The cohomological Kudla conjecture for unitary Shimura varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918529249312768 |
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| author | Greer, François Tayou, Salim |
| author_facet | Greer, François Tayou, Salim |
| contents | We construct natural extensions of the Kudla--Millson generating series of cohomology classes of special cycles in compactified unitary Shimura varieties of signature $(n+1,1)$ and prove that they are holomorphic Hermitian modular forms. This proves the cohomological version of a conjecture of Kudla and Bruinier--Rosu--Zemel, in all codimensions up to the middle. We also develop the theory of Hermitian quasi-modular forms, with a particular focus on polynomial weighted theta functions, and prove that the generating series of Zariski closures of special cycles is a Hermitian quasi-modular form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The cohomological Kudla conjecture for unitary Shimura varieties Greer, François Tayou, Salim Number Theory Algebraic Geometry 14G35, 14M27, 11F27 We construct natural extensions of the Kudla--Millson generating series of cohomology classes of special cycles in compactified unitary Shimura varieties of signature $(n+1,1)$ and prove that they are holomorphic Hermitian modular forms. This proves the cohomological version of a conjecture of Kudla and Bruinier--Rosu--Zemel, in all codimensions up to the middle. We also develop the theory of Hermitian quasi-modular forms, with a particular focus on polynomial weighted theta functions, and prove that the generating series of Zariski closures of special cycles is a Hermitian quasi-modular form. |
| title | The cohomological Kudla conjecture for unitary Shimura varieties |
| topic | Number Theory Algebraic Geometry 14G35, 14M27, 11F27 |
| url | https://arxiv.org/abs/2507.13299 |