On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated
Annales de l'Institut Fourier, Volume 65 (2015) no. 5, p. 1953-1967

Wea are interested in the regularity of a minimal singular metric of a line bundle. One main conclusion of our general result in this paper is the existence of smooth Hermitian metrics with semi-positive curvatures on the so-called Zariski’s example of a line bundle defined over the blow-up of 2 at twelve points. This is an example of a line bundle which is nef, big, not semi-ample, and whose section ring is not finitely generated. We generalize this result to the higher dimensional case when the stable base locus of a line bundle is a smooth hypersurface with a holomorphic tubular neighborhood.

On s’intéresse à la régularité d’une métrique singulière minimale d’un fibré en droites. Une des conséquences principales du résultat général de cet article est l’existence des métriques Hermitiennes lisses à courbure semi-positive sur X. Ici, X denote l’exemple de Zariski d’un fibré en droites défini sur l’éclatement du plan projectif en douze points. C’est un exemple de fibré en droites qui est nef, gros et non semi-ample et dont l’anneau des sections n’est pas de type fini. Nous généralisons ce résultat au cas de la dimension supérieure lorsque le lieu de base stable d’un fibré en droites est une hypersurface lisse avec un voisinage tubulaire holomorphe.

DOI : https://doi.org/10.5802/aif.2978
Classification:  32J25,  14C20
Keywords: minimal singular metrics, tubular neighborhoods, Zariski’s example
@article{AIF_2015__65_5_1953_0,
     author = {Koike, Takayuki},
     title = {On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {5},
     year = {2015},
     pages = {1953-1967},
     doi = {10.5802/aif.2978},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_5_1953_0}
}
Koike, Takayuki. On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated. Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 1953-1967. doi : 10.5802/aif.2978. http://www.numdam.org/item/AIF_2015__65_5_1953_0/

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