TITLE: Are holomorphic vector bundles over Kähler manifolds Kähler
QUESTION [5 upvotes]: Let $X$ be a Kähler manifold and $E\to X$ a holomorphic vector bundle. Is there a Kähler structure on $E$ compatible with its complex structure?

REPLY [6 votes]: Proposition 3.18 of Voisin's Hodge Theory and Complex Algebraic Geometry I says that, if $X$ is compact Kahler and $E$ is a holomorphic vector bundle over $X$, then $\mathbb{P}(E)$ is Kahler. Since $E$ embeds as an open submanifold of $\mathbb{P}(E \oplus \mathbb{C})$, this establishes your result for $X$ compact, and I think her proof could be simplified if you just want the vector bundle version and not the projective bundle version. But it looks to me like she actually is using compactness in a nontrivial way.