Potential theory and Lefschetz theorems for arithmetic surfaces

Bost, J.-B.

doi:10.1016/s0012-9593(99)80015-9

Bost, J.-B.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 32 (1999) no. 2, pp. 241-312.

MR Zbl | 4 citations dans Numdam

DOI : 10.1016/s0012-9593(99)80015-9

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     author = {Bost, J.-B.},
     title = {Potential theory and {Lefschetz} theorems for arithmetic surfaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {241--312},
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Bost, J.-B. Potential theory and Lefschetz theorems for arithmetic surfaces. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 32 (1999) no. 2, pp. 241-312. doi : 10.1016/s0012-9593(99)80015-9. http://www.numdam.org/articles/10.1016/s0012-9593(99)80015-9/

Bibliographie
Cité par

[B-De] A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A., Vol. 45, 1959, pp. 208-215. | MR | Zbl

[B] E. Bombieri, Canonical models of surfaces of general type, Publ. Math. IHES, Vol. 42, 1973, pp. 171-219. | EuDML | Numdam | MR | Zbl

[B-G-S] J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc., Vol. 7, 1994, pp. 903-1027. | MR | Zbl

[Br] M. Brelot, Etude et extensions du principe de Dirichlet, Ann. Inst. Fourier Grenoble, Vol. 5, 1953-1954, pp. 371-419. | EuDML | Numdam | MR | Zbl

[C] T. Chinburg, Capacity theory on varieties, Compositio Math., Vol. 80, 1991, pp. 75-84. | EuDML | Numdam | MR | Zbl

[Ch] G. Choquet, Capacitabilité en potentiel logarithmique, Acad. Roy. Belg. Bull. Cl. Sci., (5), Vol. 44, 1958, pp. 321-326. | MR | Zbl

[C-C] R. Coleman and W. Mccallum, Stable reduction of Fermat curves and local components of Jacobi sum Hecke characters, J. reine angew. Math., Vol. 385, 1988, pp. 41-101. | EuDML | MR | Zbl

[D1] P. Deligne, Intersection sur les surfaces régulières, Exposé in SGA 7 II, Groupe de Monodromie en Géometrie Algébrique, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin, 1973. | MR | Zbl

[D2] P. Deligne, Preuve des conjectures de Tate et de Shafarevitch (d'après G. Faltings), in Séminaire Bourbaki 1983/1984, Exposé n° 616, Astérisque, Vol. 121-122, 1985, pp. 25-41. | EuDML | Numdam | MR | Zbl

[D3] P. Deligne, Le déterminant de la cohomologie, Contemporary Mathematics, Vol. 67, 1987, pp. 93-178. | MR | Zbl

[De1] J. Deny, Les potentiels d'énergie finie, Acta Math., Vol. 82, 1950, pp. 107-183. | MR | Zbl

[De2] J. Deny, Méthodes hilbertiennes en théorie du potentiel, in Potential Theory (C.I.M.E., I Ciclo, Stresa, 1969), pp. 121-201. Edizioni Cremonese, Rome 1970. | Zbl

[De-L] J. Deny and J.L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier, Vol. 5, 1953-1954, pp. 305-370 (1955). | Numdam | MR | Zbl

[Do] J.L. Doob, Potential theory and its probabilistic counterpart, Springer Verlag, New York, Heidelberg, Berlin, 1984. | MR

[Fa] G. Faltings, Calculus on arithmetic surfaces, Ann. of Math., Vol. 119, 1984, pp. 387-424. | MR | Zbl

[F1] A. Franchetta, Sulle curve reducibili appartenenti ad una superficie algebrica, Univ. Roma. Ist. Naz. Alta Mat. Rend. Mat. e Appl., (5), Vol. 8, 1949, pp. 378-398. | MR | Zbl

[F2] A. Franchetta, Sul sistema aggiunto ad una curva riducibile, Atti. Acad. Naz. Lincei Rend. Cl. Sci. Mat. Nat., (8), Vol. 6, 1949, pp. 685-687. | MR | Zbl

[Fu] W. Fulton, On the topology of algebraic varieties, Proc. of Symposia in Pure Mathematics, Vol. 46, Part 1, 1987, pp. 15-46. | MR | Zbl

[Ga] C. Gasbarri, Le théorème d'annulation de Serre sur les variétés arithmétiques, J. of Algebraic Geometry, Vol. 5, 1996, pp. 571-581. | MR | Zbl

[G-S1] H. Gillet and C. Soulé, Arithmetic intersection theory, Publ. Math. I.H.E.S., Vol. 72, 1990, pp. 93-174. | Numdam | MR | Zbl

[G-S2] H. Gillet and C. Soulé, Arithmetic analogs of the standard conjectures, in Motives (Seattle 1991, U. Jannsen, S. Kleiman, J.-P. Serre ed.), Proc. Symp. Pure Math., Vol. 55, Part 1, 1994, pp. 129-140. | MR | Zbl

[Gr1] A. Grothendieck, Revêtements étales et groupe fondamental (SGA1), Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-Heidelberg-New York, 1971. | MR | Zbl

[Gr2] A. Grothendieck, Théorèmes de Lefschetz locaux et globaux (SGA2), North Holland Publ. Co., Amsterdam, 1968. | MR

[H] D. Harbater, Galois covers of an arithmetic surface, Amer. J. of Math., Vol. 110, 1988, pp. 849-885. | MR | Zbl

[Ha] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, Heidelberg, Berlin, 1977. | MR | Zbl

[H-M] H. Hironaka and H. Matsumura, Formal functions and formal embeddings, J. Math. Soc. Japan, Vol. 20, 1968, pp. 52-67. | MR | Zbl

[Hr] D. Hriljac, Heights and Arakelov intersection theory, Ann. J. of Math., Vol. 107, 1985, pp. 23-38. | MR | Zbl

[I] Y. Ihara, Horizontal divisors on arithmetic surfaces associated with Belyĭ uniformizations, in The Grothendieck theory of dessins d'enfants (Luminy 1993, L. Schneips ed.), pp. 245-254, London Math. Soc. Lecture Note Series, Vol. 200, Cambridge University Press, Cambridge, 1994. | Zbl

[L] S. Lefschetz, L'analysis situs et la géométrie algébrique, Paris, Gauthier-Villars, 1924. | JFM

[LM] R. Levin-Ménégaux, Un théorème d'annulation en caractéristique positive, in Séminaire sur les faisceaux de courbe de genre au moins deux (L. Szpiro ed.), Astérisque, Vol. 86, 1981, pp. 35-43. | Zbl

[MB1] L. Moret-Bailly, Groupes de Picard et problèmes de Skolem I, Ann. Sc. Ecole Normale Supérieure, Vol. 22, 1989, pp. 161-179. | Numdam | MR | Zbl

[MB2] L. Moret-Bailly, Métriques permises, Astérisque, Vol. 127, 1985, pp. 29-87. | Numdam | MR

[M] D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. IHES, Vol. 9, 1961, pp. 5-22. | Numdam | MR | Zbl

[R] C.P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. Indian Math. Society, Vol. 36, 1972, pp. 41-51. | MR | Zbl

[Ra] T. Ransford, Potential theory in the complex plane, Cambridge University Press, Cambridge, 1995. | MR | Zbl

[Ray] (Mme) M. Raynaud, Théorèmes de Lefschetz en cohomologie cohérente et en cohomologie étale, Bull. Soc. Math. France, Mémoire n° 41, 1975. | Numdam | MR | Zbl

[Ru1] R. Rumely, Capacity theory on algebraic curves, Lecture Notes in Mathematics, Vol. 1378, Springer-Verlag, Berlin, 1989. | MR | Zbl

[Ru2] R. Rumely, An intersection pairing for curves, with analytic contributions from nonarchimedean places, Canadian Mathematical Society Conference Proceedings, Vol. 15, 1995, pp. 325-357. | MR | Zbl

[So] C. Soulé, A vanishing theorem on arithmetic surfaces, Inventiones Math., Vol. 116, 1994, pp. 544-899. | MR | Zbl

[Sz] L. Szpiro, Sur le théorème de rigidité de Parsin et Arakelov, Journées de Géométrie Algébrique de Rennes, Astérisque, Vol. 64, 1979, pp. 169-202. | Numdam | MR | Zbl

[T] M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959. | MR | Zbl

[Z1] O. Zariski, Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Memories of Amer. Math. Soc., New York, 1951. | MR | Zbl

[Z2] O. Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publ. Math. Soc. of Japan, Vol. 4, 1958, vii + 89 pp. | MR | Zbl

[Z3] O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. Math., Vol. 76, 1962, pp. 560-615. | MR | Zbl

[Zh] S. Zhang, Positive line bundles on arithmetic surfaces, Ann. Math., Vol. 136, 1992, pp. 569-587. | MR | Zbl

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