K\"ahlerian K3 surfaces and Niemeier lattices.~I
Izvestiya. Mathematics , Tome 77 (2013) no. 5, pp. 954-997.

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Using the results obtained in [1], Remark 1.14.7, we clarify the relation between Kählerian $\mathrm{K3}$ surfaces and Niemeier lattices. We emphasize that all 24 Niemeier lattices are important in the description of $\mathrm{K3}$ surfaces, not only the one related to the Mathieu group.
Keywords: $\mathrm{K3}$ surface, Kählerian surface, integer quadratic form.
Mots-clés : automorphism group
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V. V. Nikulin. K\"ahlerian K3 surfaces and Niemeier lattices.~I. Izvestiya. Mathematics , Tome 77 (2013) no. 5, pp. 954-997. http://geodesic.mathdoc.fr/item/IM2_2013_77_5_a3/

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

[2] V. V. Nikulin, “Konechnye gruppy avtomorfizmov kelerovykh poverkhnostei tipa K3”, UMN, 31:2 (1976), 223–224 | MR | Zbl

[3] V. V. Nikulin, “Finite automorphism groups of Kahler K3 surfaces”, Trans. Mosc. Math. Soc., 2 (1980), 71–135 | MR | Zbl | Zbl

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

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

[6] Sh. Kondō, “Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of K3 surfaces”, Duke Math. J., 92:3 (1998), 593–603 | DOI | MR | Zbl

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

[8] V. V. Nikulin, Kahlerian $\mathrm{K3}$ surfaces and Niemeier lattices. I, arXiv: 1109.2879

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

[10] J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren Math. Wiss., 290, Springer-Verlag, New York, 1988 | MR | MR | MR | Zbl

[11] 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, Hermann, Paris VI, 1968 | MR | MR | Zbl | Zbl

[12] I. R. Shafarevich, B. G. Averbukh, Yu. R. Vainberg, A. B. Zhizhchenko, Yu. I. Manin, B. G. Moishezon, G. N. Tyurina, A. N. Tyurin, “Algebraicheskie poverkhnosti”, Tr. MIAN SSSR, 75, Nauka, M., 1965, 3–215 | MR | Zbl

[13] D. Burns, M. Rapoport, “On the Torelli problem for Kählerian K-3 surfaces”, Ann. Sci. École Norm. Sup. (4), 8:2 (1975), 235–273 | MR | Zbl

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

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

[16] A. N. Todorov, “Applications of the Kähler–Einstein–Calabi–Yau metric to moduli of K3 surfaces”, Invent. Math., 61:3 (1981), 251–265 | DOI | MR | Zbl

[17] I. I. Pjateckiǐ-Šapiro, I. R. Šafarevič, “A Torelli theorem for algebraic surfaces of type K3”, Math. USSR-Izv., 5:3 (1971), 547–588 | DOI | MR | Zbl | Zbl

[18] W. Barth, “K3 surfaces with nine cusps”, Geom. Dedicata, 72:2 (1998), 171–178 | DOI | MR | Zbl

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

[20] A. Taormina, K. Wendland, The overarching finite symmetry group of Kummer surfaces in the Mathieu group $M_{24}$, arXiv: 1107.3834