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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.

##### MSC:
 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