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Article: On Fano manifolds with NEF tangent bundles admitting 1-dimensional varieties of minimal rational tangents

TitleOn Fano manifolds with NEF tangent bundles admitting 1-dimensional varieties of minimal rational tangents
Authors
KeywordsMathematics
Issue Date2002
PublisherAmerican Mathematical Society.
Citation
Transactions Of The American Mathematical Society, 2002, v. 354 n. 7, p. 2639-2658 How to Cite?
AbstractLet 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 Identifierhttp://hdl.handle.net/10722/48402
ISSN
2015 Impact Factor: 1.196
2015 SCImago Journal Rankings: 2.168
ISI Accession Number ID
References

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_HK
dc.date.accessioned2008-05-22T04:11:47Z-
dc.date.available2008-05-22T04:11:47Z-
dc.date.issued2002en_HK
dc.identifier.citationTransactions Of The American Mathematical Society, 2002, v. 354 n. 7, p. 2639-2658en_HK
dc.identifier.issn0002-9947en_HK
dc.identifier.urihttp://hdl.handle.net/10722/48402-
dc.description.abstractLet 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
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dc.format.extent489302 bytes-
dc.format.mimetypeapplication/pdf-
dc.format.mimetypeapplication/pdf-
dc.languageengen_HK
dc.publisherAmerican Mathematical Society.en_HK
dc.relation.ispartofTransactions of the American Mathematical Societyen_HK
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.rightsTransactions of the American Mathematical Society. Copyright © American Mathematical Society.en_HK
dc.rightsFirst published in Transactions of the American Mathematical Society, 2002, v. 354, p. 2639-2658, published by the American Mathematical Society,en_HK
dc.subjectMathematicsen_HK
dc.titleOn Fano manifolds with NEF tangent bundles admitting 1-dimensional varieties of minimal rational tangentsen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://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+tangentsen_HK
dc.identifier.emailMok, N:nmok@hkucc.hku.hken_HK
dc.identifier.authorityMok, N=rp00763en_HK
dc.description.naturepublished_or_final_versionen_HK
dc.identifier.doi10.1090/S0002-9947-02-02953-7en_HK
dc.identifier.scopuseid_2-s2.0-0035998087en_HK
dc.identifier.hkuros66704-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-0035998087&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume354en_HK
dc.identifier.issue7en_HK
dc.identifier.spage2639en_HK
dc.identifier.epage2658en_HK
dc.identifier.isiWOS:000174806600005-
dc.publisher.placeUnited Statesen_HK
dc.identifier.scopusauthoridMok, N=7004348032en_HK

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