Page 406 -

§11. Introduction to K3 Surfaces 381

(11.8) Remark. For algebraic K3 surfaces 1 ≤ ρ(X) ≤ 20. In the case of the
4
4
4
4
Fermat surface 0 = y + y + y + y we have ρ = 20 while ρ = 1 for the generic
0 1 2 3
quadric surface. For generic complex K3 surfaces we have ρ = 0. The algebraic K3
surfaces have hyperplane sections, hence there exist curves on the surface, but they
are all homologous.
(11.9) Remark. Every complex K3 surface has a K¨ ahler form. The Hodge data and
the K¨ ahler form can be used to parameterize complex K3 surfaces. This is the Torelli
theorem, and for further details see the book by Barth, Peters, and van de Ven [1980],
Compact complex surfaces, Springer-Verlag, Ergebnisse der Mathematik und ihrer
Grenzgebiete, 3 Folge, Band 4.