K3-Surfaces and Enriques Surfaces

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).