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 $α_{1}, α_{2}, α_{3}, d \geq 2$ be integers such that $α_{1}, α_{2}, α_{3}$ are pairwise coprime and $gcd (α_{1},d-1) = 1$. Then consider the Koras-Russell threefold $Y := \{ x + x^d y^{α_{1}} + z^{α_{2}} + t^{α_{3}} = 0\} \subset \mathbb{A}^{4}_{k}$. 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 $α_{1}$ 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 $Y$.