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| Format: | Preprint |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1705.11025 |
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| _version_ | 1866909489931747328 |
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| author | Hashimoto, Yoshinori |
| author_facet | Hashimoto, Yoshinori |
| contents | Suppose that we have a compact Kähler manifold $X$ with a very ample line bundle $\mathcal{L}$. We prove that any positive definite hermitian form on the space $H^0 (X,\mathcal{L})$ of holomorphic sections can be written as an $L^2$-inner product with respect to an appropriate hermitian metric on $\mathcal{L}$. We apply this result to show that the Fubini--Study map, which associates a hermitian metric on $\mathcal{L}$ to a hermitian form on $H^0 (X,\mathcal{L})$, is injective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1705_11025 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Mapping properties of the Hilbert and Fubini--Study maps in Kähler geometry Hashimoto, Yoshinori Differential Geometry 53C55 Suppose that we have a compact Kähler manifold $X$ with a very ample line bundle $\mathcal{L}$. We prove that any positive definite hermitian form on the space $H^0 (X,\mathcal{L})$ of holomorphic sections can be written as an $L^2$-inner product with respect to an appropriate hermitian metric on $\mathcal{L}$. We apply this result to show that the Fubini--Study map, which associates a hermitian metric on $\mathcal{L}$ to a hermitian form on $H^0 (X,\mathcal{L})$, is injective. |
| title | Mapping properties of the Hilbert and Fubini--Study maps in Kähler geometry |
| topic | Differential Geometry 53C55 |
| url | https://arxiv.org/abs/1705.11025 |