Compact Complex Surfaces

Frontmatter

Introduction

Abstract

This book is mainly concerned with the classification of smooth compact complex surfaces, i.e. of compact 2-dimensional complex manifolds, which in this introduction we shall always assume to be connected*).

W. Barth, C. Peters, A. Van de Ven

Standard Notations

W. Barth, C. Peters, A. Van de Ven

I. Preliminaries

Abstract

We shall use the standard notations for some algebraic structures which frequently occur. These notations are listed on p. 9.

W. Barth, C. Peters, A. Van de Ven

II. Curves on Surfaces

Abstract

Unless stated otherwise, X denotes in this chapter a 2-dimensional complex manifold, not necessarily compact or connected.

W. Barth, C. Peters, A. Van de Ven

III. Mappings of Surfaces

Abstract

In this chapter a surface is a reduced 2-dimensional complex space, unless specified otherwise. In this respect we draw in particular attention to the convention valid for Sects. 8–18.

W. Barth, C. Peters, A. Van de Ven

IV. Some General Properties of Surfaces

Abstract

In this chapter a surface is always a connected 2-dimensional complex manifold.

W. Barth, C. Peters, A. Van de Ven

V. Examples

Abstract

In this chapter a surface will again mean a connected 2-dimensional complex manifold.

W. Barth, C. Peters, A. Van de Ven

VI. The Enriques-Kodaira Classification

Abstract

In this chapter a surface will be a compact, connected 2-dimensional complex manifold. As defined in II, Sect. 1, a curve on a surface is always a closed 1-dimensional subvariety, locally given by one equation (essentially, an effective divisor).

W. Barth, C. Peters, A. Van de Ven

VII. Surfaces of General Type

Abstract

The minimal surfaces of general type can be parametrised in a satisfactory way, namely by a countable number of quasi-projective families.More precisely we have Gieseker’s theorem([Gi]), p.236):

In this chapter the use of the words “surface” and “curve” is the same as in Chap. VI.

W. Barth, C. Peters, A. Van de Ven

VIII. K3-Surfaces and Enriques Surfaces

Abstract

What is said about the use of “surface” and “curve” (on a surface) at the beginning of Chap. VI also applies in this chapter.

W. Barth, C. Peters, A. Van de Ven

Backmatter