A converse of Berndtsson's theorem on the positivity of direct images
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909994491838464 |
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| author | Xu, Wang Yang, Hui |
| author_facet | Xu, Wang Yang, Hui |
| contents | Berndtsson's famous theorem asserts that, for a compact Kähler fibration $p:X\to Y$, the direct image bundle $p_*(K_{X/Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation of our previous work, we prove a converse of Berndtsson's theorem in the case of a projective fibration: if $p_*(K_{X/Y}\otimes L\otimes E)$ is Griffiths semi-positive for every semi-positive Hermitian holomorphic line bundle $E\to X$, then the curvature of $L$ must be semi-positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12825 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A converse of Berndtsson's theorem on the positivity of direct images Xu, Wang Yang, Hui Complex Variables 32F17, 32L05, 32U05 Berndtsson's famous theorem asserts that, for a compact Kähler fibration $p:X\to Y$, the direct image bundle $p_*(K_{X/Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation of our previous work, we prove a converse of Berndtsson's theorem in the case of a projective fibration: if $p_*(K_{X/Y}\otimes L\otimes E)$ is Griffiths semi-positive for every semi-positive Hermitian holomorphic line bundle $E\to X$, then the curvature of $L$ must be semi-positive. |
| title | A converse of Berndtsson's theorem on the positivity of direct images |
| topic | Complex Variables 32F17, 32L05, 32U05 |
| url | https://arxiv.org/abs/2601.12825 |