Compact structures on \({\mathbb C}\) *\(\times {\mathbb C}\) * (Q1093061)

By a ``surface'' we mean a connected 2-dimensional complex manifold. By a `compact structure' on a surface V we mean an equivalence class of pairs (S,C) where S is a compact surface with S-C as a Zariski-open subset biholomorphic to V; two such pairs (S,C) and (S',C') being `equivalent' if there is a chain \((S,C)=(S_ 0,C_ 0),...,(S_ i,C_ i),...,(S_ n,C_ n)=(S,C')\) where each pair \((S_{i+1},C_{i+1})\) is obtained from \((S_ i,C_ i)\) by blowing down an exceptional curve of the first kind r by blowing up a point, in \(C_ i.\) We prove: Theorem A. Let V be proper homotopy equivalent \({\mathbb{C}}\times {\mathbb{C}}^ *\). Then \(({\mathbb{P}}^ 2,2L)\) is the only compact structure on it, where 2L denotes the union of two lines on \({\mathbb{P}}^ 2.\) Theorem B. Let V be proper homotopy equivalent to \({\mathbb{C}}^ *\times {\mathbb{C}}^ *\). Then the compact structures on V are one of the following: \((i)\quad ({\mathbb{P}}^ 2;3L)\) where 3L denotes the union of any three lines in general position. \((ii)\quad (X,E),\) where X is the total space of a \({\mathbb{P}}^ 1\)-bundle over an elliptic curve and E is a section with \((E^ 2)=0\). \((iii)\quad (H,E)\), where H is a Hopf surface and E an elliptic curve on H.