Complex analytic compactifications of \({\mathbb{C}}\) 3
In this note, a complete classification of smooth connected compactifications X of \({\mathbb{C}}^ 3 \)with normal divisor Y at infinity is announced. They fall into 3 classes: the 2 classical ones \((X={\mathbb{P}}_ 3({\mathbb{C}})\) or \(Q_ 3\), the 3-dimensional quadric) and a new one \((V_ 5,H_ 5)\) constructed by the author; \(V_ 5\) is a Fano 3-fold of degree \( 5\) in \({\mathbb{P}}_ 6({\mathbb{C}})\) and \(H_ 5\) a singular Del Pezzo hyperplane section of \(V_ 5\). The aim of this note is to exclude the last remaining possibility of a Fano 3-fold X of index \(r=1\). The argument uses the Iskovskij-Shokurov classification, and Mori's theory of extremal rays. For the case of \(b_ 2(X)=1\), Y non-normal, see \textit{T. Peternell} and \textit{M. Schneider}, Math. Ann. 280, No.1, 129-146 (1988; see the following review).
classification of smooth connected compactifications
scientific article; zbMATH DE number 4061398