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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

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