Compact Complex Surfaces

Abstract

Preface to the First EditionPar une belle matinee du mois de mai, une eIegante amazone parcourait, sur une superbe jument alezane, les allees fleuries de Bois de Boulogne.(A. Camus, La Peste)Early versions of parts of this work date back to the mid-sixties, when the third author started to write a book on surfaces. But for several reasons, in particular the appearance of SafareviC's book, he postponed the projects. It was revived about ten years later, when all three authors were in Leiden. It is impossible to cover in one book the vast and rapidly developing theory of surfaces. Choices have to be made, with respect to content as well as to presentation. We have chosen for a complex-analytic point of view; this distinguishes our text from most of the existing treatments. Relations with the case of characteristic p are not discussed. We hope to have succeeded in writing a readable book; a book that can be used by non-specialists. The specialist will find very little that is new to him anyhow.As to acknowledgements, the authors certainly have to thank the Koninklijke Shellprijs, awarded to the third author in 1964. The numerous contacts with colleagues from other countries made possible by that award have had a very favourable influence on this book. Our thanks are furthermore due to

Bihermitian metrics on Hopf surfaces

“…As we explained in Remark 1, the fact that (M, J) has a standard metric with respect to which the degree of the anti-canonical bundle is positive (see Proposition 2) implies that (M, J) belongs to the class VII of the Enriques-Kodaira classification [5]. [31] or [27] for a proof), (M, J) is either a minimal Hopf surface or an Inoue surface described in [21].…”

“…(M, J) is of Kodaira dimension −∞. This means that (M, J) belongs to the class VII of the Enriques-Kodaira classification [5]. In particular, b 1 (M ) = 1.…”

Reflection groups in algebraic geometry

“…The following results follow from the Enriques-Kodaira classification of complex algebraic surfaces (see [9]). (1) X ∼ = P 2 or there exists a regular map f : X → B to some nonsingular curve B whose fibres are isomorphic to P 1 .…”

On the Kähler Rank of Compact Complex Surfaces

“…-One has [1]. Denoting the sheaf of closed differential (1, 0)-forms by dO X and looking at the long exact cohomology sequence of…”

“…These are surfaces whose minimal models have positive second Betti number and admit a global spherical shell (see Definition 9) and are the only "known" compact complex surfaces not considered in [8]. (We refer the reader to [1] for the general theory of compact complex surfaces.) It turns out that the modified Kähler rank does not coincide with the Kähler rank in general.…”