Compact Kähler surfaces with trivial canonical bundle

Abstract

The classical conjectures of Weil on K3 surfaces – that the set of such surfaces is connected; that a version of the Torelli theorem holds; that each such surface is Kähler; and that the period map is surjective – are reconsidered in the light of a generalisation of the Nakai-Moishezon criterion, and short proofs of all the conjectures are given. Most of the proofs apply equally or with minor variation to complex 2-tori, the only other compact Kähler surfaces with trivial canonical bundle.