Convex bodies associated to linear series
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 5, p. 783-835

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give some applications and examples.

Dans son travail sur la log-concavité des multiplicités, Okounkov montre au passage que l'on peut associer un corps convexe à un système linéaire sur une variété projective, puis utiliser la géométrie convexe pour étudier ces systèmes linéaires. Bien qu'Okounkov travaille essentiellement dans le cadre classique des fibrés en droites amples, il se trouve que sa construction s'étend au cas d'un grand diviseur arbitraire. De plus, ce point de vue permet de rendre transparentes de nombreuses propriétés de base des invariants asymptotiques des systèmes linéaires, et ouvre la porte à de nombreuses extensions. Le but de cet article est d'initier un développement systématique de la théorie et de donner quelques applications et exemples.

DOI : https://doi.org/10.24033/asens.2109
Classification:  14F05,  52C99
Keywords: algebraic varieties, linear series, convex bodies
@article{ASENS_2009_4_42_5_783_0,
author = {Lazarsfeld, Robert and Musta\c t\u a, Mircea},
title = {Convex bodies associated to linear series},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {Ser. 4, 42},
number = {5},
year = {2009},
pages = {783-835},
doi = {10.24033/asens.2109},
zbl = {1182.14004},
mrnumber = {2571958},
language = {en},
url = {http://www.numdam.org/item/ASENS_2009_4_42_5_783_0}
}

Lazarsfeld, Robert; Mustață, Mircea. Convex bodies associated to linear series. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 5, pp. 783-835. doi : 10.24033/asens.2109. http://www.numdam.org/item/ASENS_2009_4_42_5_783_0/

 V. Alexeev & M. Brion, Toric degenerations of spherical varieties, Selecta Math. (N.S.) 10 (2004), 453-478. | MR 2134452 | Zbl 1078.14075

 K. Ball, An elementary introduction to modern convex geometry, in Flavors of geometry, Math. Sci. Res. Inst. Publ. 31, Cambridge Univ. Press, 1997, 1-58. | MR 1491097 | Zbl 0901.52002

 T. Bauer, A. Küronya & T. Szemberg, Zariski chambers, volumes, and stable base loci, J. reine angew. Math. 576 (2004), 209-233. | MR 2099205 | Zbl 1055.14007

 D. Bayer & D. Mumford, What can be computed in algebraic geometry?, in Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., XXXIV, Cambridge Univ. Press, 1993, 1-48. | MR 1253986 | Zbl 0846.13017

 C. Birkar, P. Cascini, C. Hacon & J. Mckernan, Existence of minimal models for varieties of log general type, preprint arXiv:math/0610203. | MR 2601039 | Zbl 1210.14019

 S. Boucksom, On the volume of a line bundle, Internat. J. Math. 13 (2002), 1043-1063. | MR 1945706 | Zbl 1101.14008

 S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. 37 (2004), 45-76. | Numdam | MR 2050205 | Zbl 1054.32010

 S. Boucksom, J.-P. Demailly, M. Păun & T. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, preprint arXiv:math/0405285. | Zbl 1267.32017

 S. Boucksom, C. Favre & M. Jonsson, Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom. 18 (2009), 279-308. | MR 2475816 | Zbl 1162.14003

 M. Brion, Sur l'image de l'application moment, in Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986), Lecture Notes in Math. 1296, Springer, 1987, 177-192. | MR 932055 | Zbl 0667.58012

 F. Campana & T. Peternell, Algebraicity of the ample cone of projective varieties, J. reine angew. Math. 407 (1990), 160-166. | MR 1048532 | Zbl 0728.14004

 H. Chen, Arithmetic Fujita approximation, preprint arXiv:0810.5479. | Numdam | MR 2722508 | Zbl 1202.14024

 A. Della Vedova & R. Paoletti, Moment maps and equivariant volumes, Acta Math. Sin. 23 (2007), 2155-2188. | MR 2357452 | Zbl 1136.14034

 J.-P. Demailly, L. Ein & R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137-156. | Zbl 1077.14516

 L. Ein, R. Lazarsfeld, M. Mustață, M. Nakamaye & M. Popa, Asymptotic invariants of line bundles, Pure Appl. Math. 1 (2005), 379-403. | Zbl 1139.14008

 L. Ein, R. Lazarsfeld, M. Mustață, M. Nakamaye & M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), 1701-1734. | Numdam | Zbl 1127.14010

 L. Ein, R. Lazarsfeld, M. Mustață, M. Nakamaye & M. Popa, Restricted volumes and base loci of linear series, to appear in Amer. J. Math.

 L. Ein, R. Lazarsfeld & K. E. Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), 409-440. | Zbl 1033.14030

 T. De Fernex, A. Küronya & R. Lazarsfeld, Higher cohomology of divisors on a projective variety, Math. Ann. 337 (2007), 443-455. | Zbl 1127.14012

 T. Fujita, Approximating Zariski decomposition of big line bundles, Kodai Math. J. 17 (1994), 1-3. | Zbl 0814.14006

 W. Fulton, Introduction to toric varieties, Annals of Math. Studies 131, Princeton Univ. Press, 1993. | Zbl 0813.14039

 P. M. Gruber, Convex and discrete geometry, Grund. Math. Wiss. 336, Springer, 2007. | Zbl 1139.52001

 C. D. Hacon & J. Mckernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), 1-25. | Zbl 1121.14011

 J. Harris & I. Morrison, Moduli of curves, Graduate Texts in Math. 187, Springer, 1998. | Zbl 0913.14005

 Y. Hu & S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331-348. | MR 1786494 | Zbl 1077.14554

 K. Kaveh, Note on the cohomology ring of spherical varieties and volume polynomial, preprint arXiv:math/0312503. | MR 2828718 | Zbl 1222.14108

 A. G. Khovanskiĭ, The Newton polytope, the Hilbert polynomial and sums of finite sets, Funct. Anal. Appl. 26 (1993), 276-281. | MR 1209944 | Zbl 0809.13012

 A. Khovanskii & K. Kaveh, Convex bodies and algebraic equations on affine varieties, preprint arXiv:0804.4095.

 R. Lazarsfeld, Positivity in algebraic geometry. I & II, Ergebnisse Math. Grenzg. 48 & 49, Springer, 2004. | MR 2095471 | Zbl 1093.14500

 A. Moriwaki, Continuity of volumes on arithmetic varieties, preprint arXiv:math/0612269. | MR 2496453 | Zbl 1167.14014

 M. Mustață, On multiplicities of graded sequences of ideals, J. Algebra 256 (2002), 229-249. | MR 1936888 | Zbl 1076.13500

 M. Nakamaye, Base loci of linear series are numerically determined, Trans. Amer. Math. Soc. 355 (2003), 551-566. | MR 1932713 | Zbl 1017.14017

 T. Oda, Convex bodies and algebraic geometry, Ergebnisse Math. Grenzg. 15, Springer, 1988. | MR 922894 | Zbl 0628.52002

 A. Okounkov, Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996), 405-411. | MR 1400312 | Zbl 0893.52004

 A. Okounkov, Note on the Hilbert polynomial of a spherical manifold, Funct. Anal. Appl. 31 (1997), 133-140. | MR 1475330 | Zbl 0928.14032

 A. Okounkov, Why would multiplicities be log-concave?, in The orbit method in geometry and physics (Marseille, 2000), Progr. Math. 213, Birkhäuser, 2003, 329-347. | MR 1995384 | Zbl 1063.22024

 R. Paoletti, Szegő kernels and finite group actions, Trans. Amer. Math. Soc. 356 (2004), 3069-3076. | MR 2052941 | Zbl 1055.53065

 R. Paoletti, The asymptotic growth of equivariant sections of positive and big line bundles, Rocky Mountain J. Math. 35 (2005), 2089-2105. | MR 2210648 | Zbl 1121.32008

 S. Takagi, Fujita's approximation theorem in positive characteristics, J. Math. Kyoto Univ. 47 (2007), 179-202. | MR 2359108 | Zbl 1136.14004

 S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), 551-587. | MR 2242627 | Zbl 1108.14031

 H. Tsuji, Effective birationality of pluricanonical systems, preprint arXiv:math/0011257.

 A. Wolfe, Cones and asymptotic invariants of multigraded systems of ideals, J. Algebra 319 (2008), 1851-1869. | MR 2392582 | Zbl 1142.14004

 X. Yuan, Big line bundles over arithmetic varieties, Invent. Math. 173 (2008), 603-649. | MR 2425137 | Zbl 1146.14016

 X. Yuan, On volumes of arithmetic line bundles, preprint arXiv:0811.0226. | MR 2575090 | Zbl 1197.14023