标题： 沿奇点叶层的可统一的叶层射影结构 显示英文标题

标题： Uniformizable foliated projective structures along singular foliations

摘要： 我们考虑在紧复流形上的全纯叶状结构，对于这些结构，我们研究沿叶面变化的全纯的射影结构（叶状射影结构）的存在性，这些结构满足特定的可统一性质。 我们的结果表明，叶状结构的奇点对这种结构的存在施加了严格的限制。 一个叶状射影结构将叶状结构的奇点分为抛物型和非抛物型。 对于紧致凯勒流形上的强可统一叶状射影结构，存在一个非退化、非抛物型奇点意味着叶状结构是完全可积的。 我们建立了一个指标定理，该定理对仅具有支持叶状射影结构使其全部为抛物型的非退化奇点的叶状结构施加强烈的上同调限制。 作为我们结果的应用，我们证明，在任何维数的射影空间上，仅具有非退化奇点的次数至少为二的曲线叶状结构不具有强可统一的叶状射影结构。 显示英文摘要

摘要： We consider holomorphic foliations by curves on compact complex manifolds, for which we investigate the existence of projective structures along the leaves varying holomorphically (foliated projective structures), that satisfy particular uniformizability properties. Our results show that the singularities of the foliation impose severe restrictions for the existence of such structures. A foliated projective structure separates the singularities of a foliation into parabolic and non-parabolic ones. For a strongly uniformizable foliated projective structure on a compact K\"ahler manifold, the existence of a single non-degenerate, non-parabolic singularity implies that the foliation is completely integrable. We establish an index theorem that imposes strong cohomological restrictions on the foliations having only non-degenerate singularities that support foliated projective structures making all of them parabolic. As an application of our results, we prove that, on a projective space of any dimension, a foliation by curves of degree at least two, with only non-degenerate singularities, does not admit a strongly uniformizable foliated projective structure.