Mapping properties of the Hilbert and Fubini–Study maps in Kähler geometry

Hashimoto, Yoshinori

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

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

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