In this paper, we define and study the Weil–Petersson geometry. Under the framework of the Weil–Petersson geometry, we study the Weil–Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil–Petersson metric and the Ricci curvature of the Weil–Petersson metric for Calabi–Yau fourfold moduli. We also prove that the Hodge volume of the moduli space is finite. Finally, we proved that the curvature of the Hodge metric is bounded if the Hodge metric is complete and the dimension of the moduli space is 1.
AMS 2000 Mathematics subject classification: Primary 53A30. Secondary 32C16
