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The Nikulin congruence for four-dimensional \(M\)-varieties. (English. Russian original) Zbl 1064.14071
Math. Notes 76, No. 2, 191-199 (2004); translation from Mat. Zametki 76, No. 2, 205-215 (2004).
It is proven that the Euler characteristic of a real non-singular algebraic \(M\)-variety (i.e., such that the total \({\mathbb Z}/2\) Betti number of the real point set is equal to the total \({\mathbb Z}/2\) Betti number of the complexification) is divisible by \(2^{m+3}\), provided that the Euler characteristic of any connected component of the real point set is divisible by \(2^m\), and a few more conditions are fulfilled. The claim is generalized to smooth \(8\)-dimensional manifolds with an orientation-preserving involution. Such a statement for real non-singular \(M\)-surfaces has been established by V. Nikulin [Math. USSR, Izv. 22, 99–172 (1984; Zbl 0547.10021)]. The proof is based on the technique of the equivariant cohomology of topological spaces with involution.
MSC:
14P25 Topology of real algebraic varieties
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
55N91 Equivariant homology and cohomology in algebraic topology
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