Projective manifolds whose tangent bundles are numerically effective

Campana, Frédéric; Peternell, Thomas

doi:10.1007/bf01446566

Cited by 113 publications

(116 citation statements)

References 16 publications

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“…Let X be a Fano manifold with nef tangent bundle. Campana and Peternell [CP1] conjectured that X is biholomorphic to a rational homogeneous manifold. Using the classification theory of Fano manifolds, they solved the conjecture for dimensions up to 3.…”

Section: Statement Of Results and Background On Universal Familiesmentioning

confidence: 99%

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

Mok

2002

Trans. Amer. Math. Soc.

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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 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 Cx, and (b) recovering the structure of a rational homogeneous manifold from Cx. 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 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 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 = P 2 resp. Q 3 resp. K(G 2 ). 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 P 1 -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 c 2 1 (V ) ≤ 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 2 1 (V ) = 4c 2 (V ).

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Section: Statement Of Results and Background On Universal Familiesmentioning

confidence: 99%

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

Mok

2002

Trans. Amer. Math. Soc.

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“…On the other hand, by Proposition A.5 again, the index of X is m − z = dim X/2 + 1, then the only possibility is X = Q 2 , whose Picard number is not one. Now a simple computation shows that the only possible cases are (α, z, m) = (3, 2k, 3k + 1), (4,3,5), (4,6,9), (5,4,6). …”

Section: Proof Of Theorem 11 (C)mentioning

confidence: 99%

“…For the types (4, 3, 5) and (5,4,6) the VMRT at the general point is a finite set of points; an example of type (4,3,5) appears by considering the congruence of 4-secant lines to a Palatini threefold in P 5 (see [7, Theorem 0.1]), but there are not known examples of type (5, 4, 6) with Z smooth. As for (4,6,9), an anonymous referee informed us that C. Peskine and F. Zak have provided and example.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

See 1 more Smart Citation

A classification theorem on Fano bundles

Muñoz¹,

Conde²,

Occhetta³

2014

Ann. inst. Fourier

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In this paper we classify rank two Fano bundles E on Fano manifolds satisfying H 2 (X, Z) ∼ = H 4 (X, Z) ∼ = Z. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization P(E), that allows us to obtain the cohomological invariants of X and E. As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

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“…The classification of the complex projective varieties or compact Kähler manifolds whose tangent bundle is numerically effective [9,6] yields a far-reaching generalization of the Hartshorne-Frankel-Mori theorem, the latter stating that the complex projective n-space is the only projective n-variety whose tangent bundle is ample. Manifolds whose cotangent bundle is nef were studied by Kratz [19].…”

Section: Introductionmentioning

confidence: 99%

Metrics on semistable and numerically effective Higgs bundles

Bruzzo

Grana-Otero

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, establishing semistability criteria for Higgs bundles on projective manifolds of any dimension.

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Projective manifolds whose tangent bundles are numerically effective

Cited by 113 publications

References 16 publications

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

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

A classification theorem on Fano bundles

Metrics on semistable and numerically effective Higgs bundles

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