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 $P \subset G$ 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 $\mathcal {C}_x$, 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 $K(G_2)$.

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 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 special case where $c_1^2(V) = 4c_2(V)$.