Pluriharmonic maps into buildings and symmetric differentials
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910890405658624 |
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| author | Brotbek, Damian Daskalopoulos, Georgios Deng, Ya Mese, Chikako |
| author_facet | Brotbek, Damian Daskalopoulos, Georgios Deng, Ya Mese, Chikako |
| contents | Given a complex smooth quasi-projective variety $X$, a semisimple algebraic group $G$ defined over some non-archimedean local field $K$ and a Zariski dense representation $\varrho:π_1(X)\to G(K)$, we construct a $\varrho$-equivariant (pluri-)harmonic map from the universal cover of $X$ into the Bruhat-Tits building $Δ(G)$ of $G$, with some suitable asymptotic behavior. This theorem generalizes the previous work by Gromov-Schoen to the quasi-projective setting.
As an application, we prove that $X$ has nonzero global logarithmic symmetric differentials if there exists a linear representation $π_1(X)\to {\rm GL}_N(\mathbb{K})$ with infinite image, where $ \mathbb{K}$ is any field. This theorem generalizes the previous work by Brunebarbe, Klingler and Totaro to the quasi-projective setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_11835 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pluriharmonic maps into buildings and symmetric differentials Brotbek, Damian Daskalopoulos, Georgios Deng, Ya Mese, Chikako Algebraic Geometry Complex Variables Differential Geometry Given a complex smooth quasi-projective variety $X$, a semisimple algebraic group $G$ defined over some non-archimedean local field $K$ and a Zariski dense representation $\varrho:π_1(X)\to G(K)$, we construct a $\varrho$-equivariant (pluri-)harmonic map from the universal cover of $X$ into the Bruhat-Tits building $Δ(G)$ of $G$, with some suitable asymptotic behavior. This theorem generalizes the previous work by Gromov-Schoen to the quasi-projective setting. As an application, we prove that $X$ has nonzero global logarithmic symmetric differentials if there exists a linear representation $π_1(X)\to {\rm GL}_N(\mathbb{K})$ with infinite image, where $ \mathbb{K}$ is any field. This theorem generalizes the previous work by Brunebarbe, Klingler and Totaro to the quasi-projective setting. |
| title | Pluriharmonic maps into buildings and symmetric differentials |
| topic | Algebraic Geometry Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2206.11835 |