Cylindrical Homomorphisms and Lawson Homology

TL;DR

This paper investigates Lawson homology of certain hypersurfaces using cylindrical homomorphisms, computes semi-topological K-theory for specific cubic hypersurfaces, and proves Suslin's conjecture for these cases.

Contribution

It introduces a geometric approach to study Lawson homology and applies it to compute K-theory and verify Suslin's conjecture for particular hypersurfaces.

Findings

Computed rational semi-topological K-theory for cubic hypersurfaces of dimensions 5, 6, and 8.

Proved Suslin's conjecture for these hypersurfaces using the Bloch-Kato conjecture.

Demonstrated existence of varieties with infinitely generated s-filtration steps undetected by Abel-Jacobi map.

Abstract

We use the cylindrical homomorphism and a geometric construction introduced by J. Lewis to study the Lawson homology groups of certain hypersurfaces $X\subset \mathbb{P}^{n+1}$ of degree $d\leq n+1$. As an application, we compute the rational semi-topological K-theory of a generic cubic of dimension 5, 6 and 8 and, using the Bloch-Kato conjecture, we prove Suslin's conjecture for these varieties. Using the generic cubic sevenfolds, we show that there are smooth projective varieties with the lowest non-trivial step in their s-filtration infinitely generated and undetected by the Abel-Jacobi map.