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A bound on the order of \(H^{(a)}_{n-1}(X,{\mathbb{Z}}/2)\) on a real algebraic variety. (English) Zbl 0558.14003
Géométrie algébrique réelle et formes quadratiques, Journ. S.M.F., Univ. Rennes 1981, Lect. Notes Math. 959, 443-450 (1982).
[For the entire collection see Zbl 0487.00005.]
The author proves a bound on the order of the algebraic part \(H^{(a)}_{n-1}\) of \(H_{n-1}(X({\mathbb{R}}),{\mathbb{Z}}/2)\) for a geometrical integral, projective and smooth scheme X over \({\mathbb{R}}\) in terms of \(\dim_{{\mathbb{R}}}H^ 1(<{\mathcal O}_ X)\), the Comessatti characteristic of \(X_{{\mathbb{C}}}({\mathbb{C}})\) (where \(X_{{\mathbb{C}}}=X\times_{{\mathbb{R}}}({\mathbb{C}}))\), the rank of \(NS(X_{{\mathbb{C}}})\) \((NS=\)Néron-Severi-group, \(G=Gal({\mathbb{C}}/{\mathbb{R}}))\), and of \(\dim_{{\mathbb{Z}}/2}[NS(X_{{\mathbb{C}}})^ G]_ 2\). This allows him to construct Abelian varieties X over \({\mathbb{R}}\) satisfying \(H^{(a)}_{n- 1}(X)\neq H_{n-1}(X({\mathbb{R}}),{\mathbb{Z}}/2)\).
Reviewer: H.W.Schülting

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C99 Cycles and subschemes
14Pxx Real algebraic and real-analytic geometry
Zbl 0487.00005