Compact Kähler surfaces with trivial canonical bundle (Q1811009)

In this paper, the author considers the classical conjectures of Weil and Andreotti on \(K3\) surfaces: The set of all \(K3\) surfaces constitutes one connected family; two \(K3\) surfaces defining the same period point are isomorphic, which is normally referred to as the Torelli theorem for \(K3\) surfaces; each point in the period domain is the period point of some \(K3\) surface; all \(K3\) surfaces admit Kähler metrics. A proof of the last result was given by the author [\textit{N. Buchdahl}, Ann. Inst. Fourier 49, No. 1, 287--302 (1999; Zbl 0926.32025)] and, independently, by \textit{A. Lamari} [Ann. Inst. Fourier 49, No. 1, 263--285 (1999; Zbl 0926.32026)], while for the other results detailed and complete proofs appear in the exposés of \textit{A. Beauville} [in: Géométrie des surfaces \(K3\); modules et périodes, Sém. Palaiseau 1981--1982, Astérisque 126 (1985; Zbl 0607.14007)] and the standard book of \textit{W. Barth, C. Peters} and \textit{A. van de Ven} [``Compact complex surfaces'' (Berlin 1984; Zbl 0718.14023)]. In this paper, the author gives new and short proofs of the above results, which are based on a generalization, given by the author, of the Nakai-Moishezon criterion for elements in \(H^{1,1}_{\mathbb R}(X)\) to be representable by positive closed \((1,1)\)-forms and a result of Matumoto and Donaldson about the group \(G\) of automorphisms of \(H^2(X,{\mathbb Z})\) induced by the (orientation-preserving) self-diffeomorphisms. Most of the proofs apply, with minor variation, to complex 2-tori.