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 CP² #9CP̄². Firstly, for an elliptic curve C₀ embedded in CP², let S CP² #9CP̄² be the blow up of CP² at nine points on the image of C₀ 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 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 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.