The Curvature of the Petersson-Weil Metric on the Moduli Space of Kähler-Einstein Manifolds

Abstract

The Petersson-Weil metric is a main tool for investigating the geometry of moduli spaces. When A. Weil considered the classical Teichmuller space from the viewpoint of deformation theory, he suggested, in 1958, investigating the Petersson inner product on the space of holomorphic quadratic differentials. He conjectured that it induced a Kahler metric on the Teichmuller space. After proving this property, Ahlfors showed, in 1961, that the holomorphic sectional and Ricci curvatures were negative. Royden’s conjecture of a precise upper bound for the holomorphic sectional curvature was proven by Wolpert and Tromba in 1986 along with the negativity of the sectional curvature.