AUGMENTED WEIL-PETERSSON METRICS ON MODULI SPACES OF POLARIZED RICCI-FLAT KAHLER MANIFOLDS AND ORBIFOLDS

We show that the base complex manifold of an effectively parametrized family of compact polarized Ricci-flat KAhler orbifolds, and in particular manifolds, admits a smooth augmented Weil-Petersson metric whose holomorphic sectional curvature is bounded above by a negative constant. As a consequence, we conclude that such base manifold is Kobayashi hyperbolic.