# Blow up of a Kähler manifold is Kähler

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?

• Have you looked at pp. 185-187 of Griffiths/Harris for an analogous discussion of positive line bundles on the blowup? – Ted Shifrin Jun 24 '19 at 17:24
• @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 =( – Michael Jun 25 '19 at 9:04
• 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 – Michael Jun 25 '19 at 9:10
• See also the reference here: math.stackexchange.com/questions/2537880/… – Moishe Kohan Jun 25 '19 at 23:11