Linear Shafarevich Conjecture in positive characteristic, Hyperbolicity and Applications
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914081913438208 |
|---|---|
| author | Deng, Ya Yamanoi, Katsutoshi |
| author_facet | Deng, Ya Yamanoi, Katsutoshi |
| contents | Given a complex quasi-projective normal variety $X$ and a linear representation $\varrho:π_1(X)\to {\rm GL}_{N}(K)$ with $K$ any field of positive characteristic, we mainly establish the following results:
1. the construction of the Shafarevich morphism ${\rm sh}_\varrho:X\to {\rm Sh}_\varrho(X)$ associated with $\varrho$.
2. In cases where $X$ is projective, $\varrho$ is faithful and the $Γ$-dimension of $X$ is at most two (e.g. $\dim X=2$), we prove that the Shafarevich conjecture holds for $X$.
3. In cases where $\varrho$ is big, we prove that the Green-Griffiths-Lang conjecture holds for $X$.
4. When $\varrho$ is big and the Zariski closure of $\varrho(π_1(X))$ is a semisimple algebraic group, we prove that $X$ is pseudo Picard hyperbolic, and strongly of log general type.
5. If $X$ is special or $h$-special, then $\varrho(π_1(X))$ is virtually abelian.
We also prove Claudon-Höring-Kollár's conjecture for complex projective manifolds with linear fundamental groups of any characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear Shafarevich Conjecture in positive characteristic, Hyperbolicity and Applications Deng, Ya Yamanoi, Katsutoshi Algebraic Geometry Complex Variables Given a complex quasi-projective normal variety $X$ and a linear representation $\varrho:π_1(X)\to {\rm GL}_{N}(K)$ with $K$ any field of positive characteristic, we mainly establish the following results: 1. the construction of the Shafarevich morphism ${\rm sh}_\varrho:X\to {\rm Sh}_\varrho(X)$ associated with $\varrho$. 2. In cases where $X$ is projective, $\varrho$ is faithful and the $Γ$-dimension of $X$ is at most two (e.g. $\dim X=2$), we prove that the Shafarevich conjecture holds for $X$. 3. In cases where $\varrho$ is big, we prove that the Green-Griffiths-Lang conjecture holds for $X$. 4. When $\varrho$ is big and the Zariski closure of $\varrho(π_1(X))$ is a semisimple algebraic group, we prove that $X$ is pseudo Picard hyperbolic, and strongly of log general type. 5. If $X$ is special or $h$-special, then $\varrho(π_1(X))$ is virtually abelian. We also prove Claudon-Höring-Kollár's conjecture for complex projective manifolds with linear fundamental groups of any characteristic. |
| title | Linear Shafarevich Conjecture in positive characteristic, Hyperbolicity and Applications |
| topic | Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2403.16199 |