Geodesic
Classification of Degenerations and Picard Lattices of K\"ahlerian K3 Surfaces with Symplectic Automorphism Group $C_4$
Informatics and Automation, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 148-179.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the author's papers of 2013–2018, the degenerations and Picard lattices of Kählerian K3 surfaces with finite symplectic automorphism groups of high order were classified. For the remaining groups of small order—$D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$, and $C_1$—the classification was not completed because each of these cases requires very long and difficult considerations and calculations. The case of $D_6$ was recently completely studied in the author's paper of 2019. In the present paper an analogous complete classification is presented for the cyclic group $C_4$ of order $4$. 
@article{TRSPY_2019_307_a7,
     author = {Viacheslav V. Nikulin},
     title = {Classification of {Degenerations} and {Picard} {Lattices} of {K\"ahlerian} {K3} {Surfaces} with {Symplectic} {Automorphism} {Group} $C_4$},
     journal = {Informatics and Automation},
     pages = {148--179},
     publisher = {mathdoc},
     volume = {307},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2019_307_a7/}
}
TY  - JOUR
AU  - Viacheslav V. Nikulin
TI  - Classification of Degenerations and Picard Lattices of K\"ahlerian K3 Surfaces with Symplectic Automorphism Group $C_4$
JO  - Informatics and Automation
PY  - 2019
SP  - 148
EP  - 179
VL  - 307
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2019_307_a7/
LA  - ru
ID  - TRSPY_2019_307_a7
ER  - 
%0 Journal Article
%A Viacheslav V. Nikulin
%T Classification of Degenerations and Picard Lattices of K\"ahlerian K3 Surfaces with Symplectic Automorphism Group $C_4$
%J Informatics and Automation
%D 2019
%P 148-179
%V 307
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2019_307_a7/
%G ru
%F TRSPY_2019_307_a7
Viacheslav V. Nikulin. Classification of Degenerations and Picard Lattices of K\"ahlerian K3 Surfaces with Symplectic Automorphism Group $C_4$. Informatics and Automation, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 148-179. http://geodesic.mathdoc.fr/item/TRSPY_2019_307_a7/

[1] Bourbaki N., Groupes et algèbres de Lie. Chs. IV, V et VI: Groupes de Coxeter et systèmes de Tits. Groupes engendrés par des réflexions. Systèmes de racines, Éléments de mathématique, Hermann, Paris, 1968 | MR

[2] Burns D., \textup {Jr.}, Rapoport M., “On the Torelli problem for kählerian K-3 surfaces”, Ann. sci. Éc. norm. supér. Sér. 4, 8:2 (1975), 235–273 | MR | Zbl

[3] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Grundl. Math. Wiss., 290, Springer, New York, 1988 | MR | Zbl

[4] The GAP Group., GAP — groups, algorithms, programming — a system for computational discrete algebra, Vers. 4.6.5, 2013 http://www.gap-system.org

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

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

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

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

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

[10] V. V. Nikulin, “Finite automorphism groups of Kählerian surfaces of type K3”, Usp. Mat. Nauk, 31:2 (1976), 223–224 | MR | Zbl

[11] V. V. Nikulin, “Finite automorphism groups of Kähler K3 surfaces”, Trans. Moscow Math. Soc., 38 (1980), 71–135 | MR | Zbl

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

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

[14] Nikulin V.V., “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 ; arXiv: 1109.2879 [math.AG] | DOI | MR | Zbl

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

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

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

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

[19] V. V. Nikulin, “Classification of degenerations and Picard lattices of Kählerian K3 surfaces with symplectic automorphism group $D_6$”, Izv. Math., 83:6 (2019), 1201–1233 | DOI | DOI | MR | Zbl

[20] Nikulin V.V., Classification of degenerations and Picard lattices of Kählerian K3 surfaces with small finite symplectic automorphism groups, E-print, 2019, arXiv: 1804.00991v2 [math.AG] | MR

[21] I. I. Pjatecki\u i-Šapiro and I. R. Šafarevič, “A Torelli theorem for algebraic surfaces of type $K3$”, Math. USSR, Izv., 5:3 (1971), 547–588 | DOI | MR

[22] I. R. Šafarevič, B. G. Averbuh, Ju. R. Vainberg, A. B. Žižčenko, Ju. I. Manin, B. G. Moišezon, G. N. Tjurina, and A. N. Tjurin, Algebraic Surfaces, Nauka, Moscow, 1965 | MR

[23] Proc. Steklov Inst. Math., 75, Am. Math. Soc., Providence, RI, 1967 | MR

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

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

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