Lefschetz decompositions of Kudla-Millson theta functions
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911936370704384 |
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| author | Bruinier, Jan Hendrik Zuffetti, Riccardo |
| author_facet | Bruinier, Jan Hendrik Zuffetti, Riccardo |
| contents | In the 80's Kudla and Millson introduced a theta function in two variables. It behaves as a Siegel modular form with respect to the first variable, and is a closed differential form on an orthogonal Shimura variety with respect to the other variable. We prove that the Lefschetz decomposition of the cohomology class of that theta function is also its modular decomposition in Eisenstein, Klingen and cuspidal parts. We also show that the image of the Kudla-Millson lift is contained in the primitive cohomology of X. As an application, we deduce dimension formulas for the cohomology of X in low degree. Our results cover the cases of moduli spaces of compact hyperkähler manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lefschetz decompositions of Kudla-Millson theta functions Bruinier, Jan Hendrik Zuffetti, Riccardo Number Theory Algebraic Geometry 11F27 (Primary), 14J10, 11F46, 14C25 (Secondary) In the 80's Kudla and Millson introduced a theta function in two variables. It behaves as a Siegel modular form with respect to the first variable, and is a closed differential form on an orthogonal Shimura variety with respect to the other variable. We prove that the Lefschetz decomposition of the cohomology class of that theta function is also its modular decomposition in Eisenstein, Klingen and cuspidal parts. We also show that the image of the Kudla-Millson lift is contained in the primitive cohomology of X. As an application, we deduce dimension formulas for the cohomology of X in low degree. Our results cover the cases of moduli spaces of compact hyperkähler manifolds. |
| title | Lefschetz decompositions of Kudla-Millson theta functions |
| topic | Number Theory Algebraic Geometry 11F27 (Primary), 14J10, 11F46, 14C25 (Secondary) |
| url | https://arxiv.org/abs/2406.19921 |