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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents
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by Ngaiming Mok PDF
Trans. Amer. Math. Soc. 354 (2002), 2639-2658 Request permission

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 $c_1^2(V) = 4c_2(V)$.

References
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Additional Information
  • Ngaiming Mok
  • Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
  • MR Author ID: 191186
  • Received by editor(s): December 31, 2000
  • Published electronically: February 4, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2639-2658
  • MSC (2000): Primary 14J60, 53C07
  • DOI: https://doi.org/10.1090/S0002-9947-02-02953-7
  • MathSciNet review: 1895197