zbMATH — the first resource for mathematics

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.
14J45 Fano varieties
32J05 Compactification of analytic spaces
14J30 \(3\)-folds
14E30 Minimal model program (Mori theory, extremal rays)
Full Text: DOI
[1] Abe, M., Furushima, M.: On non-normal del Pezzo surfaces Math. Nach. 260, 3-13 (2003) · Zbl 1036.14002
[2] Barthel, G., Kaup, L.: Topologie des Espaces Complexes Compactes Singulières. Montreal Lecture Notes, Les Presses de l?Université de Montréal 80, (1982)
[3] Campana, F., Flenner, H.: Projective threefolds containing a smooth rational surface with ample normal bundle. J. reine und angew. Math. 440, 77-98 (1993) · Zbl 0769.14014
[4] Fujita, T.: Projective varieties of ?-genus one. Algebraic and Topological Theories, Kinokuniya, Tokyo Japan. pp. 149-175 1985 · Zbl 0800.14020
[5] Fujita, T.: On singular del Pezzo varieties, in Algebraic Geometry, Proceedings L?Aquila 1988 Lect. Note in Math. Springer 1417, 117-128 (1990)
[6] Furushima, M.: Singular del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space . Nagoya Math. J. 104, 1-28 (1986) · Zbl 0612.14037
[7] Furushima, M.: The complete classification of compactifications of which are projective manifolds with second Betti number one Math. Ann. 297, 627-662 (1993) · Zbl 0788.32022
[8] Furushima, M.: An example of a non-projective smooth compactification of with second Betti number equal to one. Math. Ann. 300, 89-96 (1994) · Zbl 0803.32016
[9] Furushima, M.: Non-projective compactifications of (I) Kyushu J. Math. 50, 221-239 (1996) · Zbl 0886.32021
[10] Furushima, M.: Non-projective compactifications of (II) (New Examples) Kyushu J. Math. 52, 149-162 (1998) · Zbl 0898.32017
[11] Furushima, M.: Non-projective compactifications of III: A remark on indices. Hiroshima Math. J. 29, 295-298 (1999) · Zbl 0946.32009
[12] Furushima, M.: Birationa construction of projective compactifications of with second Betti number equal to one. Annali di Matematica pura ed applicata 178, 115-128 (2000) · Zbl 1027.14020
[13] Furushima, M., Nakayama, N.: The family of lines on the Fano threefold V5. Nagoya Math. J. 116, 111-122 (1989) · Zbl 0731.14025
[14] Furushima, M., Nakayama, N.: A new construction of a compactification of . Tohoku Math. J. 41, 543-560 (1989) · Zbl 0703.14025
[15] Hirzebruch, F.: Some problems on differentiable and complex manifolds. Ann. Math. 60, 213-236 (1954) · Zbl 0056.16803
[16] Iskovskih, V.A.: Fano 3-folds I. Math. USSR Izvestiya 11, 485-527 (1977) · Zbl 0382.14013
[17] Iskovskih, V.A.: Fano 3-folds II. Math. USSR Izvestiya 12, 469-506 (1978) · Zbl 0424.14012
[18] Iskovskih, V.A.: Algebraic Geometry V. Encyclopaedia of Mathematical sciences, Springer-Verlag. 47, (1999)
[19] Katz, S.: Small resolution of Gorenstein threefold singularities. Contemporary Math. AMS 116, 61-70 (1991) · Zbl 0755.14002
[20] Kollar, J.: Flips, Flops, Minimal models, etc. Survey in Differential Geometry. 1, 113-199 (1991)
[21] Laufer, H.: On minimally elliptic singularities. Amer. J. Math. 99, 1257-1295 (1977) · Zbl 0384.32003
[22] Laufer, H.: On as an exceptional set. Ann. Math. Studies, Princeton University Press, Princeton, NJ 100, 261-275 (1981) · Zbl 0523.32007
[23] Miyanishi, M.: Open Algebraic Surfaces, CRM Monograph series vol.12 Amer. Math. Sci. 2001
[24] Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. 116, 133-176 (1982) · Zbl 0557.14021
[25] Mukai, S.: New development in Fano manifold theory related to the vector bubdle method and moduli problem. Exposition of Sugaku 116, 133-176 (1995)
[26] Nakamura, I.: Moishezon 3-folds homeomorphic to a cubic hypersurface in J. Algebraic Geometry 5, 537-569 (1996) · Zbl 0883.14020
[27] Peternell, T., Schneider, M.: Compactifications of , I. Math. Ann. 280, 129-146 (1988) · Zbl 0651.14025
[28] Peternell, T., Schneider, M.: Compactifications of : A Survey. Several Complex Variables and Complex Geometry. ed. E. Bedford et al. Proc.Symp. Pure Math.Amer. Math. Soc., Providence, RI 52, 455-466 (1991) · Zbl 0745.32012
[29] Peternell, T.: Compactifications of , II. Math. Ann. 283, 121-137 (1989) · Zbl 0671.14020
[30] Reid, M.: Minimal model of canonical 3-folds. Algebraic Varieties and Analytic Varieties (Advanced Studies in Pure Math.). Ed. S. Iitaka. Kinokuniya, Japan 1, 131-180 (1983)
[31] Reid, M.: Nonnormal del Pezzo surfaces. Publ. Res. Inst. Math. Sci. 30, 695-727 (1994) · Zbl 0867.14015
[32] Remmert, R., Van de Ven, T.: Zwei Sätze über die komplex-projektive Ebene. Nieuw Arch. Wisk. 8(3) 147-157 (1960) · Zbl 0136.20703
[33] Shin, K.-H.: 3-dimensional Fano varieties with canonical singularities. Tokyo J. Math. 12, 375-385 (1989) · Zbl 0708.14025
[34] Smith, P.A.: Fixed points of periodic transformations, appendix B in Algebraic Topology by Solomon Lefschetz. American Math. Sco. Coll. Publ. XXVII, (1942 )
[35] Takagi, H.: On the classification of ?Fano 3-folds of Gorenstein index 2. I, Nagoya Math. J. 167, 117-155 (2002) · Zbl 1048.14022
[36] Takagi, H.: On the classification of ?Fano 3-folds of Gorenstein index 2. II, Nagoya Math. J. 167, 157-216 (2002) · Zbl 1048.14023
[37] Tsuji, M.: Dimca hypersurfaces and Nagata automorphisms. Math. Nachr. 218, 175-184 (2000) · Zbl 0965.14018
[38] Umezu, Y.: Quartic surfaces of elliptic ruled type. Amer. Math. Soc. 283, 127-143 (1984) · Zbl 0575.14035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.