no. 6
Multipoint Okounkov bodies
[Corps d’Okounkov à plusieurs points]
Annales de l'Institut Fourier, Tome 71 (2021) no. 6, pp. 2595-2646.

Étant donné un grand fibré en droites gros L sur une variété projective X et le choix de N≥1 points différents sur X, on donne une nouvelle construction de N corps d’Okounkov qui donne des informations géométriques importantes sur (L→X;p 1 ,⋯,p N ) comme, par example, le volume de L, la constante de Seshadri de L aux points p 1 ,⋯,p N et la possibilité de construire des « Kähler packings » centrés en p 1 ,⋯,p N . Les cas des variétés toriques et des surfaces sont examinés en détail.

Starting from the data of a big line bundle L on a projective manifold X with a choice of N≥1 different points on X we provide a new construction of N Okounkov bodies that encode important geometric features of (L→X;p 1 ,⋯,p N ) such as the volume of L, the (moving) multipoint Seshadri constant of L at p 1 ,⋯,p N , and the possibility to construct Kähler packings centered at p 1 ,⋯,p N . Toric manifolds and surfaces are examined in detail.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3462
Classification : 14C20, 32Q15, 57R17
Keywords: Okounkov body, Seshadri constant, packings problem, projective manifold, ample line bundle.
Mot clés : Corps d’Okounkov, constante de Seshadri, problème du « packing », variété projective, fibré en droite ample. 
@article{AIF_2021__71_6_2595_0,
     author = {Trusiani, Antonio},
     title = {Multipoint {Okounkov} bodies},
     journal = {Annales de l'Institut Fourier},
     pages = {2595--2646},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {6},
     year = {2021},
     doi = {10.5802/aif.3462},
     zbl = {07554455},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3462/}
}
TY  - JOUR
AU  - Trusiani, Antonio
TI  - Multipoint Okounkov bodies
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 2595
EP  - 2646
VL  - 71
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.3462/
DO  - 10.5802/aif.3462
LA  - en
ID  - AIF_2021__71_6_2595_0
ER  - 
%0 Journal Article
%A Trusiani, Antonio
%T Multipoint Okounkov bodies
%J Annales de l'Institut Fourier
%D 2021
%P 2595-2646
%V 71
%N 6
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.3462/
%R 10.5802/aif.3462
%G en
%F AIF_2021__71_6_2595_0
Trusiani, Antonio. Multipoint Okounkov bodies. Annales de l'Institut Fourier, Tome 71 (2021) no. 6, pp. 2595-2646. doi : 10.5802/aif.3462. http://archive.numdam.org/articles/10.5802/aif.3462/

[1] Anderson, Dave Okounkov bodies and toric degenerations, Math. Ann., Volume 356 (2013) no. 3, pp. 1183-1202 | DOI | MR | Zbl

[2] Bauer, Thomas; Di Rocco, Sandra; Harbourne, Brian; Kapustka, Michał; Knutsen, Andreas; Syzdek, Wioletta; Szemberg, Tomasz A primer on Seshadri constants, Interactions of classical and numerical algebraic geometry (Contemp. Math.), Volume 496, Amer. Math. Soc., Providence, RI, 2009, pp. 33-70 | DOI | MR | Zbl

[3] Bauer, Thomas; Küronya, Alex; Szemberg, Tomasz Zariski chambers, volumes, and stable base loci, J. Reine Angew. Math., Volume 576 (2004), pp. 209-233 | DOI | MR | Zbl

[4] Biran, Paul Symplectic packing in dimension 4, Geom. Funct. Anal., Volume 7 (1997) no. 3, pp. 420-437 | DOI | MR | Zbl

[5] Boucksom, Sébastien Corps d’Okounkov, Séminaire Bourbaki, Volume 65 (2012), pp. 1-38 | Zbl

[6] Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011, xxiv+841 pages | DOI | MR

[7] Demailly, Jean-Pierre Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) (Lecture Notes in Math.), Volume 1507, Springer, Berlin, 1992, pp. 87-104 | DOI | MR | Zbl

[8] Demailly, Jean-Pierre A numerical criterion for very ample line bundles, J. Differential Geom., Volume 37 (1993) no. 2, pp. 323-374 | DOI | MR | Zbl

[9] Dumnicki, Marcin; Küronya, Alex; Maclean, Catriona; Szemberg, Tomasz Rationality of Seshadri constants and the Segre–Harbourne–Gimigliano–Hirschowitz conjecture, Adv. Math., Volume 303 (2016), pp. 1162-1170 | DOI | MR | Zbl

[10] Eckl, Thomas Kähler packings and Seshadri constants on projective complex surfaces, Differential Geom. Appl., Volume 52 (2017), pp. 51-63 | DOI | MR | Zbl

[11] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 6, pp. 1701-1734 | DOI | Numdam | MR | Zbl

[12] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Restricted volumes and base loci of linear series, Amer. J. Math., Volume 131 (2009) no. 3, pp. 607-651 | DOI | MR | Zbl

[13] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993, xii+157 pages (The William H. Roever Lectures in Geometry) | DOI | MR | Zbl

[14] Ito, Atsushi Okounkov bodies and Seshadri constants, Adv. Math., Volume 241 (2013), pp. 246-262 | DOI | MR | Zbl

[15] Kaveh, Kiumars Toric degenerations and symplectic geometry of smooth projective varieties, J. Lond. Math. Soc. (2), Volume 99 (2019) no. 2, pp. 377-402 | DOI | MR | Zbl

[16] Kaveh, Kiumars; Khovanskii, Askold G. Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), Volume 176 (2012) no. 2, pp. 925-978 | DOI | MR | Zbl

[17] Khovanskii, Askold Georgievich Newton polyhedron, Hilbert polynomial, and sums of finite sets, Funct. Anal. Appl., Volume 26 (1992) no. 4, p. 1 | DOI | Zbl

[18] Küronya, Alex; Lozovanu, Victor Infinitesimal Newton–Okounkov bodies and jet separation, Duke Math. J., Volume 166 (2017) no. 7, pp. 1349-1376 | DOI | MR | Zbl

[19] Küronya, Alex; Lozovanu, Victor Positivity of line bundles and Newton–Okounkov bodies, Doc. Math., Volume 22 (2017), pp. 1285-1302 | DOI | MR | Zbl

[20] Küronya, Alex; Lozovanu, Victor; Maclean, Catriona Convex bodies appearing as Okounkov bodies of divisors, Adv. Math., Volume 229 (2012) no. 5, pp. 2622-2639 | DOI | MR | Zbl

[21] Lazarsfeld, Robert Positivity in algebraic geometry I: Classical setting: line bundles and linear series, 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, xviii+387 pages | DOI | MR | Zbl

[22] Lazarsfeld, Robert; Mustaţă, Mircea Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009) no. 5, pp. 783-835 | DOI | Numdam | MR | Zbl

[23] McDuff, Dusa; Polterovich, Leonid Symplectic packings and algebraic geometry, Invent. Math., Volume 115 (1994) no. 3, pp. 405-434 (With an appendix by Yael Karshon) | DOI | MR | Zbl

[24] Nagata, Masayoshi On the 14-th problem of Hilbert, Amer. J. Math., Volume 81 (1959), pp. 766-772 | DOI | MR | Zbl

[25] Nakamaye, Michael Base loci of linear series are numerically determined, Trans. Amer. Math. Soc., Volume 355 (2003) no. 2, pp. 551-566 | DOI | MR | Zbl

[26] Okounkov, Andrei Brunn–Minkowski inequality for multiplicities, Invent. Math., Volume 125 (1996) no. 3, pp. 405-411 | DOI | MR | Zbl

[27] Okounkov, Andrei Why would multiplicities be log-concave?, The orbit method in geometry and physics (Marseille, 2000) (Progr. Math.), Volume 213, Birkhäuser Boston, Boston, MA, 2003, pp. 329-347 | DOI | MR | Zbl

[28] Shin, Jaesun Extended Okounkov bodies and multi-point Seshadri constants (2017) (https://arxiv.org/abs/1710.04351)

[29] Takayama, Shigeharu Pluricanonical systems on algebraic varieties of general type, Invent. Math., Volume 165 (2006) no. 3, pp. 551-587 | DOI | MR | Zbl

[30] Witt Nyström, David Okounkov bodies and the Kähler geometry of projective manifolds (2015) (https://arxiv.org/abs/1510.00510)

Cité par Sources :