On Deformations of Fano Manifolds

In this paper we provide new necessary and sufficient conditions for the existence of K"ahler-Einstein metrics on small deformations of a Fano K"ahler-Einstein manifold. We also show that the Weil-Petersson metric can be approximated by the Ricci curvatures of the canonical $L^2$ metrics on the direct image bundles. In addition, we describe the plurisubharmonicity of the energy functional of harmonic maps on the Kuranishi space of the deformation of compact K"ahler-Einstein manifolds of general type.