Hodge theory for local systems and cohomological support loci
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918193429217280 |
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| author | Cao, Junyan Deng, Ya Hacon, Christopher D. Paun, Mihai |
| author_facet | Cao, Junyan Deng, Ya Hacon, Christopher D. Paun, Mihai |
| contents | In this article, we pursue two main objectives. The first is to show that the fundamental results of Green-Lazarsfeld (1987, 1991) on generic vanishing theorems, and works of Budur-Wang (2015, 2020) on cohomology jumping loci, can be established within a unified framework based on suitable versions of the $\partial\bar{\partial}$-lemma. Our second-and primary-goal is to develop the technical tools required for this approach, namely an $L^2$-Hodge theory for the cohomology of rank-one local systems on quasi-compact Kähler manifolds. Further developments concerning higher-rank local systems, as well as several geometric applications, will be presented in a companion paper and are briefly outlined in the introduction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge theory for local systems and cohomological support loci Cao, Junyan Deng, Ya Hacon, Christopher D. Paun, Mihai Algebraic Geometry Complex Variables In this article, we pursue two main objectives. The first is to show that the fundamental results of Green-Lazarsfeld (1987, 1991) on generic vanishing theorems, and works of Budur-Wang (2015, 2020) on cohomology jumping loci, can be established within a unified framework based on suitable versions of the $\partial\bar{\partial}$-lemma. Our second-and primary-goal is to develop the technical tools required for this approach, namely an $L^2$-Hodge theory for the cohomology of rank-one local systems on quasi-compact Kähler manifolds. Further developments concerning higher-rank local systems, as well as several geometric applications, will be presented in a companion paper and are briefly outlined in the introduction. |
| title | Hodge theory for local systems and cohomological support loci |
| topic | Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2511.06773 |