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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.19229 |
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| _version_ | 1866913335982686208 |
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| author | Chen, Kuan-Wen |
| author_facet | Chen, Kuan-Wen |
| contents | Let $f:X \rightarrow Δ$ be a one-parameter semistable degeneration of $m$-dimensional compact complex manifolds. Assume that each component of the central fiber $X_0$ is Kähler. Then, we provide a criterion for a general fiber to satisfy the $\partial\overline{\partial}$-lemma and a formula to compute the Hodge index on the middle cohomology of the general fiber in terms of the topological conditions/invariants on the central fiber.
We apply our theorem to several examples, including the global smoothing of $m$-fold ODPs, Hashimoto-Sano's non-Kähler Calabi-Yau threefolds, and Sano's non-Kähler Calabi-Yau $m$-folds.
To deal with the last example, we also prove a Lefschetz-type theorem for the cohomology of the fiber product of two Lefschetz fibrations over $\mathbb{P}^1$ with disjoint critical locus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19229 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Hodge Structures of Global Smoothings of Normal Crossing Varieties Chen, Kuan-Wen Algebraic Geometry Let $f:X \rightarrow Δ$ be a one-parameter semistable degeneration of $m$-dimensional compact complex manifolds. Assume that each component of the central fiber $X_0$ is Kähler. Then, we provide a criterion for a general fiber to satisfy the $\partial\overline{\partial}$-lemma and a formula to compute the Hodge index on the middle cohomology of the general fiber in terms of the topological conditions/invariants on the central fiber. We apply our theorem to several examples, including the global smoothing of $m$-fold ODPs, Hashimoto-Sano's non-Kähler Calabi-Yau threefolds, and Sano's non-Kähler Calabi-Yau $m$-folds. To deal with the last example, we also prove a Lefschetz-type theorem for the cohomology of the fiber product of two Lefschetz fibrations over $\mathbb{P}^1$ with disjoint critical locus. |
| title | On the Hodge Structures of Global Smoothings of Normal Crossing Varieties |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2404.19229 |