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ISSN 1088-6834(e) ISSN 0894-0347(p)

Heights of projective varieties and positive Green forms

Author(s): J.-B. Bost; H. Gillet; C. Soulé
Journal: J. Amer. Math. Soc. 7 (1994), 903-1027.
MSC: Primary 14G40; Secondary 11G35, 14C17
MathSciNet review: 1260106
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Abstract: Using arithmetic intersection theory, a theory of heights for projective varieties over rings of algebraic integers is developed. These heights are generalizations of those considered by Weil, Schmidt, Nesterenko, Philippon, and Faltings. Several of their properties are proved, including lower bounds and an arithmetic Bézout theorem for the height of the intersection of two projective varieties. New estimates for the size of (generalized) resultants are derived. Among the analytic tools used in the paper are ``Green forms'' for analytic subvarieties, and the existence of positive Green forms is discussed.

References:

[A-K]: A. B. Altman and S. L. Kleiman, Joins of schemes, linear projections, Compositio Math. 31 (1975), 309-343. MR 0396560 (53:423)
[Ar1]: S. J. Arakelov, Intersection theory of divisors on an arithmetic surface, Math. USSR Izv. 8 (1974), 1167-1180. MR 0472815 (57:12505)
[Ar2]: -, Theory of intersections on an arithmetic surface, Proc. Internat. Congr. Math., Vancouver, vol. 1, 1975, pp. 405-408. MR 0466150 (57:6031)
[B-B-E-M]: B. Beauzamy, E. Bombieri, P. Enflo, and H. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (1990), 219-245. MR 1072467 (91m:11015)
[Bi-V]: J.-M. Bismut and E. Vasserot, The asymptotic of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), 355-367. MR 1016875 (91c:58141)
[Bl]: A. Blanchard, Sur les variétés analytiques complexes, Ann. Sci. École Norm. Sup. 73 (1956), 157-202. MR 0087184 (19:316e)
[Bo-V]: E. Boda and W. Vogel, On a system of parameters, local intersection multiplicity and Bézout's theorem, Proc. Amer. Math. Soc. 78 (1980), 1-7. MR 548071 (80k:14011)
[Bo-Va]: E. Bombieri and J. Vaaler, On Siegel's lemma, Invent. Math. 73 (1983), 11-32. MR 707346 (85g:11049a)
[B-R]: A. Borel and R. Remmert, Über kompakte homogene Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1962), 429-439. MR 0145557 (26:3088)
[B1]: J.-B. Bost, Green's currents and height pairing on complex tori, Duke Math. J. 61 (1990), 899-912. MR 1084464 (91m:14034)
[B2]: -, Théorie de l'intersection et théorème de Riemann-Roch arithmétiques, Séminaire Bourbaki no 731, 1990/1991, Astérisque 201-203 (1991), 43-88.
[BGS]: J.-B. Bost, H. Gillet, and C. Soulé, Une analogue arithmétique du théorème de Bézout, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 845-848. MR 1108504 (92d:14014)
[B-C]: R. Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of zeroes of their holomorphic sections, Acta Math. 114 (1968), 71-112. MR 0185607 (32:3070)
[C-Wa]: W.-L. Chow and B. L. van der Waerden, Zur algebraischen Geometrie IX. Über zugeordnete Formen und algebraische Systeme von algebraischen Mannigfaltigkeiten, Math. Ann. 113 (1937), 692-704. MR 1513117
[Co-G]: C. Cornalba and P. Griffiths, Analytic cycles and vector bundles on non-compact algebraic varieties, Invent. Math. 28 (1975), 1-206. MR 0367263 (51:3505)
[Fa1]: G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), 387-424. MR 740897 (86e:14009)
[Fa2]: -, Diophantine approximation on Abelian varieties, Ann. of Math. (2) 133 (1991), 549-576. MR 1109353 (93d:11066)
[Fa3]: -, Lectures on the arithmetic Riemann-Roch theorem, notes by S. Zhang, Ann. of Math. Stud., no. 127, Princeton Univ. Press, Princeton, NJ, 1992. MR 1158661 (93f:14006)
[Fa-W]: G. Faltings, G. Wüstholz, et al., Rational points, Aspects of Mathematics, vol. E6, Vieweg-Verlag, Braunschweig, 1984. MR 766568 (87h:14016)
[Fo]: J. Fogarty, Truncated Hilbert functors, J. Reine Angew. Math. 234 (1969), 65-88. MR 0244268 (39:5585)
[Fu1]: W. Fulton, Rational equivalence on singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 147-167. MR 0404257 (53:8060)
[Fu2]: -, Intersection theory, Ergeb. Math. Grenzgeb. (3), Band 2, Springer-Verlag, Berlin, Heidelberg, and New York, 1984.
[Fu3]: -, Introduction to intersection theory in algebraic geometry, CBMS Regional Conf. Ser. in Math., vol. 54, Amer. Math. Soc., Providence, RI, 1984. MR 735435 (85j:14008)
[Ga]: F. Gaeta, Sul calculo effectivo della forma associata $F(W_{\alpha + \beta - n}^{g\ell })$ all'intersezione de duo cicle effetivi pure $U_\alpha ^g,\;V_\beta ^\ell$ di ${S_n}$ in funzione delle $F(U_\alpha ^g),\;F(V_\beta ^\ell )$ relative ai cicli secanti, Atti Accad. Naz. Lincei (8) 24 (1958), 269-276. MR 0100597 (20:7027)
[Ge]: I. M. Gelfand, Collected papers (S. G. Gindikin, V. W. Guillemin, A. A. Kirillov, B. Kostant, and S. Sternberg, eds.), Springer-Verlag, Berlin, 1987-1989.
[Gi]: H. Gillet, An introduction to higher dimensional Arakelov theory, Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 209-288. MR 902594 (88h:14005)
[G-S1]: H. Gillet and C. Soulé, Intersection theory using Adams operations, Invent. Math. 90 (1987), 243-277. MR 910201 (89d:14005)
[G-S2]: -, Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 94-174. MR 1087394 (92d:14016)
[G-S3]: -, Characteristic classes for algebraic vector bundles with Hermitian metric, I, II, Ann. of Math. (2) 131 (1990), 163-203, 205-238. MR 1038362 (91m:14032a)
[G-S4]: -, Amplitude arithmétique, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 887-890. MR 974432 (90a:14028)
[G-S5]: -, Un théorème de Riemann-Roch-Grothendieck arithmétique, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), 929-932. MR 1055224 (91m:14031)
[G-S6]: -, An arithmetic Riemann-Roch Theorem, Invent. Math. 110 (1992), 473-543. MR 1189489 (94f:14019)
[G-S7]: -, On the number of lattice points in convex symmetric bodies and their duals, Israel J. Math. 74 (1991), 347-357. MR 1135244 (92k:11069)
[Gr]: D. R. Grayson, Reduction theory using semistability, Comment. Math. Helv. 59 (1984), 600-634. MR 780079 (86h:22018)
[Gu]: W. Gubler, Höhentheorie, Dissertation, ETH, Zurich, 1992.
[H-W]: B. Harris and B. Wang, Archimedean height pairing of intersecting cycles, Internat. Math. Research Notices 4 (1993), 107-111. MR 1214701 (94g:14010)
[Ha-K]: R. Harvey and A. W. Knapp, Positive ( $p,\;p$ ) forms, Wirtinger's inequality, and currents, Value Distribution Theory (Proc. Tulane Univ. Program, 1972-73), Part A (R. O. Kujala and A. L. Vitter, eds.), Dekker, New York, 1974, pp. 43-62. MR 0355096 (50:7573)
[He]: S. Helgason, Groups and geometric analysis, Academic Press, New York, 1984. MR 754767 (86c:22017)
[Hi]: M. Hindry, Letter to J.-B. Bost, January 1990.
[K]: J. R. King, The currents defined by analytic varieties, Acta Math. 127 (1971), 186-220. MR 0393550 (52:14359)
[K-M]: F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves I, Preliminaries on det aud div, Math. Scand. 39 (1976), 19-55. MR 0437541 (55:10465)
[Kr]: J. Kramer, Néron-Tate height for cycles on Abelian varieties (d'après Faltings), preprint.
[Kro]: L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraischen Grössen, Crelle J. Reine Angew. Math. 92 (1881-1882), 1-122; Werke, Vol. II, Leipzig, Teubner, 1897, pp. 237-287.
[La]: S. Lang, Introduction to Arakelov theory, Springer-Verlag, Berlin, Heidelberg, and New York, 1988. MR 969124 (89m:11059)
[Lau]: M. Laurent, Hauteur de matrices d'interpolation, Approximations Diophantiennes et Nombres Transcendants (Luminy 1990) (P. Philippon, ed.), de Gruyter, Berlin, 1992, pp. 215-238. MR 1176532 (93i:11072)
[Le1]: P. Lelong,Plurisubharmonic functions and positive differential forms, Gordon and Breach, New York, 1969.
[Le2]: -, Mesure de Mahler des polynômes et majoration par convexité, C.R. Acad. Sci. Paris 315 (1992), 139-142. MR 1197225 (94a:32026)
[Lev]: H. Levine, A theorem on holomorphic mappings into complex projective space, Ann. of Math. (2) 71 (1960), 529-535. MR 0117757 (22:8531)
[MB1]: L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129 (1985).
[MB2]: -, Métriques permises, Séminaire sur les Pinceaux Arithmétiques; la Conjecture de Mordell, Astérisque 127 (1985), 29-87.
[MB3]: -, Hauteurs et classes de Chern sur les surfaces arithmétiques, Astérisque 183 (1990), 37-58. MR 1065154 (92g:14018a)
[Me]: F. Mertens, Über die bestimmenden Eigenschaften der Resultante von -Formen mit Veränderlichen, Sitzungsber. Wiener Akad. 93 (1886), 527-566.
[M-F]: D. Mumford and J. Fogarty, Geometric invariant theory, Second ed., Springer-Verlag, Berlin, 1982. MR 719371 (86a:14006)
[N1]: Yu. V. Nesterenko, Estimates for the orders of zeros of functions of a certain class and their applications in the theory of transcendental numbers, Izv. Akad. Nauk. USSR Ser. Mat. 41 (1977); English transl., Math. USSR-Izv. 11 (1977), 239-270. MR 0491535 (58:10774)
[N2]: -, Estimates for the characteristic function of a prime ideal, Mat. Sbornik 123 (1984); English transl., Math. USSR-Sb. 51 (1985), 9-32. MR 728927 (85e:11046)
[No]: D. G. Northcott, An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. 45 (1949), 502-509. MR 0033094 (11:390a)
[P1]: P. Philippon, Critères pour l'indépendance algébrique, Inst. Hautes Études Sci. Publ. Math. 64 (1986), 5-52. MR 876159 (88h:11048)
[P2]: -, Sur des hauteurs alternatives I, Math. Ann. 289 (1991), 255-283. MR 1092175 (92m:11061)
[P3]: -, Sur des hauteurs alternatives II, preprint, 1991. MR 1092175 (92m:11061)
[Sa]: P. Samuel, Méthodes d'algèbre abstraite en géométrie algébrique, Second ed., Springer-Verlag, Berlin, 1967. MR 0213347 (35:4211)
[Sch]: W. M. Schmidt, On heights of algebraic subspaces and diophantine approximations, Ann. of Math. (2) 85 (1967), 430-472. MR 0213301 (35:4165)
[Sh]: B. V. Shabat, Distribution values of holomorphic mappings, Transl. Math. Mono., vol. 61, Amer. Math. Soc., Providence, RI, 1985. MR 807367 (86k:32023)
[Si]: C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abhandlungen Preussischen Akad. Wiss. (1929), 41-69.
[So1]: C. Soulé, Théorie de Nevanlinna et théorie d'Arakelov, Astérisque 183 (1990), 127-135. MR 1065158 (91j:32028)
[So2]: -, Géométrie d'Arakelov et théorie des nombres transcendants, J. Arithmétiques de Luminy (17-21 juillet 1989), Astérisque 198-200 (1991), 355-371.
[So3]: -, Opérations en -théorie algébrique, Canad. J. Math. 27 (1985), 488-550.
[So-A-B-K]: C. Soulé, D. Abramovich, J.-F. Burnol, and J. Kramer, Lectures on Arakelov geometry, Cambridge Stud. Adv. Math., vol. 33, Cambridge Univ. Press, Cambridge, 1992. MR 1208731 (94e:14031)
[St1]: W. Stoll, The continuity of the fiber integral, Math. Z. 95 (1967), 87-138. MR 0243113 (39:4437)
[St2]: -, About the value distribution of holomorphic maps into projective space, Acta Math. 123 (1969), 83-114. MR 0259173 (41:3815)
[St3]: -, Value distribution of holomorphic maps into compact complex manifolds, Lecture Notes in Math., vol. 135, Springer-Verlag, Berlin, Heidelberg, and New York, 1970. MR 0267138 (42:2040)
[St4]: -, Aspects of value distribution theory in several complex variables, Bull. Ann. Math. Soc. 83 (1977), 166-183. MR 0427692 (55:723)
[Stu]: U. Stuhler, Eine Bemerkung zur Reduktionstheorie quadratischen Formen, Arch. Math. 27 (1976), 604-610. MR 0424707 (54:12666)
[S-V]: T. Struppeck and J. D. Vaaler, Inequalities for heights of algebraic subspaces and the Thue-Siegel principle, Analytic Number Theory (B. C. Berndt, H. G. Diamond, H. Halberstam, and A. Hildegrand, eds.), Progr. Math., vol. 85, Birkhäuser, Boston, 1983, pp. 493-527. MR 1084199 (92f:11092)
[Sz]: L. Szpiro, Degrés, intersections, hauteurs, Séminaire sur les Pinceaux Arithmétiques, la Conjecture de Mordell, Astérisque 127 (1985), 11-28. MR 801917
[V]: W. Vogel, Lectures on results on Bézout's theorem, Tata Inst. Fund. Res., Bombay, 1984. MR 743265 (86f:14003)
[Vo]: P. Vojta, Applications of arithmetic algebraic geometry to diophantine approximations, CIME Proceedings Trento, 1991 (to appear). MR 1338861 (96c:11067)
[Wa]: B. L. van der Waerden, Moderne algebra II, zweite verbesserte Auflage, Springer, Berlin, 1940. MR 0002841 (2:120b)
[We1]: A. Weil, Number-theory and algebraic geometry, Vol. II, Proc. Internat. Math. Congress, Cambridge, MA, 1950, pp. 90-100. MR 0045416 (13:579d)
[We2]: -, Arithmetic on algebraic varieties, Ann. of Math. (2) 53 (1951), 412-444. MR 0042169 (13:66d)
[Wu]: G. Wüstholz, Uber das abelsche Analogen des Lindemannschen Satzes, Invent. Math. 72 (1983), 363-388. MR 704397 (85i:11064)
[Zh1]: S. Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), 569-587. MR 1189866 (93j:14024)
[Zh2]: -, Positive line bundles on arithmetic varieties, preprint, February 1992.

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