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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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