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

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 ${\mathrm{CP}}^2\mathrm{\#}9{\stackrel{}{\mathrm{CP}}}^2$. Firstly, for an elliptic curve $C_0$ embedded in ${\mathrm{CP}}^2$, let $S\cong {\mathrm{CP}}^2\mathrm{\#}9{\stackrel{}{\mathrm{CP}}}^2$ be the blow up of ${\mathrm{CP}}^2$ 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 $S\mathrm{\backslash }C$ 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 $S\mathrm{\backslash }C$. 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.