Lawson homology, morphic cohomology and Chow motives

Abstract

In this paper, the Lawson homology and morphic cohomology are defined on the Chow motives. We also define the rational coefficient Lawson homology and morphic cohomology of the Chow motives of finite quotient projective varieties. As a consequence, we obtain a formula for the Hilbert scheme of points on a smooth complex projective surface. Further discussion concerning generic finite maps is given. As a result, we give examples of self-product of smooth projective curves with nontrivial Griffiths groups by using a result of Ceresa.

The Lawson homology for Fulton–MacPherson configuration spaces

“…This statement follows from Theorem 1.3 and a result of the authors [11] where we have found smooth algebraic curves C such that L p H 2p .C n / is not finitely generated for 1 Ä p Ä n 2 if n 3.…”

On the Hard Lefschetz Conjecture for Lawson Homology

“…which at the level of π 0 induces the usual intersection pairing on algebraic cycles modulo algebraic equivalence ( [6]). This intersection product is graded-commutative and associative ([11,Prop.2.4]). There exists a canonical cycle map Φ p,k : L p H k (X) → H k (X) which is compatible with pull back, push forward and intersection product.…”

“…Now for two smooth projective varieties X, Y of dimension n, m respectively, denote by Corr d (X, Y ) := CH n+d (X × Y ) the Chow group of algebraic cycles of dimension n + d on X × Y . By [11], each Γ ∈ Corr d (X × Y ) determines a homomorphism of Lawson homology Γ * :…”

“…Proof. This comes directly from the hard Lefschetz theorem for singular homology and the following known results on Lawson homology: if p < 0, then L p H k (X) ∼ = H k (X) ( [7,11,16]). If p = 0, L p H k (X) = H k (X).…”

“…Furthermore, there is a canonical duality homomorphism D : L q H l (X) → L n−q H 2n−l (X) which is compatible with the cycle maps. For smooth projective varieties, the duality homomorphism is an isomorphism ( [8]) and L * H * (X) := l≤2q,q∈Z L q H l (X) is a bigraded ring with the intersection product as multiplication and being anti-commutative for the second grading ( [11,Prop.2.4]). Since we are always interested in smooth projective varieties, every statement about Lawson homology has an equivalent dual statement about morphic cohomology.…”