Nonvanishing results for Kähler varieties
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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_ | 1866917028910071808 |
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| author | Höring, Andreas Lazić, Vladimir Lehn, Christian |
| author_facet | Höring, Andreas Lazić, Vladimir Lehn, Christian |
| contents | Nonvanishing theorems play a central role in birational geometry, since they derive geometric consequences from numerical information and constitute a crucial step towards abundance and semiampleness problems. General nonvanishing statements remain rare, especially in the Kähler setting.
We present two types of nonvanishing results for compact Kähler varieties. First, on non-uniruled varieties with nonzero Euler-Poincaré characteristic, we prove nonvanishing for adjoint bundles of numerical dimension one on Kähler klt pairs, as well as nonvanishing for nef line bundles of numerical dimension one on $K$-trivial varieties. Second, on hyperkähler manifolds we study line bundles $\mathcal L$ which are nef but not big, and establish a dichotomy: either nonvanishing holds for $\mathcal L$, or any closed positive current in the cohomology class of $\mathcal L$ has maximal Lelong components with a rather restricted geometry. We obtain much stronger abundance-type results in dimension $4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_14634 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonvanishing results for Kähler varieties Höring, Andreas Lazić, Vladimir Lehn, Christian Algebraic Geometry Complex Variables Differential Geometry 14E30, 32J27, 32Q15, 14J42, 32U40, 53C26 Nonvanishing theorems play a central role in birational geometry, since they derive geometric consequences from numerical information and constitute a crucial step towards abundance and semiampleness problems. General nonvanishing statements remain rare, especially in the Kähler setting. We present two types of nonvanishing results for compact Kähler varieties. First, on non-uniruled varieties with nonzero Euler-Poincaré characteristic, we prove nonvanishing for adjoint bundles of numerical dimension one on Kähler klt pairs, as well as nonvanishing for nef line bundles of numerical dimension one on $K$-trivial varieties. Second, on hyperkähler manifolds we study line bundles $\mathcal L$ which are nef but not big, and establish a dichotomy: either nonvanishing holds for $\mathcal L$, or any closed positive current in the cohomology class of $\mathcal L$ has maximal Lelong components with a rather restricted geometry. We obtain much stronger abundance-type results in dimension $4$. |
| title | Nonvanishing results for Kähler varieties |
| topic | Algebraic Geometry Complex Variables Differential Geometry 14E30, 32J27, 32Q15, 14J42, 32U40, 53C26 |
| url | https://arxiv.org/abs/2508.14634 |