Albanese map for GM\({\mathbb{Z}}\)-varieties (Q1060255)

The author proves a conjecture from his previous paper [cf. Math. USSR, Izv. 22, 247-275 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No.2, 268-297 (1983; Zbl 0537.14035)]. Let X be a real algebraic variety and \(\alpha\) : \(X\to A\) be its Albanese mapping. If X is a GM\({\mathbb{Z}}\)-variety, then the set \(\alpha\) (X(\({\mathbb{R}}))\) generates the group A(\({\mathbb{R}})\). By definition, X is a GM\({\mathbb{Z}}\)-variety, iff some analogues of the Thom-Harnack inequality become equalities for X (this notion was introduced by the author, loc. cit.).