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

Abstract

Let X be a Fano manifold of Picard number 1 admitting a rational curve with trivial normal bundle and f : X′ → 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 : X′ → 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.