In this paper, we obtain a number of results related to the hard Lefschetz theorem for pseudoeffective line bundles, due to Demailly, Peternell and Schneider. Our first result states that the holomorphic sections produced by the theorem are in fact parallel, when the Chern connection associated with the singular metric is computed in the sense of currents, and the corresponding multiplier ideal sheaves are taken into account. Our proof is based on a control of the covariant derivative in the delicate approximation process used in the construction of these sections. Then we show that there is an isomorphism between the space of such parallel sections and the sheaf cohomology group of appropriate degree. As an application, we show that the closedness property of the sections produces a singular holomorphic foliation on the tangent bundle. Finally, we discuss some questions related to the optimality of the multiplier ideal sheaves involved in the generalized hard Lefschetz theorem.

中文翻译：

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关于伪有效线丛的硬 Lefschetz 定理

在本文中，由于 Demailly、Peternell 和 Schneider，我们获得了许多与伪有效线束的硬 Lefschetz 定理相关的结果。我们的第一个结果表明，当在电流的意义上计算与奇异度量相关的陈连接时，该定理产生的全纯截面实际上是平行的，并考虑了相应的乘数理想滑轮。我们的证明是基于在这些部分的构造中使用的精细近似过程中对协变导数的控制。然后我们证明了这种平行截面的空间与适当度数的层上同调群之间存在同构。作为一个应用，我们展示了截面的闭合特性在切丛上产生了一个奇异的全纯叶理。最后，