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

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 $\text{CP}^2 \text{$#$9} \overline{\text{CP}}^2$. Firstly, for an elliptic curve $C_0$ embedded in $\text{CP}^2$, let $S \cong \text{CP}^2 \text{$#$9} \overline{\text{CP}}^2$ be the blow up of $\text{CP}^2$ 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. What's more, with an ample line bundle fixed on $S$, complete K\"ahler metrics can be constructed on the quasi-projective variety $S\backslash C$. So complete K\"ahler 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 analogue deformation family.