Classification of Degenerations of Codimension ${\le }\,5$ and Their Picard Lattices for K\"ahlerian K3 Surfaces with the Symplectic Automorphism Group $(C_2)^2$

Viacheslav V. Nikulin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 189-242

Résumé

In our papers of 2013–2018, we classified degenerations and Picard lattices of Kählerian K3 surfaces with finite symplectic automorphism groups of high order. 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 cases of $D_6$ and $C_4$ have been recently completely analyzed. Here we consider an analogous complete classification for the group $(C_2)^2$ of order $4$. We restrict ourselves to degenerations of codimension ${\le }\,5$. This group also has degenerations of codimension $6$ and $7$, which will be classified in a future paper.