A long-standing question in the theory of complex surfaces is: what are all compactifications of ~2 ; i.e., what are all compact complex manifolds of complex dimension two which contain an analytic subset A such that M-A is biholomorphic to the space ~2 of two complex variables? Recently Morrow [3"1 succeeded in classifying all minimal normal compactifications of IE 2, which answers the question completely in the case where the analytic set A has only "nice" singularities.In this note we use Morrows classification to prove a result on the possible singularities of A in general. As a corollary (Corollary 3 below) we have a new proof of the result of Remmert and T. Van de Ven [5] that complex projective space ]p2 is the only compactification of ¢2 with second Betti number equal to one.Definition. Let M be a compactification of C 2, M-C 2 =A. Let A = 0 F~ be the decomposition of A into its .irreducible branches.i=l
IfX is a compact complex manifold of complex dimension two, denote by K x the canonical line bundle (the second exterior power of the cotangent bundle) of X. It is immediate from the classification of surfaces [15,5] that if K x is negative then X is either the complex projective plane IP 2, or the product tP ~ x tP 1 of two projective lines (-the non-singular quadric surface Q2C IP3), or is derived from one of these by blowing up points. Here a vector bundle E on a complex space X (singularities allowed) is negative if the zero section X o is exceptional in E. Now if (X, (gx) is an analytic surface with singular points, then if each singularity is Cohen-Macaulay (the homological codimension of each stalk (gx, x equals the dimension of X = 2; this is equivalent to normal (integrally closed) in this dimension) then at least the canonical sheaf ~x is defined and Serre-Grothendieck duality holds. And if each point is Gorenstein ((gx, x has finite injective dimension) then J{x is locally trivial and so it is the sheaf of sections of a holomorphic line bundle Kx, the canonical bundle of the singular space X. The purpose of this note is to expose all compact Gorenstein surfaces (compact two-dimensional complex spaces with only Gorenstein singularities) for which K x is negative. Theorem. Let X be a compact Gorenstein surface with negative canonical bundle K x •Then either (A) X=~ '2, (B) X = the non-singular quadric hypersurface ff~2 C tP 3, (C) X = the sinoular quadric hypersurface ff)~ C IP 3, (D) X is the space obtained by blowin9 down the zero section of a IP~-bundle on a non-singular elliptic curve F, or (E) X is a rational surface with only rational double points as singularities, obtained from IP 2 by blowin9 up some number ~ < 8 points (iterations allowed) then btowin9 down some number fl <~ non-sinouIar rational curves, each with selfintersection -2.
By a (complex analytic) surface we shall mean a reduced irreducible twodimensional complex space (X, ~x), with or without singularities. We want to know when such a surface is projective algebraic.Suppose X is non-singular. Then we have at our disposal the classification of two-dimensional compact complex manifolds due principally to Kodaira ([8]). Indeed, Kodaira's classification is substantially based upon certain criteria for algebraicity involving such numerical invariants as the Betti numbers, the irregularity and geometric genus, and the transcendence degree of the meromorphic function field. Since these invariants are equally natural and often equally accessible in the case of compact spaces with singular points it is natural to ask how far the ideas of the classification theory extend to the larger class of all (possibly singular) complex surfaces. On the positive side we have, for instance, Grauert's important extension of the result of Kodaira that a Hodge manifold is algebraic: . 343). A compact analytic space admitting a positive (= weakly positive) holomorphic line bundle is projective algebraic:-and a theorem of Artin which can be rephrased as an extension to certain singular surfaces of the Chow-Kodaira criterion that a non-singular Moisheson surface is algebraic: On the other hand there are examples, the first due to Grauert ([6], p. 366) which show that the Chow-Kodaira criterion fails in general for arbitrary singular surfaces.In this paper we use Grauert's theorem quoted above together with the classification theory applied to the non-singular model to obtain numerical criteria for algebraicity of a singular surface. We will use the classical notation for the standard topological and analytic invariants of a compact surface X: b t
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