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

Author:
Ngaiming Mok

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

Published electronically:
February 4, 2002

MathSciNet review:
1895197

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Abstract: Let 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, should be biholomorphic to a rational homogeneous manifold , where is a simple Lie group, and 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 , and (b) recovering the structure of a rational homogeneous manifold from . The author proves that, when and the generic variety of minimal rational tangents is 1-dimensional, is biholomorphic to the projective plane , the 3-dimensional hyperquadric , or the 5-dimensional Fano homogeneous contact manifold of type , to be denoted by .

The principal difficulty is part (a) of the scheme. We prove that is a rational curve of degrees , and show that resp. 2 resp. 3 corresponds precisely to the cases of resp. resp. . Let be the normalization of a choice of a Chow component of minimal rational curves on . Nefness of the tangent bundle implies that is smooth. Furthermore, it implies that at *any* point , the normalization of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that is a rational curve, our principal object of study is the universal family of , giving a double fibration , which gives -bundles. There is a rank-2 holomorphic vector bundle on whose projectivization is isomorphic to . We prove that is stable, and deduce the inequality from the inequality resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on in the special case where .

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Additional Information

**Ngaiming Mok**

Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

DOI:
https://doi.org/10.1090/S0002-9947-02-02953-7

Received by editor(s):
December 31, 2000

Published electronically:
February 4, 2002

Article copyright:
© Copyright 2002
American Mathematical Society