Arithmetic geometry of toric varieties. Metrics, measures and heights
Astérisque, no. 360 (2014) , 228 p.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez le site de la revue.
@book{AST_2014__360__R1_0,
     author = {Burgos Gil, Jos\'e Ignacio and Philippon, Patrice and Sombra, Mart{\'\i}n},
     title = {Arithmetic geometry of toric varieties. {Metrics,} measures and heights},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {360},
     year = {2014},
     zbl = {1311.14050},
     mrnumber = {3222615},
     language = {en},
     url = {http://archive.numdam.org/item/AST_2014__360__R1_0/}
}
TY  - BOOK
AU  - Burgos Gil, José Ignacio
AU  - Philippon, Patrice
AU  - Sombra, Martín
TI  - Arithmetic geometry of toric varieties. Metrics, measures and heights
T3  - Astérisque
PY  - 2014
DA  - 2014///
IS  - 360
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/AST_2014__360__R1_0/
UR  - https://zbmath.org/?q=an%3A1311.14050
UR  - https://www.ams.org/mathscinet-getitem?mr=3222615
LA  - en
ID  - AST_2014__360__R1_0
ER  - 
Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín. Arithmetic geometry of toric varieties. Metrics, measures and heights. Astérisque, no. 360 (2014), 228 p. http://numdam.org/item/AST_2014__360__R1_0/

[AMR88] R. Abraham, J. E. Marsden & T. Ratiu - Manifolds, tensor analysis, and applications, Appl. Math. Sci., vol. 75, Springer-Verlag, 1988. | Article | MR 960687 | Zbl 0875.58002

[Abr03] M. Abreu - "Kähler geometry of toric manifolds in symplectic coordinates", in Symplectic and contact topology: interactions and perspectives (Y. Eliashberg, B. Khesin & F. Lalonde, eds.), Fields Inst. Commun., vol. 35, Amer. Math. Soc., 2003, p. 1-24. | MR 1969265 | Zbl 1044.53051

[AV09] F. Amoroso & E. Viada - "Small points on subvarieties of a torus", Duke Math. J. 150 (2009), no. 3, p. 407-442. | Article | MR 2582101 | Zbl 1234.11081

[Ara74] S. J. Arakelov - "An intersection theory for divisors on an arithmetic surface", Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), p. 1179-1192. | MR 472815 | Zbl 0355.14002

[AMRT75] A. Ash, D. Mumford, M. Rapoport & Y. Tai - Smooth compactification of locally symmetric varieties. Lie Groups: History, Frontiers and Applications, Vol. IV, Math. Sci. Press, 1975. | MR 457437 | Zbl 0334.14007

[AKS07] M. Avendaño, T. Krick & M. Sombra - "Factoring bivariate sparse (lacunary) polynomials", J. Complexity 23 (2007), no. 2, p. 193-216. | Article | MR 2314756 | Zbl 1170.12004

[BBD+11] V. Baldoni, N. Berline, J. De Loera, M. Köppe & M. Vergne - "How to integrate a polynomial over a simplex", Math. Comp. 80 (2011), no. 273, p. 297-325. | Article | MR 2728981 | Zbl 1216.68120

[BT95] V. V. Batyrev & Y. Tschinkel - "Rational points of bounded height on compactifications of anisotropic tori", Internat. Math. Res. Notices (1995), no. 12, p. 591-635. | Article | MR 1369408 | Zbl 0890.14008

[Ber90] V. G. Berkovich - Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr., vol. 33, Amer. Math. Soc., 1990. | MR 1070709 | Zbl 0715.14013

[Ber75] D. N. Bernstein - "The number of roots of a system of equations", Funkcional. Anal, i Priložen. 9 (1975), no. 3, p. 1-4, English translation: Functional Anal. Appl. 9 (1975), 183-185. | MR 435072 | Zbl 0328.32001

[Bil97] Y. Bilu - "Limit distribution of small points on algebraic tori", Duke Math. J. 89 (1997), no. 3, p. 465-476. | Article | MR 1470340 | Zbl 0918.11035

[BH69] T. Bloom & M. Herrera - "De Rham cohomology of an analytic space", Invent. Math. 7 (1969), p. 275-296. | Article | EuDML 141968 | MR 248349 | Zbl 0175.37301

[BGS94] J.-B. Bost, H. Gillet & C. Soulé - "Heights of projective varieties and positive Green forms", J. Amer. Math. Soc. 7 (1994), no. 4, p. 903-1027. | Article | MR 1260106 | Zbl 0973.14013

[BFJ11] S. Boucksom, C. Favre & M. Jonsson - "Singular semipositive metrics in non-Archimedean geometry", e-print arXiv:1201.0187v1, 2011. | MR 3419957 | Zbl 1346.14065

[BC11] S. Boucksom & H. Chen - "Okounkov bodies of filtered linear series", Compos. Math. 147 (2011), no. 4, p. 1205-1229. | Article | MR 2822867 | Zbl 1231.14020

[BV04] S. Boyd & L. Vandenberghe - Convex optimization, Cambridge Univ. Press, 2004. | MR 2061575 | Zbl 1058.90049

[Bri88] M. Brion - "Points entiers dans les polyèdres convexes", Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 4, p. 653-663. | Article | EuDML 82241 | Numdam | MR 982338 | Zbl 0667.52011

[BBK07] J. H. Bruinier, J. I. Burgos Gil & U. Kühn - "Borcherds products and arithmetic intersection theory on Hilbert modular surfaces", Duke Math. J. 139 (2007), no. 1, p. 1-88. | Article | MR 2322676 | Zbl 1208.11077

[Bur94] J. I. Burgos - "Green forms and their product", Duke Math. J. 75 (1994), no. 3, p. 529-574. | Article | MR 1291696 | Zbl 1044.14505

[BMPS12] J. I. Burgos Gil, A. Moriwaki, P. Philippon & M. Sombra - "Arithmetic positivity on toric varieties", e-print arXiv:1210.7692v1, to appear in J. Algebraic Geom. | MR 3466351 | Zbl 1378.14048

[BPS09] J. I. Burgos Gil, P. Philippon & M. Sombra - "Hauteur des sous-schémas toriques et dualité de Legendre-Fenchel", C.R. Math. Acad. Sci. Pans Sér. I Math. 347 (2009), no. 11-12, p. 589-594. | Article | MR 2532911 | Zbl 1171.14016

[BS11] J. I. Burgos Gil & M. Sombra - "When do the recession cones of a polyhedral complex form a fan?", Discrete Comput. Geom. 46 (2011), no. 4, p. 789-798. | Article | MR 2846179 | Zbl 1233.14031

[CM00] J. Cassaigne & V. Maillot - "Hauteur des hypersurfaces et fonctions zêta d'Igusa", J. Number Theory 83 (2000), no. 2, p. 226-255. | Article | MR 1772614 | Zbl 1001.11027

[Cha06] A. Chambert-Loir - "Mesures et équidistribution sur les espaces de Berkovich", J. Reine Angew. Math. 595 (2006), p. 215-235. | MR 2244803 | Zbl 1112.14022

[Cha11] A. Chambert-Loir, "Heights and measures on analytic spaces. A survey of recent results, and some remarks", in Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume II, London Math. Soc. Lecture Note Ser., vol. 384, Cambridge Univ. Press, 2011, p. 1-50. | MR 2885340 | Zbl 1279.14027

[CT09] A. Chambert-Loir & A. Thuillier - "Mesures de Mahler et équidistribution logarithmique", Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, p. 977-1014. | Article | EuDML 10426 | Numdam | MR 2543659 | Zbl 1192.14020

[Con07] B. Conrad - "Deligne's notes on Nagata compactifications", J. Ramanujan Math. Soc. 22 (2007), no. 3, p. 205-257. | MR 2356346 | Zbl 1142.14001

[CLS11] D. A. Cox, J. B. Little & H. K. Schenck - Toric varieties, Grad. Stud. Math., vol. 124, Amer. Math. Soc., 2011. | MR 2810322 | Zbl 1223.14001

[DKS12] X. Dahan, A. Kadri & É. Schost - "Bit-size estimates for triangular sets in positive dimension", J. Complexity 28 (2012), no. 1, p. 109-135. | Article | MR 2871788 | Zbl 1246.13039

[Dan97] N. Dan - "La hauteur des quadriques", C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 12, p. 1323-1326. | Article | MR 1457080 | Zbl 0917.11024

[DKS13] C. D'Andrea, T. Krick & A. Szanto - "Subresultants in multiple roots", Linear Algebra Appl. 438 (2013), no. 5, p. 1969-1989. | Article | MR 3005269 | Zbl 1293.13009

[DP99] S. David & P. Philippon - "Minorations des hauteurs normalisées des sous-variétés des tores", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 3, p. 489-543. | EuDML 84386 | Numdam | MR 1736526 | Zbl 1002.11055

[ELST14] E. J. Elizondo, P. Lima-Filho, F. Sottile & Z. Teitler - "Arithmetic toric varieties", Math. Nachr. 287 (2014), no. 2-3, p. 216-241. | Article | MR 3163576 | Zbl 1336.14034

[Ewa96] G. Ewald - Combinatorial convexity and algebraic geometry, Grad. Texts in Math., vol. 168, Springer-Verlag, 1996. | Article | MR 1418400 | Zbl 0869.52001

[Ful93] W. Fulton - Introduction to toric varieties, Ann. of Math. Stud., vol. 131, Princeton Univ. Press, 1993. | MR 1234037 | Zbl 0813.14039

[GKZ94] I. M. Gel'Fand, M. M. Kapranov & A. V. Zelevinsky - Discriminants, resultants, and multidimensional determinants, Math. Theory Appl., Birkhäuser, 1994. | MR 1264417 | Zbl 0827.14036

[GS90a] H. Gillet & C. Soulé - "Arithmetic intersection theory", Inst. Hautes Études Sci. Publ. Math. (1990), no. 72, p. 93-174. | EuDML 104073 | Numdam | MR 1087394 | Zbl 0741.14012

[GS90b] H. Gillet & C. Soulé - "Characteristic classes for algebraic vector bundles with Hermitian metric. II", Ann. of Math. (2) 131 (1990), no. 2, p. 205-238. | Article | MR 1043268 | Zbl 0715.14006

[GHH+97] M. Giusti, J. Heintz, K. Hägele, J. E. Morais, L. M. Pardo & J. L. Montaña - "Lower bounds for Diophantine approximations", J. Pure Appl. Algebra 117/118 (1997), p. 277-317. | Article | MR 1457843 | Zbl 0871.68101

[GH94] P. Griffiths & J. Harris - Principles of algebraic geometry, John Wiley & Sons Inc., 1994. | Article | MR 1288523 | Zbl 0836.14001

[Gub02] W. Gubler - "Basic properties of heights of subvarieties", Habilitation thesis, ETH Zürich, 2002.

[Gub03] W. Gubler - "Local and canonical heights of subvarieties", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, p. 711-760. | EuDML 84517 | Numdam | MR 2040641 | Zbl 1170.14303

[Gub07] W. Gubler, "Tropical varieties for non-Archimedean analytic spaces", Invent. Math. 169 (2007), no. 2, p. 321-376. | Article | MR 2318559 | Zbl 1153.14036

[Gubl2] W. Gubler, "A guide to tropicalizations", in Algebraic and combinatorial aspects of tropical geometry, Contemp. Math., vol. 589, Amer. Math. Soc., 2013, p. 125-189. | Article | MR 3088913 | Zbl 1318.14061

[Gui95] V. Guillemin - Moment maps and combinatorial invariants of Hamiltonian 𝕋 n -spaces, Progr. Math., vol. 122, Birkhäuser, 1995. | MR 1301331 | Zbl 0828.58001

[Har66] R. Hartshorne - "Ample vector bundles", Inst. Hautes Études Sci. Publ. Math. (1966), no. 29, p. 63-94. | EuDML 103864 | Numdam | MR 193092 | Zbl 0173.49003

[HT99] R. Horst & N. V. Thoai - "DC programming: overview", J. Optim. Theory Appl. 103 (1999), no. 1, p. 1-43. | Article | MR 1715016 | Zbl 1073.90537

[Kaj08] T. Kajiwara - "Tropical toric geometry", in Toric topology, Contemp. Math., vol. 460, Amer. Math. Soc., 2008, p. 197-207. | Article | MR 2428356 | Zbl 1202.14047

[KKMS73] G. Kempf, F. F. Knudsen, D. Mumford & B. Saint-Donat - Toroidal embeddings. I, Lecture Notes in Math., vol. 339, Springer-Verlag, 1973. | MR 335518 | Zbl 0271.14017

[Kuš76] A. G. Kouchnirenko - "Polyèdres de Newton et nombres de Milnor", Invent. Math. 32 (1976), no. 1, p. 1-31. | Article | EuDML 142365 | MR 419433 | Zbl 0328.32007

[Lic68] S. Lichtenbaum - "Curves over discrete valuation rings", Amer. J. Math. 90 (1968), p. 380-405. | Article | MR 230724 | Zbl 0194.22101

[Liu02] Q. Liu - Algebraic geometry and arithmetic curves, Oxf. Grad. Texts, vol. 6, Oxford Univ. Press, 2002. | MR 1917232 | Zbl 0996.14005

[Liu06] Q. Liu - "Stable reduction of finite covers of curves", Compos. Math. 142 (2006), no. 1, p. 101-118. | Article | MR 2196764 | Zbl 1108.14020

[Mai00] V. Maillot - "Géométrie d'Arakelov des variétés toriques et fibrés en droites intégrables", Mém. Soc. Math. France (N.S.) (2000), no. 80, p. vi+129. | EuDML 94931 | Numdam | MR 1775582 | Zbl 0963.14009

[Mav00] A. R. Mavlyutov - "Semiample hypersurfaces in toric varieties", Duke Math. J. 101 (2000), no. 1, p. 85-116. | Article | MR 1733735 | Zbl 1023.14027

[Mou06] C. Mourougane - "Analytic torsion of Hirzebruch surfaces", Math. Ann. 335 (2006), no. 1, p. 221-247. | Article | MR 2217689 | Zbl 1176.58018

[NS06] T. Nishinou & B. Siebert - "Toric degenerations of toric varieties and tropical curves", Duke Math. J. 135 (2006), no. 1, p. 1-51. | Article | MR 2259922 | Zbl 1105.14073

[Oda88] T. Oda - Convex bodies and algebraic geometry, Ergeb. Math. Grenzgeb. (3), vol. 15, Springer-Verlag, 1988. | EuDML 203658 | MR 922894 | Zbl 0628.52002

[Ovc02] S. Ovchinnikov - "Max-min representation of piecewise linear functions", Beiträge Algebra Geom. 43 (2002), no. 1, p. 297-302. | EuDML 225460 | MR 1913786 | Zbl 0996.26007

[PR04] M. Passare & H. Rullgård - "Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope", Duke Math. J. 121 (2004), no. 3, p. 481-507. | Article | MR 2040284 | Zbl 1043.32001

[Pay09] S. Payne - "Analytification is the limit of all tropicalizations", Math. Res. Lett. 16 (2009), no. 3, p. 543-556. | Article | MR 2511632 | Zbl 1193.14077

[Phi94] P. Philippon - "Sur des hauteurs alternatives. II", Ann. Inst. Fourier (Grenoble) 44 (1994), no. 4, p. 1043-1065. | Article | EuDML 75089 | Numdam | MR 1306549 | Zbl 0878.11024

[PS08a] P. Philippon & M. Sombra - "Hauteur normalisée des variétés toriques projectives", J. Inst. Math. Jussieu 7 (2008), no. 2, p. 327-373. | Article | MR 2400725 | Zbl 1147.11033

[PS08b] P. Philippon & M. Sombra, "A refinement of the Bernštein-Kušnirenko estimate", Adv. Math. 218 (2008), no. 5, p. 1370-1418. | Article | MR 2419926 | Zbl 1155.14037

[Poi12] J. Poineau - "Les espaces de Berkovich sont angéliques", Bull. Soc. Math. France 141 (2013), no. 2, p. 267-297. | Article | EuDML 272716 | Numdam | MR 3081557 | Zbl 1314.14046

[RT77] J. Rauch & B. A. Taylor - "The Dirichlet problem for the multidimensional Monge-Ampère equation", Rocky Mountain J. Math. 7 (1977), no. 2, p. 345-364. | Article | MR 454331 | Zbl 0367.35025

[Roc70] R. T. Rockafellar - Convex analysis, Princeton Math. Series, vol. 28, Princeton Univ. Press, 1970. | MR 274683

[Sch93] R. Schneider - Convex bodies: the Brunn-Minkowski theory, Encyclopedia Math. Appl, vol. 44, Cambridge Univ. Press, 1993. | MR 1216521 | Zbl 0798.52001

[Som05] M. Sombra - "Minimums successifs des variétés toriques projectives", J. Reine Angew. Math. 586 (2005), p. 207-233. | Article | MR 2180605 | Zbl 1080.14060

[Stu02] B. Sturmfels - Solving systems of polynomial equations, CBMS Reg. Conf. Ser. Math., vol. 97, Amer. Math. Soc., 2002. | MR 1925796 | Zbl 1101.13040

[SUZ97] L. Szpiro, E. Ullmo & S. Zhang - "Équirépartition des petits points", Invent. Math. 127 (1997), no. 2, p. 337-347. | Article | MR 1427622 | Zbl 0991.11035

[Yua08] X. Yuan - "Big line bundles over arithmetic varieties", Invent. Math. 173 (2008), no. 3, p. 603-649. | Article | MR 2425137 | Zbl 1146.14016

[Zha95a] S. Zhang - "Positive line bundles on arithmetic varieties", J. Amer. Math. Soc. 8 (1995), no. 1, p. 187-221. | Article | MR 1254133 | Zbl 0861.14018

[Zha95b] S. Zhang, "Small points and adelic metrics", J. Algebraic Geom. 4 (1995), no. 2, p. 281-300. | MR 1311351 | Zbl 0861.14019