Nakano positivity of singular Hermitian metrics: Approximations and applications
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911774937186304 |
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| author | Inayama, Takahiro Matsumura, Shin-ichi |
| author_facet | Inayama, Takahiro Matsumura, Shin-ichi |
| contents | This paper studies the approximation of singular Hermitian metrics on vector bundles using smooth Hermitian metrics with Nakano semi-positive curvature on Zariski open sets. We show that singular Hermitian metrics capable of this approximation satisfy Nakano semi-positivity as defined through the $\overline{\partial} $-equation with optimal $L^2$-estimates. Furthermore, for a projective fibration $f \colon X \to Y$ with a line bundle $L$ on $X$, we provide a specific condition under which the Narasimhan-Simha metric on the direct image sheaf $f_{*}\mathcal{O}_{X}(K_{X/Y}+L)$ admits this approximation. As an application, we establish several vanishing theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06883 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nakano positivity of singular Hermitian metrics: Approximations and applications Inayama, Takahiro Matsumura, Shin-ichi Complex Variables Algebraic Geometry Primary 32U05, Secondary 32A70, 32L20 This paper studies the approximation of singular Hermitian metrics on vector bundles using smooth Hermitian metrics with Nakano semi-positive curvature on Zariski open sets. We show that singular Hermitian metrics capable of this approximation satisfy Nakano semi-positivity as defined through the $\overline{\partial} $-equation with optimal $L^2$-estimates. Furthermore, for a projective fibration $f \colon X \to Y$ with a line bundle $L$ on $X$, we provide a specific condition under which the Narasimhan-Simha metric on the direct image sheaf $f_{*}\mathcal{O}_{X}(K_{X/Y}+L)$ admits this approximation. As an application, we establish several vanishing theorems. |
| title | Nakano positivity of singular Hermitian metrics: Approximations and applications |
| topic | Complex Variables Algebraic Geometry Primary 32U05, Secondary 32A70, 32L20 |
| url | https://arxiv.org/abs/2402.06883 |