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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Author: Xu, Fan
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/214174
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces. Fan Xu. SIGMA 21 (2025), 062, 22 pages

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 CP² #9CP¯². Firstly, for an elliptic curve ₀ embedded in CP², let ≅ CP² #9CP¯² be the blow up of CP² at nine points on the image of ₀, and be the strict transform of the image. Then, if the normal bundle satisfies the Diophantine condition, a tubular neighborhood of the elliptic curve can be identified through a toroidal group. Fixing the Diophantine condition, a smooth compactifiable deformation of ∖ over a 9-dimensional complex manifold is constructed. Moreover, with an ample line bundle fixed on , complete Kähler metrics can be constructed on the quasi-projective variety ∖. 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.