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Singular Fano compactifications of \(\mathbb C^3\). I. (English) Zbl 1075.14039
In 1954, F. Hirzebruch [Ann. Math. (2) 60, 213–236 (1954; Zbl 0056.16803)] proposed the problem concerning the analytic compactifications \((X,Y)\) of the complex affine n-space \({\mathbb C}^n\) with the second Betti number \(b_2(X)=1\), where the pair \((X,Y)\) of a compact complex manifold \(X\) and a closed analytic subvariety \(Y\) is said to be the analytic compactification of \({\mathbb C}^n\) if the complement \(X\backslash Y\) is biholomorphic to \({\mathbb C}^n\). In the case of \(n\leq 2\), such compactifications are well-known, namely, \((X,Y)\cong ({\mathbb P}^1, \text{pt.})\) \((n=1)\), and \((X,Y)\cong ({\mathbb P}^2, \text{line})\) \((n=2)\), in particular, \(X\) is projective automatically. Distinct from the case \(n\leq 2\), there exist non-projective analytic compactifications \((X,Y)\) of \({\mathbb C}^n\) with \(b_2(X)=1\) for the higher-dimensional case \(n\geq 3\).
The present paper is intended to the case \(n=3\), i.e., the analytic compactifications \((X,Y)\) of the affine 3-space \({\mathbb C}^3\) with \(b_2(X)=1\). Note that if such an \(X\) is projective, then \(X\) is a smooth Fano threefold, and the classification of projective compactifications of \({\mathbb C}^3\) with \(b_2=1\) are completed according to the Fano index \(\in \{ 1,2,3,4 \}\) due to M. Furushima [Nagoya Math. J. 104, 1–28 (1986; Zbl 0612.14037); Math. Ann. 297, 627–662 (1993; Zbl 0788.32022)], Th. Peternell [Math. Ann. 280, 129–146 (1988; Zbl 0651.14025), ibid. 283, 121–137 (1989; Zbl 0671.14020)]; N. Nakayama, M. Schneider and others. Let \((X,Y)\) be a non-projective smooth analytic compactification of \({\mathbb C}^3\). Provided that \(Y\) is nef, it is known that there exists a small contraction, say \(\Phi : X \to V^*\), where \(V^*\) is a Fano threefold with small Gorenstein singularities [M. Furushima, Kyushu J. Math. 50, 221–239 (1996; Zbl 0886.32021)]. The exceptional set of \(\Phi\) is composed of smooth rational curves contained in the boundary \(Y\). Hence, we have \({\mathbb C}^3 \cong X\backslash Y \cong V^*\backslash \Delta\), where \(\Delta :=\Phi (Y)\). Then \(\Delta\) is an ample generator of \(\text{Pic} (V^*)\cong {\mathbb Z}\), and the anti-canonical (Cartier) divisor \(-K_{V^*}\) is written as \(-K_{V^*}\sim r \Delta\) for some \(r \in \{ 1,2,3,4 \}\). If \(r=4\), then \(V^*\cong {\mathbb P}^3\), but this case is excluded as \(V^*\) has singularities. Thus we may and shall assume that \(r\leq 3\).
The main theorem of the paper concerns the classification of \((V^*, \Delta)\) in the case of \(r=2,3\). If \(r=3\), then it is not difficult to see that \(V^*\) is isomorphic to the quadric hypersurface \(Q_0^3\) in \({\mathbb P}^4\) with one ordinary double point, and the boundary \(\Delta_2\) is a hyperplane section passing through the vertex of \(Q_0^3\) consisting of two planes. On the other hand, the situation is more subtle in the case \(r=2\). The author then restricts the possibility of the value of \(d:= {( \Delta )}^3 \in {\mathbb N}\), and shows that \(d\) equals \(4\) or \(5\). Since the case \(d=4\) is treated in M. Furushima [Kyushu J. Math. 50, 221–239 (1996; Zbl 0886.32021); ibid. 52, 149–162 (1998; Zbl 0898.32017); Hiroshima Math. J. 29, 295–298 (1999; Zbl 0946.32009)] in detail, he restricts himself to the case \(d=5\). Consequently, he succeeds in the explicit construction of \((V^*, \Delta)\) with \(r=2\) and \(d=5\) from another compactification \((Q_0^3,\Delta_2)\) via the standard blow-up and contraction process.
MSC:
14J45 Fano varieties
32J05 Compactification of analytic spaces
14J30 \(3\)-folds
14E30 Minimal model program (Mori theory, extremal rays)
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