On metrics with minimal singularities of line bundles whose stable base loci admit holomorphic tubular neighborhoods

Hosono, Genki; Koike, Takayuki

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 $r\ge 1$. 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 $r\ge 1$. 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.

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

[2] 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

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

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

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

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

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

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

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

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

[11] Koike, Takayuki 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 | DOI | Numdam | MR | Zbl

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

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

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

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

[16] Olver, F. W.; Lozier, D. M.; Boisvert, R. F.; Clark, C. W. 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] Rossi, Hugo Strongly pseudoconvex manifolds, Lectures in Modern Analysis and Applications, I, Springer, 1969, pp. 10-29 | MR | Zbl