Homology planes with quotient singularities
By a \(\mathbb{Q}\)-homology plane \(X\), the authors mean a smooth affine surface \(X\) over \(\mathbb{C}\) with \[ {H^ i(X,\mathbb{Q})=0}\text{ for }i>0. \] \(\mathbb{Z}\)- homology planes (or simply homology planes) are defined similarly, and are fairly well understood. The authors also consider logarithmic homology (or \(\mathbb{Q}\)-homology) planes; here \(X\) is allowed to have log- terminal (normal) singularities. The problems considered are: (1) classification by the (logarithmic) Kodaira dimension \(\kappa(X)\) \((:=\kappa(X_{\text{reg}})\) in the logarithmic case), and (2) studying the group of automorphisms of \(X\). Thus the authors can classify \(\mathbb{Q}\)-homology planes with \(\kappa<2\); when \(\kappa=2\), it is still open whether such a surface has to be rational. -- The authors also give a structure theorem for logarithmic homology and \(\mathbb{Q}\)-homology planes in certain cases, and for \(\Aut(X)\) when \(X\) is a \(\mathbb{Q}\)-homology plane with \(\kappa(X)=2\). There are many interesting examples throughout the paper.
scientific article; zbMATH DE number 59741