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

Abstract

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.