Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 062, 22 pages      arXiv:2406.16208      https://doi.org/10.3842/SIGMA.2025.062

Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces

Fan Xu
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

Received October 31, 2024, in final form July 16, 2025; Published online July 28, 2025

Abstract
The main aim of this paper is to construct a complex analytic family of symmetric projective K3 surfaces through a compactifiable deformation family of complete quasi-projective varieties from $\operatorname{CP}^2 \#9\overline{\operatorname{CP}}^2$. Firstly, for an elliptic curve $C_0$ embedded in $\operatorname{CP}^2$, let $S \cong \operatorname{CP}^2 \#9\overline{\operatorname{CP}}^2$ be the blow up of $\operatorname{CP}^2$ at nine points on the image of $C_0$ and $C$ be the strict transform of the image. Then if the normal bundle satisfies the Diophantine condition, a tubular neighborhood of the elliptic curve $C$ can be identified through a toroidal group. Fixing the Diophantine condition, a smooth compactifiable deformation of $S\backslash C$ over a 9-dimensional complex manifold is constructed. Moreover, with an ample line bundle fixed on $S$, complete Kähler metrics can be constructed on the quasi-projective variety $S\backslash C$. So complete Kähler metrics are constructed on each quasi-projective variety fiber of the smooth compactifiable deformation family. Then a complex analytic family of symmetric projective K3 surfaces over a 10-dimensional complex manifold is constructed through the smooth compactifiable deformation family of complete quasi-projective varieties and an analogous deformation family.

Key words: blow up; complete quasi-projective varieties; symmetry projective K3 surfaces; deformation families.

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