On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents

Abstract

Let X be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, X should be biholomorphic to a rational homogeneous manifold G/P, where G is a simple Lie group, and P ⊂ G is a maximal parabolic subgroup. In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents C x , and (b) recovering the structure of a rational homogeneous manifold from C x . The author proves that, when b 4 (X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane P 2 , the 3-dimensional hyperquadric Q 3 , or the 5-dimensional Fano homogeneous contact manifold of type G 2 , to be denoted by K(G 2 ). The principal difficulty is part (a) of the scheme. We prove that C x C PT x (X) is a rational curve of degrees < 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = P 2 resp. Q 3 resp. K(G 2 ). Let κ be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that κ is smooth. Furthermore, it implies that at any point x ∈ X, the normalization κ x of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that κ x is a rational curve, our principal object of study is the universal family u of κ, giving a double fibration p: u → κ, μ: u → X, which gives P 1 -bundles. There is a rank-2 holomorphic vector bundle V on κ whose projectivization is isomorphic to p: u → κ. We prove that V is stable, and deduce the inequality d < 4 from the inequality c 2 1 (V) < 4c 2 (V) resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of d = 4 is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on V in the special case where c 2 1 (V) = 4c 2 (V).