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

TL;DR

This paper classifies generalized Kummer surfaces by studying their associated lattices and characterizes when K3 surfaces are of this type, also exploring the orbifold Bogomolov-Miyaoka-Yau inequality for surfaces with non-negative Kodaira dimension.

Contribution

It completes the classification of generalized Kummer surfaces by analyzing the last two unknown automorphism groups and characterizes K3 surfaces that are generalized Kummer surfaces.

Findings

Computed the Kummer lattice for the last two automorphism groups.

Established a criterion for K3 surfaces to be generalized Kummer surfaces based on their Néron-Severi group.

Characterized equality cases in the orbifold Bogomolov-Miyaoka-Yau inequality for specific surfaces.

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 $\mathbb{C}^{2}$.