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Is it possible to find a complete, non compact Kahler manifold $(X,\omega)$ with a closed, connected, non compact complex submanifold $Y\subset X$ of codimension at least 2 such that the blow-up of $X$ along $Y$ is not Kähler?

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When $Y$ is compact, the blow-up is always Kahler; see e.g. Lemma 3.4 in this paper (this is a generally known folklore theorem which we had to use, and hence written down).

For $Y$ non-compact the argument should be similar, but more cumbersome.

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    $\begingroup$ Hi, i know that if $Y$ is compact then blow up is Kahler but to construct the metric on $Bl_{Y}X$ substantially you use $\pi^{*}\omega+\varepsilon c_{1}(\mathcal{O}(-E))$ where $E$ is the exceptional divisor and $\pi: Bl_{Y}X\rightarrow X$ is the canonical surjection. But the delicate thing is that the threshold $\varepsilon$ for which the above is a metric is bounded from below when $Y$ is compact, but is no longer obvious (at least to me) when $Y$ is not compact. $\endgroup$ – Italo Jun 19 '14 at 10:46
  • $\begingroup$ Same argument works, I think (I just checked, don't see any problems) $\endgroup$ – Misha Verbitsky Jun 22 '14 at 11:24

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