The K\"ahler metric of a blow-up

After a review of the general properties of holomorphic spheres in complex surfaces we describe the local geometry in the vicinity of a CP^1 embedded with a negative normal bundle. As a by-product, we build (asymptotically locally hyperbolic) Kahler-Einstein metrics on the total spaces of the line bundles O(-m), m >= 3 over CP^1. We check that the behavior of the Kahler potential is compatible with the Chern-Weil formulas for the Euler characteristic and signature. We also describe two supersymmetric setups where relevant constructions arise.