Completion of the moduli space for polarized Calabi–Yau manifolds

Abstract

In this paper, we construct a completion of the moduli space for polarized Calabi–Yau manifolds by using Ricci-flat Kähler–Einstein metrics and the Gromov–Hausdorff topology, which parameterizes certain Calabi–Yau varieties. We then study the algebro-geometric properties and the Weil–Petersson geometry of such completion. We show that the completion can be exhausted by sequences of quasi-projective varieties, and new points added have finite Weil–Petersson distance to the interior.