On projective varieties with strictly nef tangent bundles

Li, Duo; Ou, Wenhao; Yang, Xibei

doi:10.1016/j.matpur.2019.04.007

Cited by 15 publications

(17 citation statements)

References 28 publications

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“…Propositions 6.1, 6.5 and 6.6), and hence finally give a positive answer to Question 1.2 for smooth threefolds. This also extends [LOY19, Theorem 1.2] to the pair case (X, ∆) when dim X = 3.…”

Section: Introductionsupporting

confidence: 69%

Strictly nef divisors on singular threefolds

Zhong

2021

Preprint

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Let X be a normal projective threefold with mild singularities, and L X a strictly nef Q-divisor on X. We first show the ampleness of K X + tL X with sufficiently large t if either the Kodaira dimension κ(X) = 0 or the augmented irregularity q • (X) = 0. Second, we study the rational connectedness of a projective klt pair (X, ∆) with the anti-log canonical divisor −(K X + ∆) being strictly nef.

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Section: Introductionsupporting

confidence: 69%

Strictly nef divisors on singular threefolds

Zhong

2021

Preprint

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“…The purpose of this paper is to provide a structure theorem of smooth projective varieties with nef 2 T X , which gives generalizations of some results in [6,12,33,46,52,53]. Our first result is an analogue of Theorem 1.1: Theorem 1.5.…”

Section: Introductionmentioning

confidence: 91%

“…Related to these results, following the solution of the Hartshorne conjecture [39], K. Cho and E. Sato [12] proved that a smooth projective variety X with ample 2 T X is isomorphic to a projective space or a quadric. Recently, D. Li, W. Ou and X. Yang [33,Theorem 1.5] generalized this result for varieties X with strictly nef 2 T X .…”

Section: Introductionmentioning

confidence: 95%

Positivity of the second exterior power of the tangent bundles

Watanabe

2020

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Let X be a smooth complex projective variety with nef 2 T X and dim X ≥ 3. We prove that, up to a finite étale cover X → X, the Albanese map X → Alb( X) is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety F with nef 2 T F . As a bi-product, we see that either T X is nef or X is a Fano variety. Moreover we study a contraction of a K X -negative extremal ray ϕ : X → Y . In particular, we prove that X is isomorphic to the blow-up of a projective space at a point if ϕ is of birational type. We also prove that ϕ is a smooth morphism if ϕ is of fiber type. As a consequence, we give a structure theorem of varieties with nef 2 T X .

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“…Recently, Li, the second author and the fourth author proved in [LOY19] that smooth projective varieties with strictly nef anti-canonical divisors are rationally connected ([LOY19, Theorems 1.2 and 1.3]). By the well-known result of Campana and Kollár-Mori-Miyaoka ([Cam92,KMM92]), this provides important evidences for an affirmative answer to the Campana and Peternell conjecture, i.e., the smooth case of Conjecture 1.1 in all dimensions.…”

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confidence: 99%

Algebraic fibre spaces with strictly nef relative anti-log canonical divisor

Liu¹,

Ou²,

Juanyong³

et al. 2021

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A. Let ( , ) be a projective klt pair, and ∶ → a fibration to a smooth projective variety with strictly nef relative anti-log canonical divisor −( ∕ + ). We prove that is a locally constant fibration with rationally connected fibres, and the base is a canonically polarized hyperbolic projective manifold. In particular, when is a single point, we establish that is rationally connected. Moreover, when dim = 3 and −( + ) is strictly nef, we prove that −( + ) is ample, which confirms the singular version of the Campana-Peternell conjecture for threefolds.C * (− ( ∕ + ) + )and *

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On projective varieties with strictly nef tangent bundles

Cited by 15 publications

References 28 publications

Strictly nef divisors on singular threefolds

Strictly nef divisors on singular threefolds

Positivity of the second exterior power of the tangent bundles

Algebraic fibre spaces with strictly nef relative anti-log canonical divisor

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