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**Inequalities for semistable families of arithmetic varieties.**
*(English)*
Zbl 1041.14007

In an earlier paper [A. Moriwaki, J. Am. Math. Soc. 11, No. 3, 569–600 (1998; Zbl 0893.14004)] the second author proved the relative Bogomolov’s inequality for algebraic varieties over an algebraically closed field of characteristic zero.

This capacious paper under review consisting of an introduction, 10 sections and an appendix is devoted to prove the arithmetic analogues of the mentioned result and of Cornalba-Harris-Bost’s inequality [M. Cornalba and J. Harris, Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No. 3, 455–475 (1988; Zbl 0674.14006)] in the case of semistable families of arithmetic varieties. Since the usual push-forward for arithmetical cycles is insufficient for the authors’ purposes, they need to introduce the notion of arithmetic \(L^1\)-cycle and to extend the usual Chow groups defined by Gillet-Soulé [H. Gillet and C. Soulé, Publ. Math., Inst. Hautes Étud. Sci. 72, 93–174 (1990; Zbl 0741.14012)].

In section 1 of the paper the investigation of locally integrable forms is done. Section 2 is devoted to the study of the properties of three variants of arithmetic Chow groups. In section 3 the authors introduce the notion of weak positivity of arithmetic divisors. In sections 4 and 5 the authors prove the Riemann-Roch theorems for generically finite morphisms and for stable curves. Sections 6 and 7 provide respectively an asymptotic upper bound of analytic torsion and formulae for arithmetic Chern classes.

The first main result of the paper consists of a relative Bogomolov inequality in the arithmetic case and appears in section 8.

The second main result of the paper is a relative Cornalba-Harris-Bost inequality (theorem 10.1.4).

Finally, a comparison of theorems 8.1 and 10.1.4 is given (each of them has its advantage).

This is a substantial paper which contains many deep results on the subject.

This capacious paper under review consisting of an introduction, 10 sections and an appendix is devoted to prove the arithmetic analogues of the mentioned result and of Cornalba-Harris-Bost’s inequality [M. Cornalba and J. Harris, Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No. 3, 455–475 (1988; Zbl 0674.14006)] in the case of semistable families of arithmetic varieties. Since the usual push-forward for arithmetical cycles is insufficient for the authors’ purposes, they need to introduce the notion of arithmetic \(L^1\)-cycle and to extend the usual Chow groups defined by Gillet-Soulé [H. Gillet and C. Soulé, Publ. Math., Inst. Hautes Étud. Sci. 72, 93–174 (1990; Zbl 0741.14012)].

In section 1 of the paper the investigation of locally integrable forms is done. Section 2 is devoted to the study of the properties of three variants of arithmetic Chow groups. In section 3 the authors introduce the notion of weak positivity of arithmetic divisors. In sections 4 and 5 the authors prove the Riemann-Roch theorems for generically finite morphisms and for stable curves. Sections 6 and 7 provide respectively an asymptotic upper bound of analytic torsion and formulae for arithmetic Chern classes.

The first main result of the paper consists of a relative Bogomolov inequality in the arithmetic case and appears in section 8.

The second main result of the paper is a relative Cornalba-Harris-Bost inequality (theorem 10.1.4).

Finally, a comparison of theorems 8.1 and 10.1.4 is given (each of them has its advantage).

This is a substantial paper which contains many deep results on the subject.

Reviewer: Vasyl I. Andriychuk (Lviv)

### MSC:

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

14C15 | (Equivariant) Chow groups and rings; motives |

14F45 | Topological properties in algebraic geometry |