17 - Picard Group

Summary

As remarked by Zariski in [652], ‘The evaluation of [the Picard number] for a given surface presents in general grave difficulties’. This is still valid and to a lesser extent also for K3 surfaces. The Picard number or the finer invariant provided by the Néron– Severi lattice NS(X) is the most basic invariant of a K3 surface, from which one can often read off basic properties of X, e.g. whether X admits an elliptic fibration or is projective. Line bundles also play a distinguished role in the derived category D^b(X), as the easiest kind of spherical objects, and for the description of many other aspects of the geometry of X.

In this chapter we collect the most important results on the Picard group of a K3 surface. A number of results are sensitive to the ground field, whether it is algebraically closed or of characteristic zero. Accordingly, we first deal in Section 1 with the case of complex K3 surfaces, where the description of the Picard group reduces to Hodge theory, which nevertheless may be complicated to fully understand even for explicitly given K3 surfaces. Later, in Section 2, we switch to more algebraic aspects and finally to the Tate conjecture, the analogue of the Lefschetz theorem on (1, 1)-classes for finitely generated fields. In the latter two parts we often refer to Chapter 18 on Brauer groups. These two chapters are best read together.

Picard Groups of Complex K3 Surfaces

We start out with a few recollections concerning the Picard group of complex K3 surfaces.

1.1 For any K3 surface X, complex or algebraic over an arbitrary field, the Picard group Pic(X) is isomorphic to the Néron–Severi group NS(X). In other words, any line bundle L on a K3 surface X that is algebraically equivalent to the trivial line bundle is itself trivial; see Section 1.2. For projective K3 surfaces, a stronger statement holds: any numerically trivial line bundle is trivial. So, in this case

see Proposition 1.2.4. For complex non-algebraic K3 surfaces the last isomorphism does not hold in general; see Remark 1.3.4.

Let us now consider arbitrary complex K3 surfaces, projective or not. Then the Lefschetz theorem on (1, 1)-classes yields an isomorphism

As h^1,1(X) = 20, this in particular shows that

Moreover, as we shall see, every possible Picard number is attained by some complex K3 surface.