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Real algebraic \(\text{GM}\mathbb{Z}\)-surfaces. (English. Russian original) Zbl 0931.14034
Izv. Math. 62, No. 4, 695-721 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 4, 51-80 (1998).
Let \(X\) be a non-singular real projective algebraic surface, \(g: X (\mathbb{C})\to X(\mathbb{C})\) the involution induced by complex conjugation, and \(G= \{1,g\}\). In general one has \[ \dim H^*(X(\mathbb{R}), \mathbb{F}_2)\leq\dim H^1(G,H^* (X (\mathbb{C}), \mathbb{Z}))+\dim H^2(G,H^* (X(\mathbb{C}), \mathbb{Z})) \] and those \(X\) for which the inequality is an equality are called \(\text{GM} \mathbb{Z}\)-surfaces. The paper under review presents several characterizations of \(\text{GM} \mathbb{Z}\)-surfaces, and the computation of the Néron-Severi group \(NS(X)\), the Brauer group Br\((X)\) and the first cohomology group \(H_1^{\text{alg}} (X(\mathbb{R}), \mathbb{F}_2)\) for such surfaces \(X\). This generalizes previous results of the author and a very recent one of F. Mangolte and J. van Hamel [Compos. Math. 110, No. 2, 215-237 (1998; Zbl 0920.14029)].
Moreover, the author extends an old congruence of Nikulin for the Euler characteristic of the real part of orientable \(M\)-surfaces.
This article is very clearly written, but it is convenient to study first “Real algebraic GM-varieties” of the same author in the same journal [see the preceding review Zbl 0931.14034)] since it contains some missing proofs and arguments of the paper under review.

MSC:
14P25 Topology of real algebraic varieties
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