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Analogues of the Harnack-Thom inequality for a real algebraic surface. (English. Russian original) Zbl 0977.14027
Izv. Math. 64, No. 5, 915-937 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 5, 45-68 (2000).
The Harnack-Thom inequality states that the total mod \(2\) Betti number of the real part of a real algebraic variety does not exceed the total Betti number of the complexification. The present paper contains upper bounds to the first and the second mod \(2\) Betti numbers of the real part of a real algebraic surface given via the Picard and Brauer groups of the complexification and the number of real connected components of the Albanese variety. Necessary and sufficient conditions for the equalities in these relations are found. The new upper bounds are close to the other analogues of the Harnack-Thom inequality found earlier by V. A. Krasnov [Math. USSR, Izv. 22, 247-275 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 2, 268-297 (1983; Zbl 0537.14035)].

14P25 Topology of real algebraic varieties
14C22 Picard groups
14F22 Brauer groups of schemes
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