Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles

Let $X$ be a Fano manifold of Picard number 1 admitting a rational curve with trivial normal bundle and $f\colon X’\to X$ be a generically finite surjective holomorphic map from a projective manifold $X’$ onto $X$. When the domain manifold $X’$ is fixed and the target manifold $X$ is a priori allowed to deform we prove that the holomorphic map $f\colon X’\to X$ is locally rigid up to biholomorphisms of target manifolds. This result complements, with a completely different method of proof, an earlier local rigidity theorem of ours (see J. Math. Pures Appl. 80 (2001), 563–575) for the analogous situation where the target manifold $X$ is a Fano manifold of Picard number $1$ on which there is no rational curve with trivial normal bundle. In another direction, given a Fano manifold $X’$ of Picard number $1$, we prove a finiteness result for generically finite surjective holomorphic maps of $X’$ onto Fano manifolds (necessarily of Picard number $1$) admitting rational curves with trivial normal bundles. As a consequence, any $3$-dimensional Fano manifold of Picard number $1$ can only dominate a finite number of isomorphism classes of projective manifolds.