On singular complex surfaces with negative canonical bundle, with applications to singular compactifications ofC 2 and to 3-dimensional rational singularities

Abstract

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 ( t h e 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.