##
**Introduction to Arakelov theory. With an appendix by Paul Vojta: Diophantine inequalities and Arakelov theory.**
*(English)*
Zbl 0667.14001

New York etc.: Springer-Verlag. x, 187 p. DM 98.00 (1988).

Let \(R\) be a Dedekind domain, and let \(\pi: X\to S=\mathrm{Spec}(R)\) be an arithmetic surface; that is, \(X\) is a regular, integral scheme, proper over \(S\), whose generic fiber is a smooth curve. For any closed point \(x\in X\) and any divisors \(D_1\), \(D_2\) on \(X\) with no common components, one can define a local intersection index \(<D_1,D_2>_x\) in the usual way as the length of \({\mathcal O}_x/(f_1,f_2)\), where the \(f_i\)’s are local equations for the \(D_i\)’s. Summing these local intersection indices leads to a satisfactory global intersection theory for divisors without common components [I. R. Shafarevich, “Lectures on minimal models and birational transformations of two dimensional schemes”, Tata Inst. Fundament. Res. Lect. Math. Phys., Math. 37 (1966; Zbl 0164.51704); Am. Math. Soc., Transl., II. Ser. 31, 25–39 (1963; Zbl 0133.29303); translation from Proc. Int. Congr. Math. 1962, 163–176 (1963; Zbl 0126.06902) and S. Lichtenbaum, Am. J. Math. 90, 380–405 (1968; Zbl 0194.22101)].

In the classical theory of complex projective surfaces, most of the major theorems rely (both for their statement and their proof) on the fact that the global intersection pairing depends only on the linear equivalence class of a divisor. The underlying reason for this is that the surface is complete. An arithmetic surface is not complete, since its base \(S=\mathrm{Spec}(R)\) is the arithmetic analogue of an affine curve. One can complete \(S\) by adding an extra point for each archimedean absolute value on \(R\). Then for each such infinite point \(v\), one adds a fiber to \(X\to S\) by taking the Riemann surface \(X_v=X\times_S \mathrm{Spec}(K_v)\). (Here \(K\) is the quotient field of \(R\), and \(K_v\) its completion.)

In 1974 Arakelov suggested how one could define an intersection index \(<D_1,D_2>_x\) for points \(x\) on the fibers “at infinity”, and he showed that his extended intersection pairing was invariant under linear equivalence. He then proved (under some hypotheses) an arithmetic analogue of the classical adjunction formula. More recently, G. Faltings [Ann. Math. (2) 119, 387–424 (1984; Zbl 0559.14005)] proved an arithmetic analogue of the Riemann-Roch theorem, and Faltings (loc. cit.) and P. Hriljac [Am. J. Math. 107, 23–38 (1985; Zbl 0593.14004)] (independently) did the same for the Hodge index theorem.

In this concise introduction to Arakelov’s intersection theory, the author covers all of the above material and gives a few applications. He begins with the theory of metrics on line bundles, and goes on to the construction of the Green’s functions which are needed to define Arakelov’s intersection on the fibers “at infinity”. Next he defines and proves the principal properties of the local intersection pairing on the finite fibers, as did Lichtenbaum (loc. cit.) and Shafarevich (loc. cit.).

The heart of the book starts in chapter IV, with the definition of the Arakelov intersection pairing, proof of invariance under linear equivalence, the Hodge index theorem, construction of the metrized canonical sheaf, and the proof of the arithmetic adjunction formula. This is followed in chapter V with a proof of Faltings’ arithmetic Riemann-Roch theorem, which expresses the (co)volume of certain cohomology groups in terms of intersection numbers. The technical material needed to construct the Faltings volume forms on cohomology is relegated to the final chapter.

Finally, a brief appendix by Paul Vojta describes Parshin’s (conjectural) arithmetic analogue of the famous \(c^2_1\leq 3c_2\) inequality and relates Parshin’s question to some of his own conjectures.

The author has written an excellent book in a new, exciting, and very active area of research. It will undoubtedly become a standard reference in the field of arithmetic geometry, since it brings together in a coherent fashion the basic material which had previously only been available in the original journal articles. However, a potential reader should be warned that this is not a textbook for beginners; it supposes the reader has a solid background in algebraic number theory and in algebraic geometry [both algebraic as in the R. Hartshorne’s book “Algebraic geometry” (1977; Zbl 0367.14001; 3rd edition 1983) and complex as in the one by P. Griffiths and J. Harris: “Principles of algebraic geometry” (1978; Zbl 0408.14001)]. It also helps to be familiar with the author’s book: “Fundamentals of diophantine geometry” (1983; Zbl 0528.14013). But for those with the necessary background, the book under review provides a welcome entree to recent advances in arithmetic geometry.

In the classical theory of complex projective surfaces, most of the major theorems rely (both for their statement and their proof) on the fact that the global intersection pairing depends only on the linear equivalence class of a divisor. The underlying reason for this is that the surface is complete. An arithmetic surface is not complete, since its base \(S=\mathrm{Spec}(R)\) is the arithmetic analogue of an affine curve. One can complete \(S\) by adding an extra point for each archimedean absolute value on \(R\). Then for each such infinite point \(v\), one adds a fiber to \(X\to S\) by taking the Riemann surface \(X_v=X\times_S \mathrm{Spec}(K_v)\). (Here \(K\) is the quotient field of \(R\), and \(K_v\) its completion.)

In 1974 Arakelov suggested how one could define an intersection index \(<D_1,D_2>_x\) for points \(x\) on the fibers “at infinity”, and he showed that his extended intersection pairing was invariant under linear equivalence. He then proved (under some hypotheses) an arithmetic analogue of the classical adjunction formula. More recently, G. Faltings [Ann. Math. (2) 119, 387–424 (1984; Zbl 0559.14005)] proved an arithmetic analogue of the Riemann-Roch theorem, and Faltings (loc. cit.) and P. Hriljac [Am. J. Math. 107, 23–38 (1985; Zbl 0593.14004)] (independently) did the same for the Hodge index theorem.

In this concise introduction to Arakelov’s intersection theory, the author covers all of the above material and gives a few applications. He begins with the theory of metrics on line bundles, and goes on to the construction of the Green’s functions which are needed to define Arakelov’s intersection on the fibers “at infinity”. Next he defines and proves the principal properties of the local intersection pairing on the finite fibers, as did Lichtenbaum (loc. cit.) and Shafarevich (loc. cit.).

The heart of the book starts in chapter IV, with the definition of the Arakelov intersection pairing, proof of invariance under linear equivalence, the Hodge index theorem, construction of the metrized canonical sheaf, and the proof of the arithmetic adjunction formula. This is followed in chapter V with a proof of Faltings’ arithmetic Riemann-Roch theorem, which expresses the (co)volume of certain cohomology groups in terms of intersection numbers. The technical material needed to construct the Faltings volume forms on cohomology is relegated to the final chapter.

Finally, a brief appendix by Paul Vojta describes Parshin’s (conjectural) arithmetic analogue of the famous \(c^2_1\leq 3c_2\) inequality and relates Parshin’s question to some of his own conjectures.

The author has written an excellent book in a new, exciting, and very active area of research. It will undoubtedly become a standard reference in the field of arithmetic geometry, since it brings together in a coherent fashion the basic material which had previously only been available in the original journal articles. However, a potential reader should be warned that this is not a textbook for beginners; it supposes the reader has a solid background in algebraic number theory and in algebraic geometry [both algebraic as in the R. Hartshorne’s book “Algebraic geometry” (1977; Zbl 0367.14001; 3rd edition 1983) and complex as in the one by P. Griffiths and J. Harris: “Principles of algebraic geometry” (1978; Zbl 0408.14001)]. It also helps to be familiar with the author’s book: “Fundamentals of diophantine geometry” (1983; Zbl 0528.14013). But for those with the necessary background, the book under review provides a welcome entree to recent advances in arithmetic geometry.

Reviewer: Joseph H. Silverman (Providence)

### MSC:

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14H25 | Arithmetic ground fields for curves |

14G25 | Global ground fields in algebraic geometry |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |