On the canonical line bundle and negative holomorphic sectional curvature

By Gordon Heier, Steven Lu, B. Wong. 2009

52 citations recorded by the source corpus.

We prove that a smooth complex projective threefold with a K\"ahler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differential geometry and complex algebraic geometry.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fb1c0bc87b9652c8d2b72b554088171668002ab0

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle (2016, methodology): “as extended by using essentially the same techniques to the Ka¨hler case in [TY17]. Before these achievements, only some cases were known under some extra conditions. For instance, this was proven in [HLW10] supposing the abundance conjecture to hold true (which is the case indeed in dimension less ”
On real bisectional curvature for Hermitian manifolds (2016, background): “The study of holomorphic sectional curvature in Kähler geometry has been a classic topic, and it also attracted some attention in recent years, for example, the work of Heier and collaborators on compact Kähler manifolds with positive or negative holomorphic sectional curvature ([9, 10, 11, 1]), and”
RC-positivity, rational connectedness and Yau's conjecture (2017, background): “earlier under the additional assumption of the Abundance Conjecture ([25]).”
Negative holomorphic curvature and positive canonical bundle (2015, methodology): “Recently, some important progress is made by Heier-Lu-Wong [5, 6], which combines the algebraic-geometric method and the analytic method.”