Slope zero tensors, uniformizing variations of Hodge structure and quotients of tube domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917020884271104 |
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| author | Graf, Patrick Patel, Aryaman |
| author_facet | Graf, Patrick Patel, Aryaman |
| contents | We prove an equivalence between two approaches to characterizing complex-projective varieties $X$ with klt singularities and ample canonical divisor that are uniformized by bounded symmetric domains. In order to do so, we show how to construct a uniformizing variation of Hodge structure from a slope zero tensor and vice versa. As a consequence, we generalize various uniformization results of Catanese and Di Scala to the singular setting. For example, we prove that $X$ is a quotient of a bounded symmetric domain of tube type by a group acting properly discontinuously and freely in codimension one if and only if $X$ admits a slope zero tensor. As a key step in the proof, we establish the compactness of the holonomy group of the singular Kähler--Einstein metric on $X_{\mathrm{reg}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Slope zero tensors, uniformizing variations of Hodge structure and quotients of tube domains Graf, Patrick Patel, Aryaman Algebraic Geometry Complex Variables Differential Geometry We prove an equivalence between two approaches to characterizing complex-projective varieties $X$ with klt singularities and ample canonical divisor that are uniformized by bounded symmetric domains. In order to do so, we show how to construct a uniformizing variation of Hodge structure from a slope zero tensor and vice versa. As a consequence, we generalize various uniformization results of Catanese and Di Scala to the singular setting. For example, we prove that $X$ is a quotient of a bounded symmetric domain of tube type by a group acting properly discontinuously and freely in codimension one if and only if $X$ admits a slope zero tensor. As a key step in the proof, we establish the compactness of the holonomy group of the singular Kähler--Einstein metric on $X_{\mathrm{reg}}$. |
| title | Slope zero tensors, uniformizing variations of Hodge structure and quotients of tube domains |
| topic | Algebraic Geometry Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2510.15039 |