Compact Complex Surfaces

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

In this sub-chapter we collect some basic facts from topology and algebra. Among the less elementary facts needed later on we mention the topological index theorem recalled in §3. The (elementary) facts concerning quadratic forms will be used mainly in the chapter on K 3- and Enriques-surfaces.

In this sub-chapter two basic technical themes are discussed. The first centres around the Riemann-Roch and Serre duality theorems for any compact curve on a surface and is treated in Sects. 1–6. The second theme, developed in Sects. 7–8 is embedded resolution of singularities of curves and the application to the so-called simple singularities of curves.

In the framework of analytic geometry bimeromorphic maps play the same role as birational maps in algebraic geometry. The first aspect concerning these is the process of desingularization. We have seen (Sect. I.8) that the normalization can be seen as a first step in making the singularities less complicated. So we may assume that the surfaces under consideration are normal. For these there is indeed a desingularization as shown in Sect. 6. The proof is by reduction to the case of Hirzebruch-Jung singularities treated in Sect. 5. The existence of a minimal resolution is very particular for dimension 2 and depends on the fact that one can contract a (−1)-curve, i.e., a smooth rational curve with self intersection − 1, to a smooth point. This and its applications to the existence of minimal models is treated in Sect. 4. Curves contractible to (singular) points are treated more generally in Sect. 2.

In the fundamental Sect. 1 we treat meromorphic maps induced by line bundles. Then, in Sect. 2, we deal with special features for differential forms on compact surfaces. The main point is that for surfaces 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 analytical invariants which are crucial in handling non-algebraic surfaces. For instance, in Sect. 3 we give a direct proof (due to Lamari) that a compact surface is Kähler if and only if its first Betti number is even.

A universally known and in many ways most basic compact complex surface is the projective plane. Some obvious questions concerning ℙ₂ have fascinated many a geometer, in particular the question (raised by Severi in [Sev]) whether a surface which is homeomorphic to ℙ₂, is also isomorphic to ℙ₂. The last and very difficult step towards the affirmative answer was done only in the late 1970s by S. - T. Yau. The striking point is that the only known proof uses analysis (hidden in Riemann-Roch for non-algebraic surfaces) and differential geometry as well as the methods of analytic and algebraic geometry.

In this chapter we deal with the Enriques-Kodaira classification. In the first edition of this book the case (2,1) of Iitaka’s conjecture (see the Introduction) was used in the proof of Theorem 1.1. In the meantime it has become customary to give a slightly different proof for the classification theorem. The new proof, a form of which is presented below, rests on a systematic use of nef divisors (IV. Sect. 7). A central role is played by the Rationality Theorem which we prove first. We then show how the full classification of surfaces with K _x not nef follows in an astonishingly simple way, and deduce Castelnuovo’s criterion as a corollary. After all this the classification of minimal algebraic surfaces becomes quite easy. The classification of non-algebraic surfaces which follows next, is the same as in the first edition of the book. In Sect. 8 we prove Iitaka’s results concerning the stability of the ten classes under deformations.

In the first section we state the restrictions on (c ² ₁, c ₂) for minimal surfaces of general type. In the section following it, the “easy” inequalities c² ₁ > 0 and c₂ > 0 are proven and we show that there is only a finite number of (−2)-curves.

In this chapter we consider in detail the class of K 3-surfaces and that of Enriques surfaces. We start with some notation and after that we state the main results in Sect. 2. In Chapt. IV, Sect. 3 we saw that K 3- surfaces are Kähler, a fact we use from the start. The main tool for studying moduli of K 3- surfaces is the period map and we describe these moduli spaces in terms of the corresponding period domains. This is done in Sect. 6–14 after we have proved some general facts concerning the geometry of divisors on K 3-surfaces and Kummer surfaces, collected in Sect. 3–5. The geometry of Enriques surfaces as discussed in Sect. 15–18 is then coupled with a study of the period map of their universal covers in order to arrive at a description of the moduli space in terms of certain classical bounded domains. See Sect. 19–21. We finish this chapter with more recent results for projective K 3-surfaces. First, we consider their moduli spaces. After this we discuss the construction of mirror families for K 3-surfaces. Next we present Mumford’s proof that every K 3-surface contains a (possibly singular) rational curve and a 1-dimensional algebraic family of (in general singular) elliptic curves. Then we discuss enumerative results for rational curves and we finish with an application to hyperbolic geometry (related to the Green-Griffiths and Lang conjectures).

In the first subchapter we review Freedman’s topological classification of 4-manifolds as well as the 11/8 conjecture which predicts which unimodular forms can be represented by smooth 4-folds and we discuss the implications for compact complex surfaces.