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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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