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Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents

Jun-Muk Hwang — 2010

Annales scientifiques de l'École Normale Supérieure

We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety Z , a family of minimal rational curves with Z -isotrivial varieties of minimal rational tangents...

Varieties of minimal rational tangents of codimension 1

Jun-Muk Hwang — 2013

Annales scientifiques de l'École Normale Supérieure

Let  X be a uniruled projective manifold and let  x be a general point. The main result of [2] says that if the ( - K X ) -degrees (i.e., the degrees with respect to the anti-canonical bundle of  X ) of all rational curves through x are at least dim X + 1 , then X is a projective space. In this paper, we study the structure of  X when the ( - K X ) -degrees of all rational curves through x are at least dim X . Our study uses the projective variety 𝒞 x T x ( X ) , called the VMRT at  x , defined as the union of tangent directions to the rational curves...

Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1

Jun-Muk Hwang — 2007

Annales de l’institut Fourier

Let X be a Fano manifold with b 2 = 1 different from the projective space such that any two surfaces in X have proportional fundamental classes in H 4 ( X , C ) . Let f : Y X be a surjective holomorphic map from a projective variety Y . We show that all deformations of f with Y and X fixed, come from automorphisms of X . The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of X .

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