Heights of projective varieties and positive Green forms

Bost, J.-B.; Gillet, H.; Soulé, C.

doi:10.1090/S0894-0347-1994-1260106-X

Heights of projective varieties and positive Green forms
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by J.-B. Bost, H. Gillet and C. Soulé

J. Amer. Math. Soc. 7 (1994), 903-1027

DOI: https://doi.org/10.1090/S0894-0347-1994-1260106-X

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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

Allen B. Altman and Steven L. Kleiman, Joins of schemes, linear projections, Compositio Math. 31 (1975), no. 3, 309–343. MR 396560
S. Ju. Arakelov, An intersection theory for divisors on an arithmetic surface, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1179–1192 (Russian). MR 0472815
S. J. Arakelov, Theory of intersections on the arithmetic surface, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 405–408. MR 0466150
Bernard Beauzamy, Enrico Bombieri, Per Enflo, and Hugh L. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (1990), no. 2, 219–245. MR 1072467, DOI 10.1016/0022-314X(90)90075-3
Jean-Michel Bismut and Éric Vasserot, The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), no. 2, 355–367. MR 1016875
André Blanchard, Sur les variétés analytiques complexes, Ann. Sci. École Norm. Sup. (3) 73 (1956), 157–202 (French). MR 0087184
Eduard Bod’a and Wolfgang Vogel, On system of parameters, local intersection multiplicity and Bezout’s theorem, Proc. Amer. Math. Soc. 78 (1980), no. 1, 1–7. MR 548071, DOI 10.1090/S0002-9939-1980-0548071-3
E. Bombieri and J. Vaaler, On Siegel’s lemma, Invent. Math. 73 (1983), no. 1, 11–32. MR 707346, DOI 10.1007/BF01393823
A. Borel and R. Remmert, Über kompakte homogene Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1961/62), 429–439 (German). MR 145557, DOI 10.1007/BF01471087
Jean-Benoît Bost, Green’s currents and height pairing on complex tori, Duke Math. J. 61 (1990), no. 3, 899–912. MR 1084464, DOI 10.1215/S0012-7094-90-06134-4

Théorie de l’intersection et théorème de Riemann-Roch arithmétiques

201-203

Jean-Benoît Bost, Henri Gillet, and Christophe Soulé, Un analogue arithmétique du théorème de Bézout, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 11, 845–848 (French, with English summary). MR 1108504
Raoul Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965), 71–112. MR 185607, DOI 10.1007/BF02391818
Wei-Liang Chow and B. L. van der Waerden, Zur algebraischen Geometrie. IX, Math. Ann. 113 (1937), no. 1, 692–704 (German). MR 1513117, DOI 10.1007/BF01571660
Maurizio Cornalba and Phillip Griffiths, Analytic cycles and vector bundles on non-compact algebraic varieties, Invent. Math. 28 (1975), 1–106. MR 367263, DOI 10.1007/BF01389905
Gerd Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387–424. MR 740897, DOI 10.2307/2007043
Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549–576. MR 1109353, DOI 10.2307/2944319
Gerd Faltings, Lectures on the arithmetic Riemann-Roch theorem, Annals of Mathematics Studies, vol. 127, Princeton University Press, Princeton, NJ, 1992. Notes taken by Shouwu Zhang. MR 1158661, DOI 10.1515/9781400882472
Gerd Faltings and Gisbert Wüstholz (eds.), Rational points, Aspects of Mathematics, E6, Friedr. Vieweg & Sohn, Braunschweig; distributed by Heyden & Son, Inc., Philadelphia, PA, 1984. Papers from the seminar held at the Max-Planck-Institut für Mathematik, Bonn, 1983/1984. MR 766568, DOI 10.1007/978-3-322-83918-3
John Fogarty, Truncated Hilbert functors, J. Reine Angew. Math. 234 (1969), 65–88. MR 244268, DOI 10.1515/crll.1969.234.65
William Fulton, Rational equivalence on singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 147–167. MR 404257

Intersection theory

William Fulton, Introduction to intersection theory in algebraic geometry, CBMS Regional Conference Series in Mathematics, vol. 54, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1984. MR 735435, DOI 10.1090/cbms/054
Federico Gaeta, Sul calcolo effettivo della forma associata $F(W_{\alpha +\beta -n}{}^{gl})$ all’intersezione di due cicle effettivi puri $U_{\alpha }{}^{g}$, $V_{\beta }{}^{l}$ di $S_{n}$, in funzione delle $F(U_{\alpha }{}^{g})$, $F(V_{\beta }{}^{l})$ relative ai cicli secanti. I, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 24 (1958), 269–276 (Italian). MR 100597

Collected papers

Henri Gillet, An introduction to higher-dimensional Arakelov theory, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 209–228. MR 902594, DOI 10.1090/conm/067/902594
H. Gillet and C. Soulé, Intersection theory using Adams operations, Invent. Math. 90 (1987), no. 2, 243–277. MR 910201, DOI 10.1007/BF01388705
Henri Gillet and Christophe Soulé, Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 93–174 (1991). MR 1087394
Henri Gillet and Christophe Soulé, Characteristic classes for algebraic vector bundles with Hermitian metric. I, Ann. of Math. (2) 131 (1990), no. 1, 163–203. MR 1038362, DOI 10.2307/1971512
Henri Gillet and Christophe Soulé, Amplitude arithmétique, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 17, 887–890 (French, with English summary). MR 974432
Henri Gillet and Christophe Soulé, Un théorème de Riemann-Roch-Grothendieck arithmétique, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 17, 929–932 (French, with English summary). MR 1055224
Henri Gillet and Christophe Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), no. 3, 473–543. MR 1189489, DOI 10.1007/BF01231343
H. Gillet and C. Soulé, On the number of lattice points in convex symmetric bodies and their duals, Israel J. Math. 74 (1991), no. 2-3, 347–357. MR 1135244, DOI 10.1007/BF02775796
Daniel R. Grayson, Reduction theory using semistability, Comment. Math. Helv. 59 (1984), no. 4, 600–634. MR 780079, DOI 10.1007/BF02566369

Höhentheorie

B. Wang and B. Harris, Archimedean height pairing of intersecting cycles, Internat. Math. Res. Notices 4 (1993), 107–111. MR 1214701, DOI 10.1155/S1073792893000108
Reese Harvey and A. W. Knapp, Positive $(p,\,p)$ forms, Wirtinger’s inequality, and currents, Value distribution theory (Proc. Tulane Univ. Program, Tulane Univ., New Orleans, La., 1972-1973) Dekker, New York, 1974, pp. 43–62. MR 0355096
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
James R. King, The currents defined by analytic varieties, Acta Math. 127 (1971), no. 3-4, 185–220. MR 393550, DOI 10.1007/BF02392053
Finn Faye Knudsen and David Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand. 39 (1976), no. 1, 19–55. MR 437541, DOI 10.7146/math.scand.a-11642

Néron-Tate height for cycles on Abelian varieties (d’après Faltings)

Grundzüge einer arithmetischen Theorie der algebraischen Grössen

92

Werke

Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. MR 969124, DOI 10.1007/978-1-4612-1031-3
Michel Laurent, Hauteur de matrices d’interpolation, Approximations diophantiennes et nombres transcendants (Luminy, 1990) de Gruyter, Berlin, 1992, pp. 215–238 (French, with English summary). MR 1176532

Plurisubharmonic functions and positive differential forms

Pierre Lelong, Mesure de Mahler des polynômes et majoration par convexité, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 2, 139–142 (French, with English and French summaries). MR 1197225
Harold I. Levine, A theorem on holomorphic mappings into complex projective space, Ann. of Math. (2) 71 (1960), 529–535. MR 117757, DOI 10.2307/1969942

Pinceaux de variétés abéliennes

129

Métriques permises

127

Laurent Moret-Bailly, Hauteurs et classes de Chern sur les surfaces arithmétiques, Astérisque 183 (1990), 37–58 (French). Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). MR 1065154

Über die bestimmenden Eigenschaften der Resultante von

-Formen mit

Veränderlichen

93

David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. MR 719371, DOI 10.1007/978-3-642-96676-7
Ju. V. Nesterenko, Estimate of the orders of the zeroes of functions of a certain class, and their application in the theory of transcendental numbers, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 253–284, 477 (Russian). MR 0491535
Yu. V. Nesterenko, Estimates for the characteristic function of a prime ideal, Mat. Sb. (N.S.) 123(165) (1984), no. 1, 11–34 (Russian). MR 728927
D. G. Northcott, An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. 45 (1949), 502–509. MR 33094, DOI 10.1017/s0305004100025202
Patrice Philippon, Critères pour l’indépendance algébrique, Inst. Hautes Études Sci. Publ. Math. 64 (1986), 5–52 (French). MR 876159
Patrice Philippon, Sur des hauteurs alternatives. I, Math. Ann. 289 (1991), no. 2, 255–283 (French). MR 1092175, DOI 10.1007/BF01446571
Patrice Philippon, Sur des hauteurs alternatives. I, Math. Ann. 289 (1991), no. 2, 255–283 (French). MR 1092175, DOI 10.1007/BF01446571
Pierre Samuel, Méthodes d’algèbre abstraite en géométrie algébrique, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 4, Springer-Verlag, Berlin-New York, 1967 (French). Seconde édition, corrigée. MR 0213347
Wolfgang M. Schmidt, On heights of algebraic subspaces and diophantine approximations, Ann. of Math. (2) 85 (1967), 430–472. MR 213301, DOI 10.2307/1970352
B. V. Shabat, Distribution of values of holomorphic mappings, Translations of Mathematical Monographs, vol. 61, American Mathematical Society, Providence, RI, 1985. Translated from the Russian by J. R. King; Translation edited by Lev J. Leifman. MR 807367, DOI 10.1090/mmono/061

Über einige Anwendungen diophantischer Approximationen

C. Soulé, Théorie de Nevanlinna et théorie d’Arakelov, Astérisque 183 (1990), 127–135 (French). Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). MR 1065158

Géométrie d’Arakelov et théorie des nombres transcendants

198-200

Opérations en

-théorie algébrique

27

C. Soulé, Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge University Press, Cambridge, 1992. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer. MR 1208731, DOI 10.1017/CBO9780511623950
Wilhelm Stoll, The continuity of the fiber integral, Math. Z. 95 (1967), 87–138. MR 243113, DOI 10.1007/BF01111553
Wilhelm Stoll, About the value distribution of holomorphic maps into the projective space, Acta Math. 123 (1969), 83–114. MR 259173, DOI 10.1007/BF02392386
Wilhelm Stoll, Value distribution of holomorphic maps into compact complex manifolds. , Lecture Notes in Mathematics, Vol. 135, Springer-Verlag, Berlin-New York, 1970. MR 0267138
Wilhelm Stoll, Aspects of value distribution theory in several complex variables, Bull. Amer. Math. Soc. 83 (1977), no. 2, 166–183. MR 427692, DOI 10.1090/S0002-9904-1977-14247-X
Ulrich Stuhler, Eine Bemerkung zur Reduktionstheorie quadratischer Formen, Arch. Math. (Basel) 27 (1976), no. 6, 604–610 (German). MR 424707, DOI 10.1007/BF01224726
Thomas Struppeck and Jeffrey D. Vaaler, Inequalities for heights of algebraic subspaces and the Thue-Siegel principle, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 493–528. MR 1084199
Lucien Szpiro, Degrés, intersections, hauteurs, Astérisque 127 (1985), 11–28 (French). Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). MR 801917
W. Vogel, Lectures on results on Bezout’s theorem, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 74, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1984. Notes by D. P. Patil. MR 743265, DOI 10.1007/978-3-662-00493-7
Paul Vojta, Applications of arithmetic algebraic geometry to Diophantine approximations, Arithmetic algebraic geometry (Trento, 1991) Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, pp. 164–208. MR 1338861, DOI 10.1007/BFb0084730
Bartel Leendert van der Waerden, Moderne Algebra, J. Springer, Berlin, 1940 (German). MR 0002841
André Weil, Number-theory and algebraic geometry, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R.I., 1952, pp. 90–100. MR 0045416
André Weil, Arithmetic on algebraic varieties, Ann. of Math. (2) 53 (1951), 412–444. MR 42169, DOI 10.2307/1969564
G. Wüstholz, Über das Abelsche Analogon des Lindemannschen Satzes. I, Invent. Math. 72 (1983), no. 3, 363–388 (German). MR 704397, DOI 10.1007/BF01398393
Shouwu Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), no. 3, 569–587. MR 1189866, DOI 10.2307/2946601

Positive line bundles on arithmetic varieties