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Rational and homological equivalence for real cycles. (English) Zbl 0663.14002

Let X be an algebraic variety over \({\mathbb{R}}\), which is smooth and complete. In 1961 Borel and Haeflinger defined a map from the k- dimensional cycles, \(Z_ k(X)\), to \(H_ k(X({\mathbb{R}}),{\mathbb{Z}}/2{\mathbb{Z}})\) [A. Borel and A. Haefliger, Bull. Soc. Math. Fr. 89, 461-513 (1961; Zbl 0102.385)].
In this paper the authors study the kernel of this map. Using relations between “rational equivalence” and “bordism”, as well as relating part of \(H_*(X({\mathbb{R}}),{\mathbb{Z}}/2{\mathbb{Z}})\) to the bordism groups of X, they prove that this kernel is \(P_ k(X)+Z_ k^{th}(X)\), where \(P_ k(X)\) denotes the group of k-cycles which are rationally equivalent to zero and \(Z_ k^{th}(X)\) denotes the “thin cycles”, generated by those subvarieties Y with Y(\({\mathbb{R}})\) not Zariski dense in Y. For \(k=1\) this result had already been proved by F. Ischebeck [Géométrie algébrique réelle et formes quadratiques, Journ. S. M. F., Univ. Rennes 1981, Lect. Notes Math. 959, 371-380 (1982; Zbl 0496.14016)].
Reviewer: A.F.T.W.Rosenberg

MSC:

14C15 (Equivariant) Chow groups and rings; motives
14Pxx Real algebraic and real-analytic geometry
14C25 Algebraic cycles
14F25 Classical real and complex (co)homology in algebraic geometry
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