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Singular Del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space C3

Published online by Cambridge University Press:  22 January 2016

Mikio Furushima
Mikio Furushima*
Affiliation:
Kumamoto Radio Technical College, 2659-2, Suya, Nishigoshi-machi, Kikuchi-gun, Kumamoto 861-11, Japan
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Let X be an n-dimensional connected compact complex manifold and A be an analytic subset of X. We say that the pair (X, A) is a complex analytic compactification of Cn if X − A is biholomorphic to Cn. If X admits a Kähler metric, we shall say that (X, A) is a (non-singular) Kähler compactification of Cn. For n = 1, it is easy to see that (X, A) ≃ (P1, ∞). For n = 2, Remmert-Van de Ven [17] proved that (X, A) ≃ (P2, P1) if A is irreducible, where A = P1 is linearly embedded in P2. Morrow [15] gave more detailed classifications of complex analytic compactifications of C2 For n = 3, Brenton-Morrow showed the following

THEOREM ([5]). Let (X, A) be a non-singular Kähler complex analytic compactification of C3such that the analytic subset A has only isolated singular points. Then X is projective algebraic and A is birationally equivalent to a ruled surface over an algebraic curve of genus g = b3(X)/2.

Further, Brenton [3] classified the possible types of singular points of A in the case that the canonical line bundle KA of A is not trivial.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 104 , December 1986 , pp. 1 - 28
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[ 1 ] Bombieri, E. and Husemoller, D., Classifications and embeddings of surfaces, Algebraic Geometry, Arcata 1974, Amer. Math. Soc. Proc. Symp. Math., Providence, 29 (1975), 329–420.Google Scholar
[ 2 ] Brenton, L., Some algebraicity criteria for singular surfaces, Invent. Math., 4 (1977), 129–144.Google Scholar
[ 3 ] Brenton, L., On singular complex surfaces with negative canonical bundle, with applications to singular compactifications of C2 and to 3-dimensional rational singularities, Math. Ann., 248 (1980), 117–124.CrossRefGoogle Scholar
[ 4 ] Brenton, L., Drucker, D. and Prince, G. C. E., Graph theoretic techniques in algebraic geometry II: Construction of singular complex surfaces of rational cohomology type of CP2 Comment. Math. Helv., 56 (1981), 39–58.Google Scholar
[ 5 ] Brenton, L. and Morrow, J., Compactifications of Cn Trans. Amer. Math. Soc., 246 (1979), 139–158.Google Scholar
[ 6 ] Demazure, M., Del Pezzo, Surfaces de, Lecture Note, in Math., 777 (1980), 23–69, Springer-Verlag Berlin Heidelberg New York.Google Scholar
[ 7 ] Griffiths, P. and Harris, J., Principles of Algebraic Geometry, Pure and Applied Math., John Wiley and Sons, New York, 1978.Google Scholar
[ 8 ] Hartshorne, R., Geometry, Algebraic, Graduate Texts in math. 52, Springer-Verlag, New York Heidelberg Berlin, 1977.Google Scholar
[ 9 ] Hidaka, F. and Watanabe, K., Normal Gorenstein surfaces with ample anticanonical divisor, Tokyo J. Math., 4 (1981), 319–330.CrossRefGoogle Scholar
[10] Hirzebruch, F., Topological Methods in Algebraic Geometry, Grundlehren der math. Wissenschaften 131, Springer-Verlag Berlin Heidelberg New York, 1966.Google Scholar
[11] Iskovski, V. A., Fano 3-fold I, Math. U.S.S.R. Izvestija, 11 (1977), 485–527.Google Scholar
[11a] Iskovski, V. A., Anticanonical models of three-dimensional algebraic varieties, J. Soviet Math., 13–14 (1980), 745–814.Google Scholar
[12] Kodaira, K., On compact analytic surfaces II, Ann. Math., 77 (1963), 536–626.Google Scholar
[13] Kobayashi, S. and Ochiai, T., Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ., 13 (1973), 31–47.Google Scholar
[14] Laufer, H., Normal Two-Dimensional Singularities, Ann. Math., Studies 71, 1971.Google Scholar
[15] Morrow, J., Minimal normal compactifications of C2 , Proc. conf. of Complex Analysis, Rice Univ. Studies, 59 (1973), 97–112.Google Scholar
[16] Mumford, D., Varieties defined by quadratic equations, Questions on algebraic varieties, Edizioni Cremonese, Rome, 1970, 63–82.Google Scholar
[17] Remmert, R. and Ven, T. Van de, Zwei Satze über die komplex-projektive ebene, Niew Arch. Wisk., 8 (3) (1960), 147–157.Google Scholar
[18] Saint-Donat, B., Projective model of K3 surfaces, Amer. J. Math., 96 (1974), 602–639.Google Scholar
[19] Saito, K., Einfach-elliptische Singularitäten, Invent. Math., 23 (1974), 289–325.Google Scholar
[20] Wagreich, P., Elliptic singularities of surfaces, Amer. J. Math., 92 (1970), 419–454,Google Scholar