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?