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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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