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Let $(X,\omega)$ be a Kähler manifold not necessarily compact of complex dimension $n$. Let $\pi:E\to X$ be a holomorphic vector bundle of rank $r$, then $E$ can be seen as a complex manifold of complex dimension $n+r$. I also read that the total space of the canonical bundle $K_X:=\bigwedge^n(T^{1,0}X)^*$ over $X$ has a structure of a non-compact Calabi Yau manifold. Then it is natural to ask if a (complete) Ricci-flat Kähler metric exists on such manifold.

Could you please give me some references in which the authors treat this case with full details? I'm interested also in all the local calculation starting from the manifold structure on the bundle.

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    $\begingroup$ FYR (and the reference therein). Not an expert, but it seems that existence is not known unless in some special case. Obviously Yau's solution works only in compact manifold. The more famous such metric is the Eguchi–Hanson metric. $\endgroup$ Commented Apr 29, 2021 at 16:43

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In Métriques Kähleriennes et fibrés holomorphes, Calabi showed that the total space of the canonical bundle or the cotangent bundle of a positive Einstein-Kähler manifold admits a complete, Ricci-flat Kähler metric.

Corollary B.2 in A momentum construction for circle-invariant Kähler metrics extends this to the total space of a tensor product of canonical bundles over a product of positive Einstein-Kähler manifolds. The calculations here are explicit, enough so that it's easy to plot the fibre metric as a surface of rotation in Euclidean three-space.

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