Vector bundles over certain Koras-Russell threefolds of the third kind

TL;DR

This paper proves that all algebraic vector bundles over certain Koras-Russell threefolds are trivial by showing their Chow groups are trivial, and additionally establishes triviality of Chow-Witt groups under specific conditions.

Contribution

It demonstrates the triviality of algebraic vector bundles on a class of Koras-Russell threefolds and extends results to Chow-Witt groups when a parameter is odd.

Findings

Chow groups CH^i(Y) are trivial for i=1,2,3.

All algebraic vector bundles over Y are trivial.

Chow-Witt groups are trivial if α₁ is odd.

Abstract

Let $k$ be an algebraically closed base field of characteristic $0$ and let $\alpha_{1}, \alpha_{2}, \alpha_{3}, d \geq 2$ be integers such that $\alpha_{1}, \alpha_{2}, \alpha_{3}$ are pairwise coprime and $gcd (\alpha_{1},d-1) = 1$. Then consider the Koras-Russell threefold $Y := \{ x + x^d y^{\alpha_{1}} + z^{\alpha_{2}} + t^{\alpha_{3}} = 0\} \subset \mathbb{A}^{4}_{k}$. We prove that the Chow groups $CH^{i}(Y)$ are trivial for $i=1,2,3$ and therefore all algebraic vector bundles over $Y$ are trivial. If $\alpha_{1}$ is odd, we also prove that the Chow-Witt groups $\widetilde{CH}^{i}(Y, \mathcal{L})$ are trivial for $i=1,2,3$ and any line bundle $\mathcal{L}$ over $Y$.