  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.
