Kähler–einstein Metrics on Log Del Pezzo Surfaces in Weighted Projective 3–spaces

A log del Pezzo surface is a projective surface with quotient singularities such that its anticanonical class is ample. Such surfaces arise naturally in many different contexts, for instance in connection with affine surfaces [Miyanishi81], moduli of surfaces of general type [Alexeev94], 3 and 4 dimensional minimal model program [Alexeev93]. They also provide a natural testing ground for existence results of Kähler–Einstein metrics. The presence of quotient singularities forces us to work with orbifold metrics, but this is usually only a minor inconvenience. Log del Pezzo surfaces with a Kähler–Einstein metric also lead to Sasakian– Einstein 5–manifolds by [Boyer–Galicki00]. In connection with [Demailly-Kollár99], the authors ran a computer program to find examples of log del Pezzo surfaces in weighted projective spaces. The program examined weights up to a few hundred and produced 3 examples of log del Pezzo surfaces where the methods of [Demailly-Kollár99, §6] proved the existence of a Kähler–Einstein metric. The aim of this paper is twofold. First, we determine the complete list of anticanonically embedded quasi smooth log del Pezzo surfaces in weighted projective 3-spaces. Second, we improve the methods of [Demailly-Kollár99, 6.10] to prove that many of these admit a Kähler–Einstein metric. The same method also proves that some of these examples do not have tigers (in the colorful terminology of [Keel-McKernan99]). Higher dimensional versions of these results will be considered in a subsequent paper.