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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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/214174 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces. Fan Xu. SIGMA 21 (2025), 062, 22 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862545485832650752 |
|---|---|
| author | Xu, Fan |
| author_facet | Xu, Fan |
| citation_txt | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces. Fan Xu. SIGMA 21 (2025), 062, 22 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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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| first_indexed | 2026-03-21T11:46:53Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-214174 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:46:53Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Xu, Fan 2026-02-19T16:04:46Z 2025 Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces. Fan Xu. SIGMA 21 (2025), 062, 22 pages 1815-0659 2020 Mathematics Subject Classification: 14J28; 32G05 arXiv:2406.16208 https://nasplib.isofts.kiev.ua/handle/123456789/214174 https://doi.org/10.3842/SIGMA.2025.062 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. Thanks, Professor Zhou Zhang, for his course on algebraic geometry. Thanks, Professor Sho Tanimoto, for providing some useful materials. In particular, thanks to Professor Ryoichi Kobayashi for introducing this topic and giving invaluable comments. I would also like to thank the anonymous referees for their suggestions and comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces Article published earlier |
| spellingShingle | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces Xu, Fan |
| title | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces |
| title_full | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces |
| title_fullStr | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces |
| title_full_unstemmed | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces |
| title_short | Deformation Families of Quasi-Projective Varieties and Symmetric Projective K3 Surfaces |
| title_sort | deformation families of quasi-projective varieties and symmetric projective k3 surfaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214174 |
| work_keys_str_mv | AT xufan deformationfamiliesofquasiprojectivevarietiesandsymmetricprojectivek3surfaces |