Lawson homology for varieties with small Chow groups and the induced filtration on the Griffiths groups
Let \(X\) be a complex projective variety and let \(C_m(X)\) be the disjoint union of the Chow varieties of effective \(m\)-cycles of degree \(d= 0,1,2,\dots\). The empty cycle \(0\in C_m(X)\) is a natural base point and it also acts as a zero for the addition of cycles, making \(C_m(X)\) into a monoid. Put \[ {\mathcal Z}_m(X)= C_m(X)\times C_m(X)/ \sim, \quad \text{with} \quad (x,y)\sim (x',y') \Leftrightarrow x+y'= x'+y \] (the naive group completion). The complex topology induces a natural topology on the monoid \(C_m(X)\) and one equips \({\mathcal Z}_m (X)\) with the quotient topology. Friedlander showed that this topological space is independent of the chosen embedding of \(X\) into a projective space. The induced topology is called the Chow topology. The Lawson homology groups are the homotopy groups of the space: \[ L_mH_\ell(X)= \begin{cases} \pi_{\ell-2m}{\mathcal Z}_m (x)\quad & \text{if }\ell\geq 2m \\ 0\quad & \text{if }\ell< 2m.\end{cases} \] The main results of the paper are as follows: Theorem. Let \(X\) be a smooth complex projective variety for which rational and homological equivalence coincide for \(m\)-cycles in the range \(0\leq m\leq s\). Then \(L^{\text{hom}}_m H_* (X)\otimes \mathbb{Q}=0\) in the range \(0\leq m\leq s+1\). There is an interesting translation of the result in terms of a certain filtration on the \(m\)-th Griffiths group (i.e. the group of \(m\)-cycles homologically equivalent to zero modulo the group of \(m\)-cycles algebraically equivalent to zero). This filtration, the \(S\)-filtration of \textit{E. M. Friedlander} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 3, 317-343 (1995; Zbl 0854.14006)], is very roughly defined in the following way: A Griffiths class is in \(S_r\) if it is in the subgroup generated by the images of the action of correspondences from \(Y\) to \(X\), equidimensional over \(Y\), the action being restricted to \(r\)-cycles homologically equivalent to zero. Proposition. Let \(X\) be a smooth complex projective variety for which rational and homological equivalence coincide for \(m\)-cycles in the range \(0\leq m\leq s\). Then the last \(s+1\) steps induced by the \(S\)-filtration on any of the \(\mathbb{Q}\)-Griffiths groups are constant. In particular, algebraic and homological equivalence coincide for \(s+1\)-cycles.
scientific article; zbMATH DE number 1488554