Geodesic
Classification of~degenerations and Picard lattices of~K\" ahlerian
Izvestiya. Mathematics , Tome 83 (2019) no. 6, pp. 1201-1233.

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In [1]–[6] we classified the degenerations and Picard lattices of Kählerian K3 surfaces with finite symplectic automorphism groups of high order. This classification was not considered for the remaining groups of small order ($D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$ and $C_1$) because each of these cases requires very long and difficult considerations and calculations. Here we consider this classification for the dihedral group $D_6$ of order $6$.
Keywords: K3 surface, degeneration, Picard lattice
Mots-clés : automorphism group. 
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V. V. Nikulin. Classification of~degenerations and Picard lattices of~K\" ahlerian. Izvestiya. Mathematics , Tome 83 (2019) no. 6, pp. 1201-1233. http://geodesic.mathdoc.fr/item/IM2_2019_83_6_a3/

[1] V. V. Nikulin, “Kählerian K3 surfaces and Niemeier lattices. I”, Izv. Math., 77:5 (2013), 954–997 ; arXiv: 1109.2879 | DOI | DOI | MR | Zbl

[2] V. V. Nikulin, “Kählerian K3 surfaces and Niemeier lattices. II”, Development of moduli theory – Kyoto 2013, Adv. Stud. Pure Math., 69, Math. Soc. Japan, Tokyo, 2016, 421–471 | DOI | MR | Zbl

[3] V. V. Nikulin, “Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups”, Izv. Math., 79:4 (2015), 740–794 ; arXiv: 1403.6061 | DOI | DOI | MR | Zbl

[4] V. V. Nikulin, “Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. II”, Izv. Math., 80:2 (2016), 359–402 ; arXiv: 1504.00326 | DOI | DOI | MR | Zbl

[5] V. V. Nikulin, “Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. III”, Izv. Math., 81:5 (2017), 985–1029 ; arXiv: 1608.04373 | DOI | DOI | MR | Zbl

[6] V. V. Nikulin, “Classification of Picard lattices of K3 surfaces”, Izv. Math., 82:4 (2018), 752–816 ; arXiv: 1707.05677 | DOI | DOI | MR | Zbl

[7] V. V. Nikulin, Classification of degenerations and Picard lattices of Kählerian K3 surfaces with small finite symplectic automorphism groups, 2018, 39 pp., arXiv: 1804.00991v1

[8] I. R. S̆afarevic̆, B. G. Averbuh, Ju. R. Vaĭnberg, A. B. Z̆iz̆c̆enko, Ju. I. Manin, B. G. Moĭs̆ezon, G. N. Tjurina, A. N. Tjurin, “Algebraic surfaces”, Proc. Steklov Inst. Math., 75 (1965), 1–281 | MR | MR | Zbl

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

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

[12] 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

[13] 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

[14] V. V. Nikulin, “Konechnye gruppy avtomorfizmov kelerovykh poverkhnostei tipa KZ”, UMN, 31:2(188) (1976), 223–224 | MR | Zbl

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

[16] V. V. Nikulin, “Integral symmetric bilinear forms and some of their applications”, Math. USSR-Izv., 14:1 (1980), 103–167 | 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] Sh. Kondō, “Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of $K3$ surfaces”, With an appendix by Sh. Mukai, Duke Math. J., 92:3 (1998), 593–603 | DOI | MR | Zbl

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

[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. Ch. IV: Groupes de Coxeter et systèmes de Tits. Ch. V: Groupes engendrés par des réflexions. Ch. VI: Systèmes de racines, Actualites 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 | MR | Zbl

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

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