#### Vol. 311, No. 2, 2021

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Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units

### Tomasz Pełka and Paweł Raźny

Vol. 311 (2021), No. 2, 385–422
##### Abstract

We give a corrected statement of (Gurjar and Miyanishi 1988, Theorem 2), which classifies smooth affine surfaces of Kodaira dimension zero, whose coordinate ring is factorial and has trivial units. Denote the class of such surfaces by ${\mathsc{𝒮}}_{0}$. An infinite series of surfaces in ${\mathsc{𝒮}}_{0}$, not listed in loc. cit., was recently obtained by Freudenburg, Kojima and Nagamine (2019) as affine modifications of the plane. We complete their list to a series containing arbitrarily high-dimensional families of pairwise nonisomorphic surfaces in ${\mathsc{𝒮}}_{0}$. Moreover, we classify them up to a diffeomorphism, showing that each occurs as an interior of a $4$-manifold whose boundary is an exceptional surgery on a $2$-bridge knot. In particular, we show that ${\mathsc{𝒮}}_{0}$ contains countably many pairwise nonhomeomorphic surfaces.

##### Keywords
affine surface, $\mathbb{C}^*$-fibration, log minimal model program, knot surgery, Kirby diagram
##### Mathematical Subject Classification
Primary: 14R05
Secondary: 14J26, 57M99, 57R65