Deformation of morphisms onto Fano manifolds of Picard number 1 with linear varieties of minimal rational tangents

Let X be a Fano manifold of Picard number 1, different from projective space. We study the question whether the space Hom(s)(Y, X) of surjective morphisms from a projective manifold Y to X is homogeneous under the automorphism group Aut(o)(X). An affirmative answer is given in [4] under the assumption that X has a minimal dominating family K of rational curves whose variety of minimal rational tangents C-x at a general point x is an element of X is non-linear or finite. In this paper, we study the case where C-x is linear of arbitrary dimension, which covers the cases unsettled in [4]. In this case, we will define a reduced divisor B-K subset of X and an irreducible subvariety M-K subset of Chow(X) naturally associated to K. We give a sufficient condition in terms of B-K and M-K for the homogeneity of Hom(s)(Y, X). This condition is satisfied if C-x is finite and our result generalizes [4]. A new ingredient, which is of independent interest, is a similar rigidity result for surjective morphisms to projective space in logarithmic setting.