logo AFST

On metrics with minimal singularities of line bundles whose stable base loci admit holomorphic tubular neighborhoods
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 149-175.

Nous étudions les singularités minimales des métriques d’un fibre en droites L sur une variété projective lorsque le locus de base stable Y de L est une sous-variété de codimension r1. Sous certaines hypothèses sur le fibre normal et le voisinage de Y, nous donnons une description explicite de la singularité minimale des métriques de L. Nous appliquons ce résultat pour étudier un analogue (co-dimensionnel) plus élevé de l’exemple de Zariski, dans lequel le fibre en droites L n’est pas semi-ample, mais il est nef et gros.

We investigate the minimal singularities of metrics on a big line bundle L over a projective manifold when the stable base locus Y of L is a submanifold of codimension r1. Under some assumptions on the normal bundle and a neighborhood of Y, we give a explicit description of the minimal singularity of metrics on L. We apply this result to study a higher (co-)dimensional analogue of Zariski’s example, in which the line bundle L is not semi-ample, however it is nef and big.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1628
@article{AFST_2020_6_29_1_149_0,
     author = {Genki Hosono and Takayuki Koike},
     title = {On metrics with minimal singularities of line bundles whose stable base loci admit holomorphic tubular neighborhoods},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {149--175},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {1},
     year = {2020},
     doi = {10.5802/afst.1628},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1628/}
}
Genki Hosono; Takayuki Koike. On metrics with minimal singularities of line bundles whose stable base loci admit holomorphic tubular neighborhoods. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 149-175. doi : 10.5802/afst.1628. https://afst.centre-mersenne.org/articles/10.5802/afst.1628/

[1] Sébastien Boucksom Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 1, pp. 45-76 | Article | Numdam | MR 2050205 | Zbl 1054.32010

[2] Sébastien Boucksom; Philippe Eyssidieux; Vincent Guedj; Ahmed Zeriahi Monge-Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | Article | Zbl 1213.32025

[3] César Camacho; Hossein Movasati Neighborhoods of analytic varieties, Monografías del Instituto de Matemática y Ciencias Afines, Volume 35, Instituto de Matemática y Ciencias Afines; Pontificia Universidad Católica del Perú, 2003, v+90 pages | MR 2010707 | Zbl 1243.32006

[4] Jean-Pierre Demailly Analytic methods in algebraic geometry, Surveys of Modern Mathematics, Volume 1, International Press, Somerville, MA; Higher Education Press, 2012, viii+231 pages | MR 2978333 | Zbl 1271.14001

[5] Jean-Pierre Demailly Complex Analytic and Differential Geometry, 2012 (monograph, available at http://www-fourier.ujf-grenoble.fr/~demailly)

[6] Jean-Pierre Demailly; Thomas Peternell; Michael Schneider Pseudo-effective line bundles on compact Kähler manifolds, Int. J. Math., Volume 12 (2001) no. 6, pp. 689-741 | Article | Zbl 1111.32302

[7] Takao Fujita Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, Volume 155, Cambridge University Press, 1990, xiv+205 pages | MR 1162108 | Zbl 0743.14004

[8] Hans Grauert Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., Volume 146 (1962), pp. 331-368 | Article | Zbl 0178.42702

[9] Heisuke Hironaka; Hugo Rossi On the equivalence of imbeddings of exceptional complex spaces, Math. Ann., Volume 156 (1964), pp. 313-333 | Article | MR 171784 | Zbl 0136.20801

[10] Takayuki Koike Minimal singular metrics of a line bundle admitting no Zariski decomposition, Tôhoku Math. J., Volume 67 (2015) no. 2, pp. 297-321 | Article | MR 3365374 | Zbl 1326.32031

[11] Takayuki Koike On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated, Ann. Inst. Fourier, Volume 65 (2015) no. 5, pp. 1953-1967 | Article | Numdam | MR 3449202 | Zbl 1339.32007

[12] Takayuki Koike Higher codimensional Ueda theory for a compact submanifold with unitary flat normal bundle, Nagoya Math. J. (2018), pp. 1-33 | MR 4092849 | Zbl 07206584

[13] Henry B. Laufer Normal two-dimensional singularities, Annals of Mathematics Studies, Volume 71, Princeton University Press; University of Tokyo Press, 1971, xi+161 pages | MR 320365 | Zbl 0245.32005

[14] Robert Lazarsfeld Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Volume 48, Springer, 2004 | Zbl 1093.14501

[15] Noboru Nakayama Zariski-decomposition and abundance, MSJ Memoirs, Volume 14, Mathematical Society of Japan, 2004, xiv+277 pages | MR 2104208

[16] F. W. Olver; D. M. Lozier; R. F. Boisvert; C. W. Clark Digital Library of Mathematical Functions: Online Companion to NIST Handbook of Mathematical Functions (CUP) (2010) (National Insitute of Standards and Technology, http://dlmf.nist.gov)

[17] Hugo Rossi Strongly pseudoconvex manifolds, Lectures in Modern Analysis and Applications, I, Springer, 1969, pp. 10-29 | MR 249661 | Zbl 0179.40102