$L^2$-extension indices, sharper estimates and curvature positivity
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909147203633152 |
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| author | Inayama, Takahiro |
| author_facet | Inayama, Takahiro |
| contents | In this paper, we introduce a new concept of $L^2$-extension indices. This index is a function that gives the minimum constant with respect to the $L^2$-estimate of an Ohsawa--Takegoshi-type extension at each point. By using this notion, we propose a new way to study the positivity of curvature. We prove that there is an equivalence between how sharp the $L^2$-extension is and how positive the curvature is. New examples of sharper $L^2$-extensions are also systematically given. As applications, we use the $L^2$-extension index to study Prékopa-type theorems and to study the positivity of a certain direct image sheaf. We also provide new characterizations of pluriharmonicity and curvature flatness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_08456 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $L^2$-extension indices, sharper estimates and curvature positivity Inayama, Takahiro Complex Variables 32A36, 32U05 In this paper, we introduce a new concept of $L^2$-extension indices. This index is a function that gives the minimum constant with respect to the $L^2$-estimate of an Ohsawa--Takegoshi-type extension at each point. By using this notion, we propose a new way to study the positivity of curvature. We prove that there is an equivalence between how sharp the $L^2$-extension is and how positive the curvature is. New examples of sharper $L^2$-extensions are also systematically given. As applications, we use the $L^2$-extension index to study Prékopa-type theorems and to study the positivity of a certain direct image sheaf. We also provide new characterizations of pluriharmonicity and curvature flatness. |
| title | $L^2$-extension indices, sharper estimates and curvature positivity |
| topic | Complex Variables 32A36, 32U05 |
| url | https://arxiv.org/abs/2210.08456 |