A decomposition theorem for projective manifolds with nef anticanonical bundle

Cao, Junyan; Höring, Andreas

doi:10.1090/jag/715

Cited by 33 publications

(45 citation statements)

References 29 publications

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“…One of the key ingredients for the proof of Theorem 1.2 relies on the recent breakthrough of Cao and Höring on the structure theorems for projective varieties with nef anticanonical bundle (see [Cao16] and [CH17]). Actually, based on their work, we prove the following result, which is essential for Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

On projective varieties with strictly nef tangent bundles

Ou²,

Yang

2019

Journal de Mathématiques Pures et Appliquées

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In this paper, we study smooth complex projective varieties X such that some exterior power r TX of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If TX is strictly nef, then X isomorphic to the projective space P n . If 2 TX is strictly nef and if X has dimension at least 3, then X is either isomorphic to P n or a quadric Q n .

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Section: Introductionmentioning

confidence: 99%

On projective varieties with strictly nef tangent bundles

Ou²,

Yang

2019

Journal de Mathématiques Pures et Appliquées

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“…As an application of this approach, we confirm a conjecture of Yau ([57, Problem 47]) that a compact Kähler manifold with positive holomorphic sectional curvature is a projective and rationally connected manifold. This project is motivated by a number of well-known conjectures proposed by Yau, Mumford, Demailly, Campana, Peternell and etc., and we refer to [55,57,10,33,32,16,17,38,27,7,12,14,28,6,49,11,15] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

RC-positivity, rational connectedness and Yau’s conjecture

Yang

2018

Cambridge Journal of Mathematics

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In this paper, we introduce a concept of RC-positivity for Hermitian holomorphic vector bundles and prove that, if E is an RC-positive vector bundle over a compact complex manifold X, then for any vector bundle A, there exists a positive integer cA = c(A, E) such thatfor ℓ ≥ cA(k + 1) and k ≥ 0. Moreover, we obtain that, on a compact Kähler manifold X, if Λ p TX is RC-positive for every 1 ≤ p ≤ dim X, then X is projective and rationally connected. As applications, we show that if a compact Kähler manifold (X, ω) has positive holomorphic sectional curvature, then Λ p TX is RC-positive and H p,0 ∂ (X) = 0 for every 1 ≤ p ≤ dim X, and in particular, we establish that X is a projective and rationally connected manifold, which confirms a conjecture of Yau ([57, Problem 47]).

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“…Surprisingly, it is proved in [6] that the maximal rationally connected fibration of a projective manifold with nef anti-canonical bundle can be taken as a regular morphism. In view of this result, it seems to be natural to ask the following question: Question 6.1.…”

Section: Questionsmentioning

confidence: 99%