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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.

MSC:

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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