Hyperbolicity of Moduli Spaces of Log-Canonically Polarized Manifolds

Abstract

In recent years, there are quite a lot of interests and results related to hyperbolicity properties of the base spaces of various families of projective algebraic varieties. Not much is known for families of higher dimensional quasi-projective varieties. The goal of this paper is address the problem for the case of an effectively parametrized family of log-canonically polarized manifolds. We construct a Finsler metric on the base manifold of such a family with the property that its holomorphic sectional curvature is bounded from above by a negative constant, and as a consequence, we deduce the Kobayashi hyperbolicity of the base manifold. The method relies on developing analytic tools to investigate geometry of families of quasi-projective manifolds equipped with Kahler–Einstein metrics, which leads to an appropriate modification of the Weil–Petersson metric on the base manifold.