Compact Analytic Surfaces Research Articles

On Weakly 1-Complete Surfaces without Non-Constant Holomorphic Functions

In this paper, we are interested in a class of two dimensional complex manifolds which are called weakly 1-complete surfaces. Here we call a two dimensional complex manifold X a weakly 1-complete surface if X possesses a C°°-exhausting plurisubharmonic function. This class includes two different extreme objects: compact analytic surfaces and two dimensional Stein manifolds. But at the same time, this class includes some curious examples from the function theoretic point of view i.e. there are weakly 1-complete surfaces without non-constant holomorphic functions (see [3] [6] [8] [12]) and moreover non-compact weakly 1-complete surfaces have an extreme function theoretic property i.e. a non-compact weakly 1-complete surface X is holomorphically convex if and only if X possesses a non-constant holomorphic function (see [9]). Looking back to the case of compact analytic surfaces, roughly speaking, they are classified by the existence or non-existence of meromorphic function. Hence it is natural to suppose that this aspect might give a new standpoint to analyze such a curious example in the class of weakly 1-complete surfaces as far as weakly 1-completeness is expected as a nice intermediate concept between compactness and Stein. This note is an attempt towards the problem of the existence of meromorphic function on non-compact weakly 1-complete surfaces. From now on, all weakly 1-complete surfaces are connected and non-compact and have no exceptional compact curves of the first kind unless otherwise is explicitly stated. Then we shall prove the following theorem.

Some examples of singular compact analytic surfaces which are homotopy equivalent to the complex projective plane

IN THEIR fundamental work[S] on analytic structures on the complex projective spaces Hirzebruch and Kodaira show in particular the following result in dimension two: If X is a compact complex Klhler manifold homeomorphic to the complex projective plane p = p(C) and with negative canonical line bundle, then X is analytically isomorphic to p. Indeed, it is easy to check that more is true (Proposition 4, Corollary 5 below): If X is a 2-(complex)-dimensional compact complex manifold with H,(X, Z) = 0 and second Betti number b*(X) = 1, then X is projective algebraic and II*(X, Z) is generated (over Z) by the Chern class of a positive (ample) line bundle L. If L admits a non-trivial section (which will be the case if the canonical bundle KX is negatiue) then X is biholomotphic to p. The purpose of this paper is to show by example that nevertheless there exist singular compact 2-dimensional complex spaces X which are homotopy equivalent with p, with H*(X, Z) = H*(p, Z) generated by the Chern class of the bundle of a positively embedded divisor P, and even with negative “canonical” bundle; and secondly, to give a uniqueness result in the form of an algorithm for constructing such spaces. In particular we show (Theorem 6 below): Let X be a 2-dimensional compact complex analytic space each of whose singular points is an isolated rational double point. Suppose that II*(X, Z) is isomorphic to Z[t]/(t3) and is generated by the Chem class of the line bundle of a holomorphic divisor T. Then X is biholomorphic either to p or to a singular rational projective algebraic surface obtained from p by the successive application of precisely 8 monoidal transformations followed by the collapsing of a curve with precisely 8 analytic components, each a non-singular rational curve with self-intersection 2, to 1 or more singular points. Every such space X is homotopy equivalent to p, and the generator T may always be taken to be the divisor of a non-singular elliptic curve contained in the regular points of X.

Approximation by interpolation of certain real compact analytic surfaces

Approximation by interpolation of certain real compact analytic surfaces

The Theorem of Riemann-Roch on Compact Analytic Surfaces

The Theorem of Riemann-Roch on Compact Analytic Surfaces