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Algebraic cycles and topology of real algebraic varieties. (English) Zbl 0986.14042
CWI Tracts. 129. Amsterdam: Centrum voor Wiskunde en Informatica (CWI). Leiden: Univ. Leiden (Thesis 1997), ii, 134 p. (2000).
The present work is a slightly modified version of the author’s doctoral dissertation. The main theme is the determination of the algebraic homology classes in the homology groups of a real algebraic variety: If \(X\) is an algebraic variety defined over the real numbers and \(H_k(X(\mathbb{R}), \mathbb{Z}/2)\) is the \(k\)-th homology group, then the problem is to describe the subgroup \(H^{ \text{alg}}_k(X (\mathbb{R}), \mathbb{Z}/2)\) of homology classes that are represented by \(k\)-dimensional algebraic subsets of \(X\). The group \(H^{\text{alg}}_k (X(\mathbb{R}), \mathbb{Z}/2)\) is the image of the real cycle map \(\text{cl}^\mathbb{R}: Z_k(X)\to H_k (X(\mathbb{R}), \mathbb{Z}/2)\). Closely related is the complex cycle map associated with the complexification of \(X\). The author defines equivariant Borel-Moore homology groups together with cycle maps \(\text{cl}: \mathbb{Z}_k(X)\to H_{2k} (X(\mathbb{C}); G,\mathbb{Z}(i))\) \((G\) the group generated by complex conjugation) such that both the real and the complex cycle maps factor through this new map, e.g., there is a map \[ \rho_k:H_{2k} \bigl(X(\mathbb{C}); G,\mathbb{Z}(k) \bigr)\to H_k \bigl(X (\mathbb{R}),\mathbb{Z}/2 \bigr) \] with \(\text{cl}^\mathbb{R}= \rho_k\circ \text{cl}\). Thus the construction yields restrictions for \(H^{\text{alg}}_k (X(\mathbb{R}), \mathbb{Z}/2)\) in terms of the equivariant topology of the complexification of the variety \(X\). The equivariant method is applied to the study of real algebraic cycles on real Enriques surfaces and on complex projective varieties.

14P25 Topology of real algebraic varieties
14C25 Algebraic cycles
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14P05 Real algebraic sets
14F25 Classical real and complex (co)homology in algebraic geometry