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

Roulleau, Xavier

doi:10.1090/tran/7507

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

A generalized Kummer surface $X=\operatorname {Km}(T,G)$ 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 last two 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 generalized Kummer surface of type $\operatorname {Km}(T,G)$ if and only if its Néron-Severi group contains $K_{G}$.

For smooth-orbifold surfaces $\mathcal {X}$ of Kodaira dimension $\geq 0$, Kobayashi proved the orbifold Bogomolov-Miyaoka-Yau inequality $c_{1}^{2}(\mathcal {X})\leq 3c_{2}(\mathcal {X}).$ For Kodaira dimension $2$, the case of equality is characterized 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 $\mathbb {C}^{2}$.

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