Some algebraicity criteria for singular surfaces

Abstract

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

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

“…By Proposition l- (4) and the fact that 6 + (M) = 6 T (A) = 1 by Proposition l- (2). Thus from the classification of surfaces [1], M is either P 2 or the total space of a P 1 -bundle on a non-singular algebraic curve.…”

A Harnack estimate for real normal surface singularities

“…An N-curve is a projective curve in which the irreducible components are nonsingular, intersect transversely, and no three components intersect. The topology of an TV-curve is completely determined by the topology of the irreducible components and the dual graph, as in the following result, which is easily proved by a MayerVietoris argument (or see Brenton [2]). For homology we will always use Z 2 coefficients.…”

“…Let V c C 3 be the cone over A and let (F R ,0) be the germ at 0 of the real part of V. Then (F R ,0) is connected, but the punctured variety (V R \ {0}, 0) may have two components for each connected component of A R . Thus the number of components of (F R \ (0},0) is bounded by 2 …”

Algebraicity of normal analytic compactifications of ℂ2 with one irreducible curve at infinity

“…The only previously known effective criteria for determining the algebraicity of contraction of curves on surfaces was the well-known criteria of Artin [Art62] which states that a normal surface is algebraic if all its singularities are rational. We refer the reader to [MR75], [Bre77], [FL99], [Sch00], [Bȃd01], [Pal12] for other criteria -some of these are more general, but none is effective in the above sense. Moreover, as opposed to Artin's criterion, ours is not numerical, i.e.…”