Semi-topological K-theory for certain projective varieties

Abstract

In this paper we compute Lawson homology groups and semi-topological K-theory for certain threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. The computation makes use of different techniques of decomposition of the diagonal cycle, of the Bloch-Kato conjecture and of the spectral sequence relating morphic cohomology and semi-topological K-theory.

Equivariant semi-topological invariants, Atiyah’s $KR$-theory, and real algebraic cycles

“…Together with other techniques, these allow us to prove a stronger result than is predicted in general by the real Suslin's conjecture in some cases (e.g. geometrically rationally connected threefolds), extending results from the complex case of Voineagu [Voi08]. In particular we obtain that all of the equivariant morphic cohomology groups are finitely generated in these cases.…”

Lawson homology, morphic cohomology and Chow motives

“…Remark 6.5 These results can be obtained by the method of decomposition of diagonals, used by C. Peters in 28, since any unirational variety has small Chow groups for zero cycles. Later, M. Voineagu refined Peters’ results from \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {Q}$\end{document}‐coefficient Lawson homology groups to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {Z}$\end{document}‐coefficient in many cases 29.…”

Topological and Geometric filtration for products

“…Hence the element α ∈ L 1 H k (X) hom satisfies that dα = 0. Since L 1 H k (X) hom is divisible for k ≥ 2 dim X (see [17,Prop. 3.1]), we get α = 0.…”

“…This is a generalization of a result in [11], where either the dimension of X is not more than four or the group rational coefficient. A different method using the decomposition of diagonal can be found in [16] and [17].…”

“…For example, it holds for cellular varieties for which the Lawson homology are shown to be the same as the singular homology (see [13], [15]); it holds for general abelian varieties (see [5]) or abelian varieties for which the generalized Hodge conjecture holds (see [1]); it holds for smooth projective varieties for which the Chow groups in rational coefficients are isomorphic to the corresponding singular homology groups in rational coefficients by using the technique of decomposition of diagonal (cf. [17]). It was also shown to hold for threefold X with h 2,0 (X) = 0 (see [9]).…”