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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

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