Abstract. The prolongation g (k) of a linear Lie algebra g ⊂ gl(V ) plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras g ⊂ gl(V ) with non-zero prolongations.If g is the Lie algebra aut(Ŝ) of infinitesimal linear automorphisms of a projective variety S ⊂ PV , its prolongation g (k) is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation aut(Ŝ) (k) is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety S ⊂ PV with aut(Ŝ) (k) = 0, which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano.As an application, we show that when S is linearly normal and Sec(S) = PV , the blow-up Bl S (PV ) has the target rigidity property, i.e., any deformation of a surjective morphism f : Y → Bl S (PV ) comes from the automorphisms of Bl S (PV ).

For a uniruled projective manifold, we prove that a general rational curve of minimal degree through a general point is uniquely determined by its tangent vector. As applications, among other things we give a new proof, using no Lie theory, of our earlier result that a holomorphic map from a rational homogeneous space of Picard number 1 onto a projective manifold different from the projective space must be a biholomorphic map. §1. Introduction Let X be an irreducible uniruled projective variety. Let RatCurves n (X) be the normalized space of rational curves on X in the sense of [Ko]. For an irreducible component K of RatCurves n (X), let ρ : U → K and µ : U → X be the associated universal family morphisms. In other words, ρ is a P 1 -bundle over K and for α ∈ K, the corresponding rational curve in X is µ(ρ −1 (α)). An irreducible component K of RatCurves n (X) is a minimal component if µ is dominant and for a general point x ∈ X, µ −1 (x) is projective. Members of a minimal component are called minimal rational curves. For example, rational curves of minimal degree passing through a very general point of X are minimal rational curves. Here, 'very general' means that the point is chosen outside a countable union of subvarieties of dimension strictly smaller than the dimension of X. Denote by PT (X) the projectivization of the tangent bundle of the smooth part of X. Given a minimal component K, consider the rational mapfor α ∈ U such that x := µ(α) is a smooth point of X and ρ(α) corresponds to a rational curve C on X smooth at x. Let C ⊂ PT (X) be the proper image of τ . We call τ the tangent map of K and C the total variety of minimal rational tangents of K.

No abstract

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f : M → X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.

No abstract