Theory of heights. With an appendix by Jürg Kramer: An alternative foundation of the Néron-Tate height. (Höhentheorie (mit einem Appendix von Jürg Kramer: Eine alternative Begründung der Néron-Tate Höhe).) (German) Zbl 0792.14012

Using arithmetic intersection theory, G. Faltings [Ann. Math., II. Ser. 133, No. 3, 549-576 (1991; Zbl 0734.14007)] defined a height of cycles in multiprojective space over a number field \(K\). Let \(X\) be a complete scheme over \(K\) and \({\mathcal M}_ 0,\dots, {\mathcal M}_ t\) isomorphism classes of basepoint-free line bundles on \(X\). There is a multiprojective realization of \({\mathcal M}_ 0,\dots, {\mathcal M}_ t\). Using pushforward of cycles, one gets a height of any \(t\)-dimensional effective cycle \(Z\) of \(X\). The main theorem of the paper states that the height does not depend on the choice of the realization up to \(O(\sum d_ i)\), where \(i\) ranges over \(0,\dots,t\) and \(d_ i(Z)\) is the degree of \(Z\) relative to \({\mathcal M}_ 0,\dots, {\mathcal M}_{i-1}\), \({\mathcal M}_{i+1},\dots, {\mathcal M}_ t\). The proof is based on a formula which describes the difference of the heights corresponding to different realizations defined over the ring of integers. As in the classical case of points, one gets the height machine and Néron-Tate heights. For even \({\mathcal M}_ 0= \cdots= {\mathcal M}_ t\), the latter are the same as the canonical heights of P. Philippon [Math. Ann. 289, No. 2, 255-283 (1991; Zbl 0704.14017)].
In an appendix a different approach to the above mentioned Néron-Tate height for cycles on abelian varieties (over \(\mathbb{Q})\) is given using arithmetical compactifications.


14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14K15 Arithmetic ground fields for abelian varieties
14C25 Algebraic cycles
11G99 Arithmetic algebraic geometry (Diophantine geometry)
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[1] Bost, J.-B., Gillet, H., Soul?, C.: Un analogue arithm?tique du th?or?me de B?zout. C. R. Acad. Sci. Paris, S?r. I,312, 845-848 (1991) · Zbl 0756.14012
[2] Faltings, G.: Diophantine approximation on abelian varieties. Ann. Math.133, 549-576 (1991) · Zbl 0734.14007
[3] Faltings, G., W?stholz, G. et al.: Rational points, 3. Auflage, Braunschweig: Vieweg 1992
[4] Fulton, W.: Intersection theory. Berlin Heidelberg New York: Springer 1984 · Zbl 0541.14005
[5] Gillet, H., Soul?, C.: Characteristic classes for algebraic vector bundles with hermitian metric, I and II. Ann. Math.131, 163-203 (1990) · Zbl 0715.14018
[6] Gillet, H., Soul?, C.: Intersection on arithmetic varieties. Publ. Math. Inst. Hautes Etud. Sci.72, 94-174 (1990) · Zbl 0741.14012
[7] Griffiths, P., Harris, J.: Principles of algebraic geometry, New York: Wiley 1978 · Zbl 0408.14001
[8] Grothendieck, A., Dieudonn?, J.: El?ments de g?ometrie alg?brique I. Grundl. Math. Wiss., vol. 166. Berlin Heidelberg New York: Springer 1971
[9] Hartshorne, R.: Algebraic geometry. Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[10] Hindry, M.: Autour d’une conjecture de Serge Lang. Invent. Math.94, 575-603 (1988) · Zbl 0638.14026
[11] Kramer, J.: Eine alternative Begr?ndung der N?ron-Tate H?he. Appendix zur vorliegenden Arbeit.
[12] Lang, S.: Fundamentals of diophantine geometry. Berlin Heidelberg New York: Springer 1983 · Zbl 0528.14013
[13] Milne, J.S.: Abelian varieties in: Arithmetic geometry. G. Cornell, J. Silverman (eds.), pp. 103-150 Berlin Heidelberg New York: Springer 1986
[14] N?ron, A.: Quasi-fonctions et hauteurs sur les vari?t?s ab?liennes. Ann. Math.82, 249-331 (1965) · Zbl 0163.15205
[15] Northcott, D.G.: An inequality in the theory of arithmetic on algebraic varieties. Proc. Camb. Philos. Soc.45, 502-518 (1949) · Zbl 0035.30701
[16] Philippon, P.: Crit?res pour l’ind?pandance alg?brique. Publ. Math. Inst. Ha?tes Etud. sci.64, 5-52 (1986) · Zbl 0615.10044
[17] Philippon, P.: Sur des hauteurs alternatives. I. Math. Ann.289, 255-283 (1991) · Zbl 0726.14017
[18] Stoll, W.: About the value distribution of holomorphic maps into projective space. Acta Math.123, 83-114 (1969) · Zbl 0177.11302
[19] Soul?, C.: G?om?trie d’Arakelov et nombres transcendents. Preprint
[20] Soul?, C., Abramovich, D., Burnol, J.-F., Kramer, J.: Lectures on Arakelov geometry. Monographie. Cambridge: Cambridge University Press 1992 · Zbl 0812.14015
[21] Weil, A.: Arithmetic on algebraic varieties. Ann. Math.53, 412-444 (1951) · Zbl 0043.27002
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