On deformations of Fano manifolds

In this paper, we provide new necessary and sufficient conditions for the existence of Kahler–Einstein metrics on small deformations of a Fano Kahler–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 Kahler–Einstein manifolds of general type.