On Deformations of Fano Manifolds

In this paper, we study deformations of Fano K\"ahler-Einstein manifolds and provide a new necessary and sufficient condition for the existence of K\"ahler-Einstein metrics on small deformations of a Fano K\"ahler-Einstein manifold. We also investigate under what condition is the Weil-Petersson metric on the moduli space of a Fano K\"ahler-Einstein manifold well-defined when its automorphisms group is non-discrete. Moreover, when the automorphism group of the central fiber is discrete, we are able to 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.