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Compactifications of \({\mathbb{C}}^ 3\). II. (English) Zbl 0671.14020

[For part I of this paper see the author and M. Schneider, Math. Ann. 280, No.1, 129-146 (1988; Zbl 0651.14025).]
This paper shows that Fano compactifications of \({\mathbb{C}}^ 3\) with \(b_ 2=1\) and index \(r=1\) have genus \(g(:=+(c^ 3_ 1/2)+1)=12\), and are thus rational with \(b_ 3=0\). The proof rests on the classification of Fano threefolds of Iskovskij-Shokurov. Such a compactification exists, as shown by M. Furushima.
Fano compactifications with \(b_ 2=1\) and (4\(\geq)r\geq 2\) are classified in part I of this paper. Other papers with S. Kosarew show that compactifications of \({\mathbb{C}}^ 3\) with \(b_ 2=1\) are Moishezon.
Reviewer: F.Campana

MSC:

14J30 \(3\)-folds
32J05 Compactification of analytic spaces

Citations:

Zbl 0651.14025
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References:

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