On Fano manifolds with Nef tangent bundles admitting 1dimensional varieties of minimal rational tangents
Author:
Ngaiming Mok
Journal:
Trans. Amer. Math. Soc. 354 (2002), 26392658
MSC (2000):
Primary 14J60, 53C07
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 CampanaPeternell'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 CampanaPeternell 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 JunMuk 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 1dimensional, is biholomorphic to the projective plane , the 3dimensional hyperquadric , or the 5dimensional 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 rank2 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 HermitianEinstein metrics. The case of is ruled out by studying the structure of the curvature tensor of the HermitianEinstein metric on in the special case where .
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 Hwang, J.M. and Mok, N.: CartanFubini type extension of holomorphic maps for Fano manifolds with Picard number 1, J. Math. Pures et Appliquées (9) 80 (2001), 563575.
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Additional Information
Ngaiming Mok
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
DOI:
http://dx.doi.org/10.1090/S0002994702029537
PII:
S 00029947(02)029537
Received by editor(s):
December 31, 2000
Published electronically:
February 4, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
