Convex bodies associated to linear series
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 5, pp. 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: 10.24033/asens.2109
Classification: 14F05, 52C99
Keywords: algebraic varieties, linear series, convex bodies
Mot clés : variétés algebriques, systèmes linéaires, corps convexes
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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://archive.numdam.org/articles/10.24033/asens.2109/

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

[2] 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 | Zbl

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

[4] 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 | Zbl

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

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

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

[8] 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

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

[10] 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 | Zbl

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

[12] H. Chen, Arithmetic Fujita approximation, preprint arXiv:0810.5479. | Numdam | MR | Zbl

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

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

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

[16] 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

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

[18] 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

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

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

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

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

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

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

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

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

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

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

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

[30] A. Moriwaki, Continuity of volumes on arithmetic varieties, preprint arXiv:math/0612269. | MR | Zbl

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

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

[33] T. Oda, Convex bodies and algebraic geometry, Ergebnisse Math. Grenzg. 15, Springer, 1988. | MR | Zbl

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

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

[36] 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 | Zbl

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

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

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

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

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

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

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

[44] X. Yuan, On volumes of arithmetic line bundles, preprint arXiv:0811.0226. | MR | Zbl

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