Singular Nakano positivity of direct image sheaves of adjoint bundles
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910567289061376 |
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| author | Inayama, Takahiro Matsumura, Shin-ichi Watanabe, Yuta |
| author_facet | Inayama, Takahiro Matsumura, Shin-ichi Watanabe, Yuta |
| contents | In this paper, we consider a proper Kähler fibration $f \colon X \to Y$ and a singular Hermitian line bundle $(L, h)$ on $X$ with semi-positive curvature. We prove that the direct image sheaf $f_{*}(\mathcal{O}_{X}(K_{X/Y}+L) \otimes \mathcal{I}(h))$, equipped with the Narasimhan-Simha metric, is singular Nakano semi-positive in the sense that the $\overline{\partial}$-equation can be solved with optimal $L^{2}$-estimate. Our proof does not rely on the theory of Griffiths positivity for the direct image sheaf. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_11412 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular Nakano positivity of direct image sheaves of adjoint bundles Inayama, Takahiro Matsumura, Shin-ichi Watanabe, Yuta Algebraic Geometry Complex Variables Differential Geometry Primary 32U05, Secondary 32A70, 32L20 In this paper, we consider a proper Kähler fibration $f \colon X \to Y$ and a singular Hermitian line bundle $(L, h)$ on $X$ with semi-positive curvature. We prove that the direct image sheaf $f_{*}(\mathcal{O}_{X}(K_{X/Y}+L) \otimes \mathcal{I}(h))$, equipped with the Narasimhan-Simha metric, is singular Nakano semi-positive in the sense that the $\overline{\partial}$-equation can be solved with optimal $L^{2}$-estimate. Our proof does not rely on the theory of Griffiths positivity for the direct image sheaf. |
| title | Singular Nakano positivity of direct image sheaves of adjoint bundles |
| topic | Algebraic Geometry Complex Variables Differential Geometry Primary 32U05, Secondary 32A70, 32L20 |
| url | https://arxiv.org/abs/2407.11412 |