[Paper Review] Lawson homology, morphic cohomology and Chow motives

Wenchuan Hu, Li Li|arXiv (Cornell University)|Nov 4, 2007

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper establishes Lawson homology and morphic cohomology on Chow motives, including rational coefficients for finite quotient projective varieties, and applies this framework to derive a formula for the Hilbert scheme of points on a smooth complex projective surface. A key contribution is constructing examples of self-products of smooth curves with nontrivial Griffiths groups via finite maps and results on Jacobians of generic curves.

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.

Motivation & Objective

• To define Lawson homology and morphic cohomology on Chow motives, extending these theories to motives of finite quotient projective varieties.
• To establish a functorial framework linking motives and morphic cohomology, enabling transfer of results from motivic theory to cycle-theoretic invariants.
• To apply the theory to compute Lawson homology of Hilbert schemes on surfaces and to study Griffiths groups of self-products of curves.
• To explore birational invariants and uniruled threefolds using rational coefficient Lawson homology and finite morphisms.
• To provide new examples of varieties with nontrivial Griffiths groups via finite maps and Jacobian constructions.

Proposed method

• Define Lawson homology and morphic cohomology on Chow motives using homotopy groups of cycle spaces with Chow topology.
• Use correspondences in the category of Chow motives to define morphism actions on Lawson homology and morphic cohomology via pushforwards and projectors.
• Prove that rational coefficient Lawson homology of a smooth quasi-projective variety is isomorphic to that of its quotient under a finite group action via the G-invariant subspace.
• Apply the theory to projective bundles, blow-ups, and cell-decomposable varieties using known results from Friedlander, Gabber, and Hu.
• Use Karpenko’s theorem on motives to decompose Lawson homology and morphic cohomology for quotient varieties.
• Leverage finite morphisms and birational maps to compare homological invariants and derive inequalities in rational homological dimensions.

Experimental results

Research questions

• RQ1 How can Lawson homology and morphic cohomology be extended to the category of Chow motives?
• RQ2 What is the relationship between the rational coefficient Lawson homology of a variety and its finite quotient?
• RQ3 Can the Hilbert scheme of points on a smooth projective surface be described via Lawson homology using motivic techniques?
• RQ4 Under what conditions do finite morphisms induce surjections or injections on rational Lawson homology and Griffiths groups?
• RQ5 Do self-products of generic curves of genus ≥3 admit nontrivial Griffiths groups, and how can this be shown using Jacobians and finite covers?

Key findings

• Lawson homology and morphic cohomology are well-defined functors on the category of Chow motives, with morphism actions induced by correspondences.
• There is a canonical isomorphism $\pi_*:(L_pH_k(X,\mathbb{Q}))^G \cong L_pH_k(X',\mathbb{Q})$ for a finite group action $G$ on a smooth quasi-projective variety $X$, with quotient $X'$.
• The Hilbert scheme of $n$ points on a smooth complex projective surface has a Lawson homology formula derived from the motivic framework and blow-up formulas.
• For uniruled threefolds, rational Lawson homology agrees with singular homology in all but finitely many degrees, with injective maps in critical degrees.
• For a generic curve $C$ of genus $g \geq 3$, the $g$-fold self-product $X = C^g$ has nontrivial rational Griffiths groups $\mathrm{Griff}_p(X) \otimes \mathbb{Q}$ for $1 \leq p \leq g-2$.
• The natural finite morphism $\pi: C^g \to C^{(g)}$ (symmetric product) and the birational map $C^{(g)} \to J(C)$ imply $\dim_{\mathbb{Q}} \mathrm{Griff}_p(C^g) \otimes \mathbb{Q} \geq \dim_{\mathbb{Q}} \mathrm{Griff}_p(J(C)) \otimes \mathbb{Q} > 0$ for $1 \leq p \leq g-2$.