Reductive Shafarevich Conjecture
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910461210918912 |
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| author | Deng, Ya Yamanoi, Katsutoshi Katzarkov, Ludmil |
| author_facet | Deng, Ya Yamanoi, Katsutoshi Katzarkov, Ludmil |
| contents | In this paper, we prove the holomorphic convexity of the covering of a complex projective {normal} variety $X$, which corresponds to the intersection of kernels of reductive representations $ρ:π_1(X)\to {\rm GL}_{N}(\mathbb{C})$, therefore answering a question by Eyssidieux, Katzarkov, Pantev, and Ramachandran in 2012. It is worth noting that Eyssidieux had previously proven this result in 2004 when $X$ is smooth. While our approach follows the general strategy employed in Eyssidieux's proof, it introduces several improvements and simplifications. Notably, it avoids the necessity of using the reduction mod $p$ method in Eyssidieux's original proof.
Additionally, we construct the Shafarevich morphism for complex reductive representations of fundamental groups of complex quasi-projective varieties unconditionally, and proving its algebraic nature at the function field level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_03070 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reductive Shafarevich Conjecture Deng, Ya Yamanoi, Katsutoshi Katzarkov, Ludmil Algebraic Geometry Complex Variables In this paper, we prove the holomorphic convexity of the covering of a complex projective {normal} variety $X$, which corresponds to the intersection of kernels of reductive representations $ρ:π_1(X)\to {\rm GL}_{N}(\mathbb{C})$, therefore answering a question by Eyssidieux, Katzarkov, Pantev, and Ramachandran in 2012. It is worth noting that Eyssidieux had previously proven this result in 2004 when $X$ is smooth. While our approach follows the general strategy employed in Eyssidieux's proof, it introduces several improvements and simplifications. Notably, it avoids the necessity of using the reduction mod $p$ method in Eyssidieux's original proof. Additionally, we construct the Shafarevich morphism for complex reductive representations of fundamental groups of complex quasi-projective varieties unconditionally, and proving its algebraic nature at the function field level. |
| title | Reductive Shafarevich Conjecture |
| topic | Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2306.03070 |