On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated
[Sur les métriques singulières minimales d’une certaine classe de fibrés en droites dont l’anneau des sections n’est pas de type fini]
Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 1953-1967.

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.

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.

DOI : 10.5802/aif.2978
Classification : 32J25, 14C20
Keywords: minimal singular metrics, tubular neighborhoods, Zariski’s example
Mot clés : métriques singulières minimales, voisinages tubulaires, l’exemple de Zariski
Koike, Takayuki 1

1 Graduate School of Mathematical Sciences, University of Tokyo 3-8-1 Komaba, Meguro-ku Tokyo, 153-8914 (Japan)
@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},
     pages = {1953--1967},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {5},
     year = {2015},
     doi = {10.5802/aif.2978},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2978/}
}
TY  - JOUR
AU  - Koike, Takayuki
TI  - On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 1953
EP  - 1967
VL  - 65
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2978/
DO  - 10.5802/aif.2978
LA  - en
ID  - AIF_2015__65_5_1953_0
ER  - 
%0 Journal Article
%A Koike, Takayuki
%T On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated
%J Annales de l'Institut Fourier
%D 2015
%P 1953-1967
%V 65
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2978/
%R 10.5802/aif.2978
%G en
%F 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, Tome 65 (2015) no. 5, pp. 1953-1967. doi : 10.5802/aif.2978. http://www.numdam.org/articles/10.5802/aif.2978/

[1] Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Monge-Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | DOI | Zbl

[2] Demailly, Jean-Pierre Complex Analytic and Differential Geometry (https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/analmeth_book.pdf)

[3] Demailly, Jean-Pierre Analytic methods in algebraic geometry, Surveys of Modern Mathematics, 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012, pp. viii+231 | Zbl

[4] Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom., Volume 3 (1994) no. 2, pp. 295-345 | Zbl

[5] Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math., Volume 12 (2001) no. 6, pp. 689-741 | DOI | Zbl

[6] Fujino, Osamu A transcendental approach to Kollár’s injectivity theorem II, J. Reine Angew. Math., Volume 681 (2013), pp. 149-174 | Zbl

[7] Grauert, Hans Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., Volume 146 (1962), pp. 331-368 | Zbl

[8] Lazarsfeld, Robert Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 48, Springer-Verlag, Berlin, 2004, pp. xviii+387 (Classical setting: line bundles and linear series) | DOI | Zbl

[9] Ohsawa, Takeo Vanishing theorems on complete Kähler manifolds, Publ. Res. Inst. Math. Sci., Volume 20 (1984) no. 1, pp. 21-38 | DOI | Zbl

[10] Rossi, Hugo Strongly pseudoconvex manifolds, Lectures in Modern Analysis and Applications, I, Springer, Berlin, 1969, pp. 10-29 | Zbl

Cité par Sources :