The equivariant cohomology groups of a real algebraic surface and their applications.

*(English. Russian original)*Zbl 0902.55002
Izv. Math. 60, No. 6, 1193-1217 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 6, 101-126 (1996).

Given a non-singular real algebraic variety \(X\) with \(G\) the group generated by the involution induced by complex conjugation the author considers \(G\)-sheaves \({\mathcal A}\) and is interested in the equivariant cohomology groups \(H^{*}(X({\mathbb C}); G, {\mathcal A})\). He extends a five term exact sequence obtained from the first spectral sequence \(I^{p,q}({\mathcal A})\) [cf. A. Grothendieck, Tôhoku Math. J., II. Ser. 9, No. 2, 119-221 (1957; Zbl 0118.26104), p. 201] to an infinite sequence that involves the equivariant cohomology groups and the cohomology of \(X/G\) and of \(X({\mathbb R})\). The sequence is exact if the first spectral sequence converges and the homomorphisms in the sequence are related to the differentials in the \(E_{2}\)-term of \(I^{p,q}({\mathcal A})\). The author gives a precise description of the homomorphisms in case \(X\) is a surface and \({\mathcal A}\) is one of the \(G\)-sheaves \({\mathbb Z/2}, {\mathbb Z}\) or \(\mathbb Z_{-}\), where \({\mathbb Z/2}\) and \({\mathbb Z}\) are the constant sheaves with trivial \(G\)-action, and \({\mathbb Z_{-}}\) is the sheaf that equals \({\mathbb Z}\) but carries the non-trivial \(G\)-action. The author uses his results to give an easy criterion for the convergence of the second spectral sequence \(II^{p,q}({\mathbb Z/2})\) in case \(X\) is a surface with \(X({\mathbb R)} \neq 0\), and he provides an example where \(II^{p,q}({\mathbb Z/2})\) converges but \(II^{p,q}({\mathbb Z})\) does not. Furthermore, using the (exact) sequence for the sheaf \({\mathbb Z_{-}}\), he can find a lower bound for the dimension of \(_{2}Br(X)\), the subgroup of the Brauer group \(Br(X)\) consisting of the elements of order less than or equal to \(2\).

Reviewer: Michael Joachim (Münster)