Consider $\Bbb C^n$ with its standard hermitian product. This space produces many example of Kähler manifolds simply by taking a smooth affine variety $X\subseteq\Bbb C^n$ with the induced metric. Now I am wondering about the converse:
Question: Suppose that $X$ is a Kähler manifold that is also a smooth affine variety over $\Bbb C$. Can we assume that $X$ has an embedding $X\subseteq\Bbb C^n$ such that the metric of $X$ is the one inherited from the hermitian product on $\Bbb C^n$?
Note that I assume that $X$ is algebraic to begin with. It would false for a general Kähler manifold.