Smooth affine surfaces properly dominated by $\mathbf{C}^*\times\mathbf{C}^*$

Abstract

We classify smooth complex affine surfaces admitting a finite surjective morphism from $\mathbf{C}^*\times\mathbf{C}^*$. Using the classification theory of smooth affine surfaces according to logarithmic Kodaira dimension, we show that every such surface has logarithmic Kodaira dimension either $-\infty$ or $0$, and determine all possibilities. More precisely, the only surfaces of logarithmic Kodaira dimension $-\infty$ are $\mathbf{C}^2$ and $\mathbf{C}\times\mathbf{C}^*$, while those of logarithmic Kodaira dimension $0$ are precisely $\mathbf{C}^*\times\mathbf{C}^*$ and Fujita's surface $H[-1,0,-1]$. This establishes the classification of smooth affine surfaces properly dominated by $\mathbf{C}^*\times\mathbf{C}^*$ anticipated by M.~Furushima in 1989.