The classification of compact, complex surfaces amounts to the classification of minimal surfaces. This is first of all a classification according to Kodaira dimension. A refinement of this very coarse classification is the Enriques-Kodaira classification, a description of which is a first purpose of this book. Apart from the Enriques-Kodaira classification, this book is mainly devoted to a deeper study of some of these classes, namely K3-surfaces, Enriques surfaces and surfaces of general type.
The content of the different chapters is as follows.
In chapter I we collect - practically without proofs - most of the definitions and results from topology, algebra, differential geometry, analytic geometry and algebraic geometry which we shall need.
Chapter II is devoted to (possibly non-reduced) curves on (not necessarily compact) surfaces: dualising sheaf, Picard variety, singularities and their resolution. The Riemann-Roch theorem is reduced to the smooth case and Serre duality derived from the reduced projective case. Simple curve singularities are classified. Analytic intersection numbers are defined for divisors and shown to be the same as the topological ones.
The first part of chapter III deals with surface singularities, their resolution and the converse of this process, the blowing down of exceptional curves. The results are applied to study bimeromorphic maps and minimal models. Rational double points and their relations with simple curve singularities are treated with care. The second part of chapter III is devoted to (proper) curve fibrations of surfaces over curves. The main achievement here is a proof of Iitaka’s conjecture \(C_{2,1}\) about the Kodaira dimension of such fibrations. We base it on properties of the period map for stable curves, the Satake compactification and the Torelli theorem for curves.
Chapter IV is not very homogeneous. We have collected in this chapter several general theorems about surfaces which will play an important role later on in the book. The first sections deal with special features of the transcendental theory (differential forms) on compact surfaces. The main point is that for a compact surface the Fröhlicher spectral sequence always degenerates. Combining the consequences of this fact with the topological index theorem we find, following Kodaira, relations between topological and analytic invariants which are crucial in handling non-algebraic surfaces. We also prove the important signature theorem (known as algebraic index theorem in the case of algebraic surfaces). From the other subjects treated in this chapter we mention projectivity criteria (with an application to almost-complex surfaces without any complex structure) and the vanishing theorems of Ramanujam and Mumford. As to chapter V (examples), we have included this chapter as a preparation for the next one.
In chapter VI we present the Enriques-Kodaira classification. At the end we apply classification to deformations of surfaces.
Chapter VII is about surfaces of general type.
Chapter VIII deals with K3-surfaces and Enriques surfaces. We fully prove the Torelli theorem for marked K3-surfaces, the surjectivity of the period map for K3-surfaces, and the bijectivity of the period map for Enriques surfaces.