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 #9CP¯². Firstly, for an elliptic curve ₀ embedded in CP², let ≅ CP² #9CP¯² be the blow up of CP² at ni...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/214174 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces. Fan Xu. SIGMA 21 (2025), 062, 22 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | 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 #9CP¯². Firstly, for an elliptic curve ₀ embedded in CP², let ≅ CP² #9CP¯² be the blow up of CP² at nine points on the image of ₀, and be the strict transform of the image. Then, if the normal bundle satisfies the Diophantine condition, a tubular neighborhood of the elliptic curve can be identified through a toroidal group. Fixing the Diophantine condition, a smooth compactifiable deformation of ∖ over a 9-dimensional complex manifold is constructed. Moreover, with an ample line bundle fixed on , complete Kähler metrics can be constructed on the quasi-projective variety ∖. 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.
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| ISSN: | 1815-0659 |