Degenerations of K\"ahlerian K3 surfaces with finite symplectic automorphism groups
Izvestiya. Mathematics , Tome 79 (2015) no. 4, pp. 740-794.

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Using the results of [1]–[3] on Kählerian K3 surfaces and Niemeier lattices, we classify degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups with emphasis on degenerations of codimension 1.
Keywords: K3 surface, Kählerian surface, degeneration, singularities, Picard lattice, integral quadratic form.
Mots-clés : automorphism group
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V. V. Nikulin. Degenerations of K\"ahlerian K3 surfaces with finite symplectic automorphism groups. Izvestiya. Mathematics , Tome 79 (2015) no. 4, pp. 740-794. http://geodesic.mathdoc.fr/item/IM2_2015_79_4_a4/

[1] V. V. Nikulin, Kahlerian K3 surfaces and Niemeier lattices, arXiv: 1109.2879

[2] V. V. Nikulin, “Kählerian K3 surfaces and Niemeier lattices. I”, Izv. Math., 77:5 (2013), 954–997 | DOI | DOI | MR | Zbl

[3] V. V. Nikulin, “Kählerian K3 surfaces and Niemeier lattices. II”, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo (to appear)

[4] V. V. Nikulin, “Integral symmetric bilinear forms and some of their geometric applications”, Math. USSR-Izv., 14:1 (1980), 103–167 | DOI | MR | Zbl

[5] A. N. Todorov, “Applications of the Kähler–Einstein–Calabi–Yau metric to moduli of K3 surfaces”, Invent. Math., 61:3 (1980), 251–265 | DOI | MR | Zbl

[6] Yum-Tong Siu, “A simple proof of the surjectivity of the period map of $K3$ surfaces”, Manuscripta Math., 35:3 (1981), 311–321 | DOI | MR | Zbl

[7] D. Burns, M. Rapoport, “On the Torelli problem for Kählerian K-3 surfaces”, Ann. Sci. École Norm. Sup. (4), 8:2 (1975), 235–274 | MR | Zbl

[8] Vic. S. Kulikov, “Degenerations of $K3$ surfaces and Enriques surfaces”, Math. USSR-Izv., 11:5 (1977), 957–989 | DOI | MR | Zbl

[9] I. I. Pjatetski\u i-S̆apiro, I. R. S̆afarevic̆, “A Torelli theorem for algebraic surfaces of type K3”, Math. USSR-Izv., 5:3 (1971), 547–588 | DOI | MR | Zbl

[10] V. V. Nikulin, Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups, arXiv: 1403.6061

[11] V. V. Nikulin, “On Kummer surfaces”, Math. USSR-Izv., 9:2 (1975), 261–275 | DOI | MR | Zbl

[12] V. V. Nikulin, “Finite automorphism groups of Kähler K3 surfaces”, Trans. Moscow Math. Soc., 38, Amer. Math. Soc., Providence, RI, 1980, 71–135 | MR | Zbl

[13] E. Brieskorn, “Rationale Singularitäten komplexer Flächen”, Invent. Math., 4:5 (1967/1968), 336–358 | DOI | MR | Zbl

[14] H. Grauert, “Über Modifikationen und exzeptionelle analytische Mengen”, Math. Ann., 146:4 (1962), 331–368 | DOI | MR | MR | Zbl | Zbl

[15] D. Mumford, “The topology of normal singularities of an algebraic surface and a criterion for simplicity”, Inst. Hautes Études Sci. Publ. Math., 9 (1961), 5–22 | DOI | MR | Zbl

[16] H.-V. Niemeier, “Definite quadratische Formen der Dimension $24$ und Diskriminante $1$”, J. Number Theory, 5:2 (1973), 142–178 | DOI | MR | Zbl

[17] Sh. Mukai, “Finite groups of automorphisms of K3 surfaces and the Mathieu group”, Invent. Math., 94:1 (1988), 183–221 | DOI | MR | Zbl

[18] Gang Xiao, “Galois covers between K3 surfaces”, Ann. Inst. Fourier (Grenoble), 46:1 (1996), 73–88 | DOI | MR | Zbl

[19] GAP – Groups, Algorithms, Programming – a system for computational discrete algebra, Version 4.6.5, 2013 {http://www.gap-system.org}

[20] K. Hashimoto, “Finite symplectic actions on the K3 lattice”, Nagoya Math. J., 206 (2012), 99–153 ; arXiv: 1012.2682 | DOI | MR | Zbl

[21] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines, Actualités Sci. Indust., 1337, Hermann, Paris, 1968, 288 pp. | MR | MR | Zbl | Zbl

[22] J. H. Conway, N. J. A. Sloane, “Sphere packings, lattices and groups”, Grundlehren Math. Wiss., 290, Springer-Verlag, New York, 1988, xxviii+663 pp. | DOI | MR | MR | Zbl

[23] V. V. Nikulin, Degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups, II, 2015, arXiv: 1504.00326