Motivic cohomology of cyclic coverings
Tariq Syed
TL;DR
This work addresses the problem of understanding motivic cohomology for cyclic and bicyclic coverings of smooth affine varieties, focusing on how a finite Galois covering with group $\mu_s(k)$ influences motivic invariants. The authors develop general results showing that, when $s$ is invertible in the coefficient ring, the motivic cohomology groups $H^{p,q}$ of the cover pull back isomorphically to the base in many degrees, leading to isomorphisms $H^{2i,i}(X,R) \cong H^{2i,i}(Y_s,R)$ and hence $CH^i(Y_s) \otimes R \cong CH^i(X) \otimes R$; consequently, CH groups become torsion or vanish under suitable hypotheses. They also prove integral vanishing for bicyclic coverings $Y_{s,t}$ under explicit hypotheses, deducing, in particular, the triviality of vector bundles in several new examples. The paper applies these general results to concrete cases, notably cyclic and bicyclic coverings of Koras-Russell threefolds and of stably $\mathbb{A}^1$-contractible fourfolds, providing computable motivic invariants and supporting the broader program linking topological contractibility, $\mathbb{A}^1$-contractibility, and Serre-type questions in low dimensions.
Abstract
Cyclic coverings produce many examples of topologically contractible smooth affine complex varieties. In this paper, we study the motivic cohomology groups of cyclic coverings over algebraically closed fields of characteristic $0$. In particular, we prove that in many situations Chow groups of cyclic coverings become trivial after tensoring with $\mathbb{Q}$. Furthermore, we can prove that the Chow groups of certain bicyclic coverings are trivial even without tensoring with $\mathbb{Q}$.