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 i 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 Cx, and (b) recovering the structure of a rational homogeneous manifold from Cx. The author proves that, when b4(X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane ℙ2, the 3-dimensional hyperquadric Q3, or the 5-dimensional Fano homogeneous contact manifold of type G2, to be denoted by K(G2). The principal difficulty is part (a) of the scheme. We prove that Cx ⊂ ℙTx(X) is a rational curve of degrees ≤ 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = ℙ2 resp. Q3 resp. K(G2). Let K be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that K is smooth. Furthermore, it implies that at any point x ∈ X, the normalization Kx of the corresponding Chow space of minimal rational curves marked at x is smooth. After proving that Kx is a rational curve, our principal object of study is the universal family U of K, giving a double fibration ρ : U → K, μ : U → X, which gives ℙ1-bundles. There is a rank-2 holomorphic vector bundle V on K whose projectivization is isomorphic to ρ : U → K. We prove that V is stable, and deduce the inequality d ≤ 4 from the inequality c1 2(V) ≤ 4c2(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 c1 2(V) = 4c2(V).