Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916865567096832 |
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| author | Xu, Fan |
| author_facet | Xu, Fan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_16208 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces Xu, Fan Complex Variables Algebraic Geometry 14J28, 32G05 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. |
| title | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces |
| topic | Complex Variables Algebraic Geometry 14J28, 32G05 |
| url | https://arxiv.org/abs/2406.16208 |