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

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 $Y$.