Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables. (Arakelov geometry of toric varieties and integrable line bundles). (French) Zbl 0963.14009

Maillot, Vincent

Mém. Soc. Math. Fr., Nouv. Sér. 80, 129 p. (2000).

The main object of the monograph under review is an arithmetic variety \(X\), that is a scheme over \(S= \text{Spec}(O_K)\) (\(K\) is a number field) which is assumed flat, projective, integral, and regular. One of general goals is to extend the intersection theory of H. Gillet and C. Soulé [Publ. Math., Inst. Hautes Étud. Sci. Publ. Math. 72, 93-174 (1990; Zbl 0741.14012)] so that in the Arakelov geometry one could consider linear bundles equipped with a metric which is not necessarily \(C^{\infty }\). To be more precise, following S. Zhang [J. Algebr. Geom. 4, No. 2, 281-300 (1995; Zbl 0861.14019)], the author introduces the notion of an admissible line bundle (generated by its global sections and equipped with a positive metric which can be uniformly approximated by positive \(C^{\infty }\) metrics), and, more generally, of an integrable line bundle (the difference of two admissible line bundles). Further, the author introduces a new notion of a generalized differential form, and for an integrable line bundle defines the first arithmetic Chern class which turns out to be a generalized differential form in the sense of this new definition. The author defines a product of such forms using a product theory for positive currents developed by E. Bedford and B. A. Taylor [Acta Math. 149, 1-40 (1982; Zbl 0547.32012)] and J.-P. Demailly [in: Complex Analysis and Geometry, Univ. Ser. Math. 115-193 (1993; Zbl 0792.32006)]. As a result, the author constructs a generalized Chow ring containing the usual one, with pairing degree functions continuing those of Gillet–Soulé and with height function compatible with that of Zhang.
The author’s next goal consists in applying the developed theory to a particular class of arithmetic varieties, namely, to smooth projective toric varieties which are among favorite proving grounds in many problems of algebraic geometry. First of all recall that the Chow ring of such a variety is classically known to be generated by the first Chern classes of linear bundles. As to the arithmetic Chow ring, one immediately encounters a problem: Any “canonical” (i.e. functorial in \(L\)) metric on a linear bundle \(L\) is, in general, not \(C^{\infty }\). This is in fact one of motivations for developing a general intersection theory described above.
Thus, according to this general approach, given a toric variety over the integers equipped with a line bundle \(\overline L\) and its canonical metric, the author defines a current \(c_1(\overline L)\) replacing the usual first Chern class. Moreover, an explicit formula for the generalized product of such currents is given, and a vanishing theorem is proved. These results lead not only to a correct arithmetic analogue of the above mentioned classical result on generating the Chow ring by the first Chern classes, but also to a simple algorithm for computing the mixed volume of convex polytopes.
Furthermore, a more detailed study of heights on arithmetic toric varieties yields some more interesting consequences, namely a relationship between the canonical height of a hypersurface in a toric variety and the Mahler measure of the corresponding polynomial, and an arithmetic analogue of the Bernstein-Koushnirenko theorem on estimating the number of common zeroes of Laurent polynomials in terms of the mixed volumes of Newton polytopes.

Reviewer: B.Kunyavskii (Ramat Gan)

MSC:

14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
11D75	Diophantine inequalities
11G50	Heights

Keywords:

Arakelov geometry; toric variety; arithmetic variety; admissible line bundle; integrable line bundle; arithmetic Chern class; generalized Chow ring; intersection theory; current; canonical height; Bernstein-Koushnirenko theorem; Newton polytopes

Citations:

Zbl 0861.14019; Zbl 0547.32012; Zbl 0792.32006; Zbl 0741.14012

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