发表机构
East China Normal University（华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决 Dinh-Sibony 列表中三个问题：证明饱和全纯叶状正闭电流有限 Poincaré 质量，构造具有稠密双曲叶和弥散极值电流的射影曲面，并给出双曲叶 Nevanlinna 特征的对数下界。
AI 中文摘要
我们解决了 Dinh--Sibony 列表中的问题 2.2(i)、2.2(ii) 和 2.4。对于紧致 Kähler 流形上具有有限多个奇点的饱和全纯曲线叶状，每个正定向 $dd^c$-闭电流具有有限的 Poincaré 质量。因此，在此背景下，几何遍历定理不需要线性化。我们构造了一个射影曲面，其具有稠密双曲叶和一个弥散极值电流，该电流的 Nevanlinna 平均值以均匀的互反对数速率收敛。这是针对选定叶状的存在性结果，而非针对任意给定叶状的速率定理。在第一个结果的假设下，即使不存在弥散定向电流，每个双曲叶的 Nevanlinna 特征也具有对数下界。未处理正维奇异集。
英文摘要
We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh--Sibony's list. For a saturated holomorphic curve foliation on a compact Kähler manifold with finitely many singularities, every positive directed $dd^c$-closed current has finite Poincaré mass. Thus linearizability is not needed for the geometric ergodic theorem in this setting. We construct a projective surface with dense hyperbolic leaves and a diffuse extremal current whose Nevanlinna averages converge at a uniform reciprocal logarithmic rate. This is an existence result for a chosen foliation, not a rate theorem for an arbitrary prescribed foliation. Under the hypotheses of the first result, every hyperbolic leaf has a logarithmic lower bound for its Nevanlinna characteristic, even if no diffuse directed current exists. Positive-dimensional singular sets are not treated.
Comments13 pages. With 3 open problems in the end
