Reflective obstructions of unitary modular varieties
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911788838158336 |
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| author | Maeda, Yota |
| author_facet | Maeda, Yota |
| contents | To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This shows in particular that the reflective obstructions are small enough in higher dimension, say greater than $138$. Our result reduces the study of the Kodaira dimension of unitary modular varieties to the construction of a cusp form of small weight in a quantitative manner. As a byproduct, we formulate and partially prove the finiteness of Hermitian lattices admitting reflective modular forms, which is a unitary analog of the conjecture by Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective obstructions uses Prasad's volume formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_01128 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Reflective obstructions of unitary modular varieties Maeda, Yota Algebraic Geometry Number Theory Representation Theory Primary:14G35, Secondary:11F03, 11E39, 14E22 To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This shows in particular that the reflective obstructions are small enough in higher dimension, say greater than $138$. Our result reduces the study of the Kodaira dimension of unitary modular varieties to the construction of a cusp form of small weight in a quantitative manner. As a byproduct, we formulate and partially prove the finiteness of Hermitian lattices admitting reflective modular forms, which is a unitary analog of the conjecture by Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective obstructions uses Prasad's volume formula. |
| title | Reflective obstructions of unitary modular varieties |
| topic | Algebraic Geometry Number Theory Representation Theory Primary:14G35, Secondary:11F03, 11E39, 14E22 |
| url | https://arxiv.org/abs/2204.01128 |