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

TL;DR

This paper constructs a complex analytic family of symmetric projective K3 surfaces by deforming quasi-projective varieties derived from blow-ups of the projective plane, with applications to Kähler metrics and complex geometry.

Contribution

It introduces a method to produce a family of symmetric projective K3 surfaces via deformation of quasi-projective varieties with specific geometric conditions.

Findings

Construction of a 10-dimensional family of symmetric projective K3 surfaces.

Existence of complete Kähler metrics on the deformed quasi-projective varieties.

Identification of tubular neighborhoods of elliptic curves under Diophantine conditions.

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.