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Compactifications of contractible affine 3-folds into smooth Fano 3-folds with \(B_2=2\). (English) Zbl 1086.14050

The paper contributes to the following classification problem: Let \(V\) be a smooth compact complex Fano threefold and \(D\subset V\) a holomorphic divisor such that \(X:=V\backslash D\) is a contractible affine algebraic threefold. Determine all possibilities for \(X\), \(D\) and \(V\). For \(D\) irreducible, i.e. \(b_2(V)=1\), and \(X=\mathbb C^3\), the complete list was given in 1993 by M. Furushima [Math. Ann. 297, No. 4, 627–662 (1993; Zbl 0788.32022)], cf. also T. Peternell and M. Schneider [Math. Ann. 280, No. 1, 129–146 (1988; Zbl 0651.14025)]. The author points out that the condition \(b_2(V)=1\) already implies \(X=\mathbb C^3\).
The main result of the paper under review is a complete solution of the classification problem under the assumption \(b_2(V)=2\), \(D=D_1\cup D_2\), and the line bundle \(K_V+D_1+D_2\) is not nef. The author presents a list of all possibilities and distinguishes 24 different cases. In general the affine threefold \(X\) is isomorphic to \(\mathbb C^3\) but in some cases \(X\) is only diffeomorphic to \(\mathbb C^3\). The proof is essentially based on Mori theory and in particular on the classification of Fano threefolds with second Betti number at least two, given by S. Mori and S. Mukai [Manuscr. Math. 36, 147–162 (1981); erratum ibid. 110, 407 (2003; Zbl 0478.14033), see also Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1, 101–129 (1983; Zbl 0537.14026)].

MSC:

14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
14E30 Minimal model program (Mori theory, extremal rays)
14J45 Fano varieties
14J30 \(3\)-folds
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