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

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.

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.

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DOI : https://doi.org/10.5802/afst.1635
@article{AFST_2020_6_29_2_371_0,
     author = {Yoshinori Hashimoto},
     title = {Mapping properties of the Hilbert and Fubini{\textendash}Study maps in K\"ahler geometry},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {371--389},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {2},
     year = {2020},
     doi = {10.5802/afst.1635},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1635/}
}
Yoshinori Hashimoto. Mapping properties of the Hilbert and Fubini–Study maps in Kähler geometry. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 371-389. doi : 10.5802/afst.1635. https://afst.centre-mersenne.org/articles/10.5802/afst.1635/

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