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 ${\alpha }_1,{\alpha }_2,{\alpha }_3,d\ge 2$ 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 $Y:=\{x+x^d{y}^{{\alpha }_1}+z^{{\alpha }_2}+t^{{\alpha }_3}=0\}\subset {\mathbb{A}}_k^4$. We prove that the Chow groups $C{H}^i(Y)$ 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 $Y$.

10 pages; comments welcome!