Arakelov geometry (Q2922359)

Arakelov geometry is an approach to the study of Diophantine problems from the viewpoint of algebraic geometry. This subject has its origin in the classical problem of developing an appropriate intersection theory on a scheme \(X\) defined over the ring of integers of an algebraic number field \(k\), thereby generalizing the well-known intersection theory for smooth, complete varieties over the ground field \(\mathbb{C}\) of complex numbers. \textit{S. J. Arakelov} solved this long-standing problem in the case of arithmetic surfaces in his pioneering paper [Math. USSR, Izv. 8(1974), 1167--1180 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179--1192 (1974; Zbl 0355.14002)], basically by a suitable compactification of that surface, on the one hand, and by endowing all arithmetic line bundles (divisors) on that compactified surface (over \(\mathbb{C}\)) with certain Hermitian metrics on the other. In the sequel, Arakelov's work had been extended by \textit{G. Faltings} [Ann. Math. (2) 119, 387--424 (1984; Zbl 0559.14005)], \textit{P. Deligne} [Le déterminant de la cohomologie. Contemp. Math. 67, 93--117 (1987; Zbl 0629.14008)], and others, while \textit{H. Gillet} and \textit{C. Soulé} succeeded in establishing both an arithmetic intersection theory of general arithmetic varieties and a Hermitian \(K\)-theory for them in their seminal work [Publ. Math., Inst. Hautes Étud. Sci. 72, 93--174 (1990; Zbl 0741.14012)]. Many of these developments were explained in the books on Arakelov geometry by \textit{S. Lang} [Introduction to Arakelov theory. New York etc.: Springer-Verlag. x, 187 p. (1988; Zbl 0667.14001)], \textit{G. Faltings} [Lectures on the arithmetic Riemann-Roch theorem. Annals of Mathematics Studies. 127. Princeton, NJ: Princeton University Press, x, 100 p. (1992; Zbl 0744.14016)] and \textit{C. Soulé} et al. [Lectures on Arakelov geometry. Cambridge: Cambridge University Press. 177 p. (1992; Zbl 0812.14015)], respectively.NEWLINENEWLINE In the meantime, further crucial progress in general Arakelov geometry and its applications has been achieved, in particular with a view toward the birational aspects, that is, the study of the asymptotic behavior of powers of arithmetic line bundles, the continuity of arithmetic volumes, and arithmetic analogue of the so-called Bogomolov inequality, a general Hodge index theorem for arithmetic varieties, and other related topics.NEWLINENEWLINE The book under review is an English translation of the author's original Japanese monograph [Arakerofu kika. Iwanami Shoten, Publishers, Tokyo (2008; Zbl 1304.14002)]. Its main goal is to provide a modern, comprehensive and largely self-contained introduction to Arakelov geometry, with the focus on presenting the recent developments is birational Arakelov geometry as characterized above. As for the prerequisites, the reader is assumed to be familiar with the basics of algebraic geometry and the language of algebraic schemes.NEWLINENEWLINE With regard to the precise contents, the text consists of nine chapters, each of which is divided into several sections.NEWLINENEWLINE Chapter 1 is devoted to preliminaries for all the following chapters, i.e., expositions of elementary results from several areas in mathematics are given with full proofs. This includes topics such as normed vector spaces, the length of modules, principal divisors and Weil's reciprocity law, determinant bundles, Hodge theory on complex manifolds, the Poincaré-Lelong formula for Hermitian inversible sheaves on smooth, quasi-projective complex varieties, and many more.NEWLINENEWLINE Chapter 2 discusses several topics concerning the geometry of numbers, including Minkowski's convex body theorem, Mahler's inequality, the Brunn-Minkowski theorem, and an estimate of the number of points in a convex lattice to \textit{H. Gillet} and \textit{C. Soulé} [Isr. J. Math. 74, No. 2--3, 347--357 (1991); erratum ibid. 171, 443--444 (2009; Zbl 0752.52008)].NEWLINENEWLINE Chapter 3 develops Arakelov geometry on arithmetic curves, which serves as a first introduction to the subject at all. Everything is done for a reduced order as ground ring, and the reader meets here arithmetic Chow groups, Hermitian modules, the arithmetic Riemann-Roch formula for arithmetic curves, effective estimates of the number of small sections of Hermitian modules, arithmetic degree formulae, and ample invertible sheaves on arithmetic curves.NEWLINENEWLINE Chapter 4 gives a very detailed presentation of the basic theory of Arakelov geometry on arithmetic surfaces. Starting with the Deligne intersection pairing for invertible sheaves, the author introduces Green functions on Riemann surfaces, arithmetic Chow groups, the arithmetic intersection theory on arithmetic surfaces, the Arakelov metric on dualizing sheaves, determinant bundles for curves, Falting's Riemann-Roch theorem for arithmetic surfaces, and the Faltings metric on an admissible Hermitian sheaf over a compact Riemann surface. This chapter already points to the development of Arakelov geometry for general arithmetic varieties in an exemplary way, however without proofs at this stage.NEWLINENEWLINE Chapter 5 then turns to the detailed description of higher-dimensional Arakelov theory. The focus is on concrete problems, and therefore the author develops the intersection theory only for special arithmetic cycles. More precisely, the topics of this more general chapter are: intersection theory of Cartier divisors on excellent schemes, the higher-dimensional generalization of Weil's reciprocity law in complex geometry, the intersection theory on general arithmetic varieties, arithmetic characteristic classes, the statement of the arithmetic Riemann-Roch theorem in the general case (after Gillet-Soulé), a multi-index version of Gromov's inequality, the arithmetic Hilbert-Samuel formula, positive line bundles, and some technical results used in the later sections.NEWLINENEWLINE Chapter 6 presents some of the author's own recent results on the continuity of volumes on arithmetic varieties as published in his paper [J. Algebr. Geom. 18, No. 3, 407--457 (2009; Zbl 1167.14014)]. In this context, the generalized Hodge index theorem for projective arithmetic varieties is deduced as an application of the continuity of volumes.NEWLINENEWLINE Chapter 7 is devoted to an analogue of the Nakai-Moishezon criterion of ampleness in classical projective algebraic geometry, this time with a view toward certain Hermitian invertible sheaves on a generically smooth, projective arithmetic variety. Combining this criterion with the generalized Hodge index theorem in Chapter 6, the Hilbert-Samuel formula for a nef \(C^\infty\)-Hermitian invertible sheaf on an arithmetic variety is derived as a consequence. As for the proofs, several earlier results of \textit{S. Zhang} on positive line bundles on arithmetic varieties [J. Am. Math. Soc. 8, No. 1, 187--221 (1995; Zbl 0861.14018)] are used. Another important result in algebraic geometry, namely the so-called Bogomolov inequality for vector bundles on a smooth projective surface, is proved for arithmetic projective surfaces in Chapter 8, following the author's approach published in [Duke Math. J. 74, No. 3, 713--761 (1994; Zbl 0854.14012)]. As for the ingredients of the proof, the generalized Hodge index theorem as well as existence of Hermite-Einstein metrics on stable complex vector bundles play a crucial role.NEWLINENEWLINE The book concludes with Chapter 9, in which a proof of the so-called Lang-Bogomolov conjecture is presented. The latter is a generalization of the formous Lang conjecture on abelian varieties over number fields, which was verified by \textit{G. Faltings} [Ann. Math. (2) 133, No. 3, 549--576 (1991; Zbl 0734.14007)]. More precisely, the Lang-Bogomolov conjecture is a mixed version of Lang's conjecture and a related conjecture by F. Bogomolov concerning symmetric, ample invertible sheaves on an abelian variety over an algebraic number field.NEWLINENEWLINE Bogomolov's conjecture was independently proved by \textit{E. Ullmo} [Ann. Math. (2) 147, No. 1, 167--179 (1998; Zbl 0934.14013)] and by \textit{S. Zhang} [Ann. Math. (2) 147, No. 1, 159--165 (1998; Zbl 0991.11034)], and the combined Lang-Bogomolov conjecture was affirmatively answerd by \textit{B. Poonen} [Invent. Math. 137, No. 2, 413--425 (1999; Zbl 0995.11040)], \textit{S. Zhang} [Duke Math. J. 103, No. 1, 39--46 (2000; Zbl 0972.11053)], and the author [Duke Math. J. 107, No. 1, 85--102 (2001; Zbl 1009.11039)]. The proofs given in this chapter are based on the author's approach, with suitable introduction to height functions, the \textit{L. Szpiro} et al. equidistribution theorem on small points [Invent. Math. 127, No. 2, 337--347 (1997; Zbl 0991.11035)], and cubic metrics on complex abelian varieties.NEWLINENEWLINE Compared to the earlier books on Arakelov geometry, the current monograph is much more up-to-date, detailed, comprehensive and self-contained. The exposition stands out of its high degree of clarity, completeness, rigour and topicality, which also makes the volume an excellent textbook on the subject for seasoned graduate students and young researchers in arithmetic algebraic geometry. The rich bibliography of seventy-eight references certainly serves as a useful guide to further reading with regard to the more recent research literature in the field.NEWLINENEWLINE Perhaps it should be mentioned that the present book may be seen as a largely elaborated version of the previous text [the author et al., Introduction to Arakelov geometry. River Edge, NJ: World Scientific. 1--74 (2002; Zbl 1051.14028)], which actually is a short version of Chapter 9.