Negative holomorphic curvature and positive canonical bundle

By Damin Wu, Shing-Tung Yau. 2015

116 citations recorded by the source corpus.

In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7291fde5d4f7cd882139e88cc467ff0254e88e34

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle (2016, result): “etric ω′ with negative Ricci curvature. Then, KX should be ample and, in particular, X should be projective. This conjecture has been proved only very recently in the projective case by Wu and Yau in [WY16], and extended shortly after to the Kahler case by [TY15]. Before these major breakthroughs, o”
K\"ahler manifolds of semi-negative holomorphic sectional curvature (2014, background): “ot use it and in fact works for Kahler manifolds. Remark7.2. After this paper had been completed and gone to press at the Journal of Diﬀerential Geometry, Wu and Yau gave a proof of Conjecture 1.2 in [WY15]. Subsequently, Tosatti and Yang in [TY15] extended that proof to the K¨ahler case. However, s”
On real bisectional curvature for Hermitian manifolds (2016, methodology): “As an attempt to push the results of [19] to conjecture (b) or (c) above, and also to study holomorphic sectional curvature on Hermitian manifolds in general, we introduce the following curvature term for Hermitian manifolds.”
RC-positivity, rational connectedness and Yau's conjecture (2017, result): “For the negative holomorphic sectional curvature case, in the recent breakthrough paper [46] of Wu and Yau, they proved that any projective Kähler manifold with negative holomorphic sectional curvature must have ample canonical line bundle.”
An extension of a theorem of Wu-Yau (2015, methodology): “ions. We also acknowledge an intellectual debt of gratitude to the work of Wu-Yau. 2. Proof ofthe main theorem In this section we give the proof of Theorem 1.1, by modifying the method used by Wu-Yau [16] to prove Corollary 1.2 in the projective case. The following is the key lemma, from Wu-Yau [16]”
A remark on our paper "Negative Holomorphic curvature and positive canonical bundle" (2016, methodology): “ATURE AND POSITIVE CANONICAL BUNDLE” DAMIN WU AND SHING–TUNG YAU This is a continuation of our ﬁrst paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical eﬀectiveness. The other one is to p”