On generalized Kummer surfaces and the orbifold Bogomolov-Miyaoka-Yau inequality

Abstract

A generalized Kummer surface X = Km(T, G) is the resolution of a quotient of a torus T by a finite group of symplectic automorphisms G. We complete the classification of generalized Kummer surfaces by studying the two last groups which have not been yet studied. For these surfaces we compute the associated Kummer lattice K G , which is the minimal primitive sub-lattice containing the exceptional curves of the resolution X → T /G. We then prove that a K3 surface is a generalized Kummer surface of type Km(T, G) if and only if its Neron-Severi group contains K G. For smooth-orbifold surfaces X of Kodaira dimension ≥ 0, Kobayashi proved the orbifold Bogomolov Miyaoka Yau inequality c 2 1 (X) ≤ 3c 2 (X). For Kodaira dimension 2, the case of equality is characterized as X being uniformized by the complex 2-ball B 2. For smooth-orbifold K3 and Enriques surfaces we characterize the case of equality as being uniformized by C 2 .