Deformation rigidity of Fano manifolds

A smooth projective variety $X$ is called rigid if any deformation of $X$ is isomorphic to itself. A first example is the projective space, but in general it is a subtle and difficult problem to prove the deformation rigidity. I'll report some recent progress in this problem. Part of this talk is based on joint works with Yifei Chen (Chinese Academy of Sciences) and Qifeng Li (Shandong University).