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On generalized Kummer surfaces and the orbifold Bogomolov-Miyaoka-Yau inequality. (English) Zbl 07062694
Summary: 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 $$\mathbb{C}^{2}$$.

##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 14L30 Group actions on varieties or schemes (quotients) 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 14J50 Automorphisms of surfaces and higher-dimensional varieties
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