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 $X\to T/G$. We then prove that a K3 surface is a generalised Kummer surface of type $Km(T,G)$ if and only if its N\'eron-Severi group contains $K_{G}$. For smooth-orbifold surfaces $\mathcal{X}$ of Kodaira dimension $\geq 0$, Kobayashi proved the orbifold Bogomolov Miyaoka Yau inequality $c_{1}^{2}(\mathcal{X})\leq3c_{2}(\mathcal{X}).$ For Kodaira dimension $2$, the case of equality is characterised as $\mathcal{X}$ being uniformized by the complex $2$-ball $\mathbb{B}_{2}$. For smooth-orbifold K3 and Enriques surfaces we characterize the case of equality as being uniformized by $\mathbb{C}^{2}$.