}}X\geq 2$, $K$ a compact subset of $X$, and $\Omega$ an open subset of $X$ containing $K$ such that $\Omega \diagdown K$ is connected. Suppose that $\Omega \diagdown K$ carries a complete Kähler metric of bounded bisectional curvature, and locally of finite volume near $K$. If $K$ admits a Stein neighborhood $V$, $V\subseteq \Omega$, such that $V\diagdown K$ is connected and $H^{2}\left( V,\mathbb{R}\right) =0,$ then $K$ is a complex analytic subvariety of $X,$ hence reduced to a finite number of points.