A tribute to a work of C. P. Ramanujam
Let V be a smooth affine surface with coordinate ring \(\Gamma\) (V) a U.F.D. This happens for example if V is contractible. Then: \(V\cong {\mathbb{C}}^ 2\) if either: (1) \(\pi_ 1^{\infty}(V)\), the fundamental group at infinity, is trivial, or: (2) \({\bar \kappa}\)(V)\(=-\infty\), where \({\bar \kappa}\):\(=\log arithmic\) Kodaira dimension. Ramanujam (essentially) proved (1) by considering the fundamental group of the real 3-dimensional boundary \(\partial T\) of a tubular neighborhood of the divisor at infinity of a compactification \(\bar V\) of V. Fujita, Miyanishi and Sugie proved (2) by purely geometrical arguments resting on the existence of Zariski decomposition of pseudo-effective divisors. This expository article describes these approaches, as well as the direct proof of the relationship: (2)\(\Rightarrow (1)\) (the author). The existence of V with \({\bar \kappa}(V)=1\quad or\quad 2\) which are homology cells (the author and Miyanishi), as well as the rationality of these (the author and Shastri) are related questions.
scientific article; zbMATH DE number 4136109