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Let $X$ be a compact Kähler manifold and $x\in X$. Denoty $Bl_x X$ the blow-up of $X$ at $x$ and $E$ the exceptional divisor.

I want to see why $Bl_x X$ is Kähler.

I know the idea is to take the Kähler form $\omega$ of $X$ and a representative $\alpha$ of the Chern class $c_1(\mathcal O(-E))$ and combine them to the form $\omega' = k \cdot p^* \omega + \alpha$ for $k>>0$.

My question is, how ensure that there is a right representative of $c_1(\mathcal O(-E))$ . I guess, we want to use that $c_1(\mathcal O(-E)|_E)$ has a positive representative and then extend it to the whole space $Bl_x X$ in a way, such that the extension represents $c_1(\mathcal O(-E))$. But why can I do this?

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    $\begingroup$ Have you looked at pp. 185-187 of Griffiths/Harris for an analogous discussion of positive line bundles on the blowup? $\endgroup$ – Ted Shifrin Jun 24 '19 at 17:24
  • $\begingroup$ @TedShifrin: I did now. I see, they give an explicit construction of the metric,so this is the straight forward way. I was hoping to save some work by using the naturality of chern classes though =( $\endgroup$ – Michael Jun 25 '19 at 9:04
  • $\begingroup$ At a closer look, it seems that chern classes are somewhere hidden in the argument; at least they calculate the curvature of the chern connection $\endgroup$ – Michael Jun 25 '19 at 9:10
  • $\begingroup$ See also the reference here: math.stackexchange.com/questions/2537880/… $\endgroup$ – Moishe Kohan Jun 25 '19 at 23:11

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