Mapping properties of the Hilbert and Fubini–Study maps in Kähler geometry
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 2, pp. 371-389.

Suppose that we have a compact Kähler manifold X with a very ample line bundle . We prove that any positive definite hermitian form on the space H 0 (X,) of holomorphic sections can be written as an L 2 -inner product with respect to an appropriate hermitian metric on . We apply this result to show that the Fubini–Study map, which associates a hermitian metric on to a hermitian form on H 0 (X,), is injective.

Soit X une variété compacte kählerienne avec un fibré en droites qui est très ample. Nous prouvons que toute forme hermitienne définie positive sur H 0 (X,) peut être écrite comme produit scalaire L 2 associé à une métrique hermitienne sur . Nous appliquons ce résultat pour montrer que l’application de Fubini–Study, des formes hermitiennes sur H 0 (X,) vers les métriques hermitiennes sur , est injective.

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Accepted:
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DOI: 10.5802/afst.1635
Hashimoto, Yoshinori 1

1 Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551 (Japan)
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     title = {Mapping properties of the {Hilbert} and {Fubini{\textendash}Study} maps in {K\"ahler} geometry},
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Hashimoto, Yoshinori. Mapping properties of the Hilbert and Fubini–Study maps in Kähler geometry. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 2, pp. 371-389. doi : 10.5802/afst.1635. http://archive.numdam.org/articles/10.5802/afst.1635/

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