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

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 $X\to T\mathrm{/}G$. 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 $\ge 0$, Kobayashi proved the orbifold Bogomolov Miyaoka Yau inequality $c_1^2(\mathcal{X})\le 3c_2(\mathcal{X})\mathrm{.}$ 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 ${\mathbb{C}}^2$.