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

• Published in: Transactions of the American Mathematical Society 2002-07, Vol.354 (7), p.2639-2658
• Main Author: Mok, Ngaiming
• Format: Article
• Language: English
• Subjects: A priori knowledge; Algebra; Algebraic geometry; Curvature; Differential geometry; Exact sciences and technology; Geometry; Hypersurfaces; Integers; Lie groups; Mathematical manifolds; Mathematical vectors; Mathematics; Sciences and techniques of general use; Symmetry; Tangents
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-02-02953-7

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 \subset G is a maximal parabolic subgroup. \par 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 \mathcal{C}_x, and (b) recovering the structure of a rational homogeneous manifold from \mathcal{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 \mathbb{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). \par The principal difficulty is part (a) of the scheme. We prove that \mathcal{C}_x \subset \mathbb{P}T_x(X) is a rational curve of degrees \leq 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = \mathbb{P}^2 resp. Q^3 resp. K(G_2). Let \mathcal{K} be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that \mathcal{K} is smooth. Furthermore, it implies that at \emph{any} point x \in X, the normalization \mathcal{K}_x of the corresponding Chow space of minimal rational curves marked at x is smooth. After proving that \mathcal{K}_x is a rational curve, our principal object of study is the universal family \mathcal{U} of \mathcal{K}, giving a double fibration \rho: \mathcal{U} \to \mathcal{K}, \mu: \mathcal{U} \to X, which gives \mathbb{P}^1-bundles. There is a rank-2 holomorphic vector bundle V on \mathcal{K} whose projectivization is isomorphic to \rho: \mathcal{U} \to \mathcal{K}. We prove that V is stable, and deduce the inequality d \leq 4 from the inequality c_1^2(V) \leq 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 speci