Rational and homological equivalence of real algebraic cycles (Q1771164)

The paper is devoted to algebraic geometry. The paper is dealing with the smooth \(n\)-dimensional quasiprojective variety \(X\). \textit{F. Ischebeck} and \textit{H.-W. Schülting} [Invent. Math. 94, No. 2, 307--316 (1988; Zbl 0663.14002)] proved that a \(k\)-cycle on \(x\) \((0\leq k\leq n-1)\) is homologous to zero iff it is rationally equivalent to a cycle \(Z'\) with \(\dim(x(\mathbb{R}))\cap\text{supp}(z'))<k\). Following their ideas, the author proves in the case \(n=3\) that if a 1-cycle \(Z\) is homologous to zero, then \(Z\) is rationally equivalent to a cycle supporting no real points. Furthermore, the author states that if \(2k+1\leq n\), then there exists a positive integer \(r\), depending only on \(X\), such that for every \(k\)-cycle \(Z\) on \(X\) the multiple \(rZ\) is rationally equivalent to a cycle supporting no real points.