The differential geometry of blow-ups

We discuss the local geometry in the vicinity of a sphere $\mathbb P^1$ embedded with a negative normal bundle. We show that the behavior of the Kähler potential near a sphere embedded with a given normal bundle can be determined using the adjunction formula. As a by-product, we construct (asymptotically locally complex-hyperbolic) Kähler–Einstein metrics on the total spaces of the line bundles $\mathcal O(-m)$, $m\ge3$, over $\mathbb P^1$.