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Article: On Fano manifolds with NEF tangent bundles admitting 1-dimensional varieties of minimal rational tangents
Title | On Fano manifolds with NEF tangent bundles admitting 1-dimensional varieties of minimal rational tangents |
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Authors | |
Keywords | Mathematics |
Issue Date | 2002 |
Publisher | American Mathematical Society. |
Citation | Transactions Of The American Mathematical Society, 2002, v. 354 n. 7, p. 2639-2658 How to Cite? |
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 ⊂ 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 i 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 Cx, and (b) recovering the structure of a rational homogeneous manifold from Cx. The author proves that, when b4(X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane ℙ2, the 3-dimensional hyperquadric Q3, or the 5-dimensional Fano homogeneous contact manifold of type G2, to be denoted by K(G2). The principal difficulty is part (a) of the scheme. We prove that Cx ⊂ ℙTx(X) is a rational curve of degrees ≤ 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = ℙ2 resp. Q3 resp. K(G2). Let K be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that K is smooth. Furthermore, it implies that at any point x ∈ X, the normalization Kx of the corresponding Chow space of minimal rational curves marked at x is smooth. After proving that Kx is a rational curve, our principal object of study is the universal family U of K, giving a double fibration ρ : U → K, μ : U → X, which gives ℙ1-bundles. There is a rank-2 holomorphic vector bundle V on K whose projectivization is isomorphic to ρ : U → K. We prove that V is stable, and deduce the inequality d ≤ 4 from the inequality c1 2(V) ≤ 4c2(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 c1 2(V) = 4c2(V). |
Persistent Identifier | http://hdl.handle.net/10722/48402 |
ISSN | 2015 Impact Factor: 1.196 2015 SCImago Journal Rankings: 2.168 |
ISI Accession Number ID | |
References |
DC Field | Value | Language |
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dc.contributor.author | Mok, N | en_HK |
dc.date.accessioned | 2008-05-22T04:11:47Z | - |
dc.date.available | 2008-05-22T04:11:47Z | - |
dc.date.issued | 2002 | en_HK |
dc.identifier.citation | Transactions Of The American Mathematical Society, 2002, v. 354 n. 7, p. 2639-2658 | en_HK |
dc.identifier.issn | 0002-9947 | en_HK |
dc.identifier.uri | http://hdl.handle.net/10722/48402 | - |
dc.description.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 ⊂ 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 i 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 Cx, and (b) recovering the structure of a rational homogeneous manifold from Cx. The author proves that, when b4(X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane ℙ2, the 3-dimensional hyperquadric Q3, or the 5-dimensional Fano homogeneous contact manifold of type G2, to be denoted by K(G2). The principal difficulty is part (a) of the scheme. We prove that Cx ⊂ ℙTx(X) is a rational curve of degrees ≤ 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = ℙ2 resp. Q3 resp. K(G2). Let K be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that K is smooth. Furthermore, it implies that at any point x ∈ X, the normalization Kx of the corresponding Chow space of minimal rational curves marked at x is smooth. After proving that Kx is a rational curve, our principal object of study is the universal family U of K, giving a double fibration ρ : U → K, μ : U → X, which gives ℙ1-bundles. There is a rank-2 holomorphic vector bundle V on K whose projectivization is isomorphic to ρ : U → K. We prove that V is stable, and deduce the inequality d ≤ 4 from the inequality c1 2(V) ≤ 4c2(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 c1 2(V) = 4c2(V). | en_HK |
dc.format.extent | 288178 bytes | - |
dc.format.extent | 489302 bytes | - |
dc.format.mimetype | application/pdf | - |
dc.format.mimetype | application/pdf | - |
dc.language | eng | en_HK |
dc.publisher | American Mathematical Society. | en_HK |
dc.relation.ispartof | Transactions of the American Mathematical Society | en_HK |
dc.rights | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | - |
dc.rights | Transactions of the American Mathematical Society. Copyright © American Mathematical Society. | en_HK |
dc.rights | First published in Transactions of the American Mathematical Society, 2002, v. 354, p. 2639-2658, published by the American Mathematical Society, | en_HK |
dc.subject | Mathematics | en_HK |
dc.title | On Fano manifolds with NEF tangent bundles admitting 1-dimensional varieties of minimal rational tangents | en_HK |
dc.type | Article | en_HK |
dc.identifier.openurl | http://library.hku.hk:4550/resserv?sid=HKU:IR&issn=0002-9947&volume=354&spage=2639&epage=2658&date=2002&atitle=On+Fano+manifolds+with+Nef+tangent+bundles+admitting+1-dimensional+varieties+of+minimal+rational+tangents | en_HK |
dc.identifier.email | Mok, N:nmok@hkucc.hku.hk | en_HK |
dc.identifier.authority | Mok, N=rp00763 | en_HK |
dc.description.nature | published_or_final_version | en_HK |
dc.identifier.doi | 10.1090/S0002-9947-02-02953-7 | en_HK |
dc.identifier.scopus | eid_2-s2.0-0035998087 | en_HK |
dc.identifier.hkuros | 66704 | - |
dc.relation.references | http://www.scopus.com/mlt/select.url?eid=2-s2.0-0035998087&selection=ref&src=s&origin=recordpage | en_HK |
dc.identifier.volume | 354 | en_HK |
dc.identifier.issue | 7 | en_HK |
dc.identifier.spage | 2639 | en_HK |
dc.identifier.epage | 2658 | en_HK |
dc.identifier.isi | WOS:000174806600005 | - |
dc.publisher.place | United States | en_HK |
dc.identifier.scopusauthorid | Mok, N=7004348032 | en_HK |