AI 中文总结
研究允许具有非正全纯截面曲率的紧致凯勒流形，证明其典范丛是nef，复二维时负曲率意味着典范丛丰富，全纯截面曲率消失则第一陈类消失，且在特定紧致复流形上非负曲率度量的全纯截面曲率必消失。
AI 中文摘要
我们研究允许具有非正全纯截面曲率的埃尔米特度量的紧致凯勒流形。我们证明了这样一个流形的典范丛是数值有效（nef）的，去除了布罗德 - 斯坦菲尔德（Broder - Stanfield）相应数值有效性结果中的多重闭假设。在复二维中，我们进一步表明负全纯截面曲率意味着典范丛的丰富性。我们还证明了全纯截面曲率消失迫使第一陈类消失。作为部分逆命题，我们表明在具有零第一博特 - 陈类的紧致复流形上，每个具有非负全纯截面曲率的埃尔米特度量必定全纯截面曲率消失。
英文摘要
We study compact Kähler manifolds admitting Hermitian metrics with nonpositive holomorphic sectional curvature. We prove that the canonical bundle of such a manifold is nef, removing the pluriclosed assumption from the corresponding nefness result of Broder-Stanfield \cite{BroderStanfield}. In complex dimension two, we further show that negative holomorphic sectional curvature implies the ampleness of the canonical bundle. In complex dimension three, the same conclusion holds if the Hermitian metric is additionally assumed to be balanced. We also prove that vanishing holomorphic sectional curvature forces the first Chern class to vanish.
CommentsThis version includes a new result: an ampleness criterion for the canonical bundle of compact Kähler threefolds admitting balanced Hermitian metrics with negative holomorphic sectional curvature
