Polarized period map for generalized K3 surfaces and the moduli of Einstein metrics
This paper unifies and concludes previous works on its subject. \textit{D. R. Morrison} [in Classification of algebraic and analytic manifolds, Proc. Symp. Katata/Jap. 1982, Prog. Math. 39, 303-332 (1983; Zbl 0526.14025)] defined a generalized K3 surface to be a compact complex surface X which has at worst simple singular points and for which its minimal resolution is a usual K3 surface. He defined a marking and polarization for such surfaces and proved for them the surjectivity of the period mapping. In the present paper it is shown (theorem 3) that every polarization on X contains a Kähler metric in the sense of Fujiki-Moishezon [\textit{A. Fujiki}, Publ. Res. Inst. Math. Sci. 14, 1-52 (1978; Zbl 0409.32016)]; \textit{B. Moishezon} [in Manifolds, Proc. internat. Conf. Manifolds relat. Top. Topology, Tokyo 1973, 343-351 (1975; Zbl 0344.32018)]). It is shown also by Yau's method the existence of a Ricci-flat Einstein-Kähler orbifold metric in the same class (theorem 1). It follows (corollary 4) that the coarse moduli spaces for the isomorphism classes of polarized generalized K3 surfaces, or for such Kähler surfaces in the Fujiki- Moishezon sense, or for such Einstein-Kähler surfaces with volume 1, coincide with \([SO_ 0(3,19)/SO(2)\times SO(19)]/\Gamma\), where \(\Gamma\) is an automorphism group of a K3-lattice which acts properly discontinuously. It coincides also with the set of isomorphism classes of Ricci-flat K3-orbifolds, that are Ricci-flat orbifolds diffeomorphic to generalized K3 surfaces (theorem 7).- D. Morrison (loc. cit.) showed that the subset in this moduli space which corresponds to the singular surfaces is the countable union of submanifolds of real codimension 3. In the last section the previous results are applied to a new proof of the known inequality for the number of singular points on generalized K3 surfaces. There is given a precise description of the equality cases.
number of singular points on generalized K3 surfaces
scientific article; zbMATH DE number 4053797
scientific article