On the fundamental homology classes of a real algebraic variety (Q1972523)

The connected components of a non-singular real algebraic variety define mod \(2\) homology classes in the complexification. It is proven that these classes are either independent or satisfy only one relation, their sum equal to zero, provided that the mod \(2\) cohomology of the complex variety is generated by the image of the Chow ring. This includes the examples known before, such as curves, surfaces with vanishing first cohomology mod \(2\) group, complete intersections. The proof uses some properties of the cycle map.