复解析空间芽上余维1叶状结构的全纯与形式首次积分
Holomorphic and Formal First Integrals for Foliations of Codimension One on Complex Analytic Space Germs
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中文总结 AI 辅助
该研究将Mattei-Moussu经典可积性定理推广到奇异复解析空间，证明二维余维1全纯叶状结构存在全纯首次积分的充要条件，给出非正规例子说明正规性的必要性，结合多种技术推导高维可积性结果。
中文摘要 AI 辅助
我们研究正规复解析空间上余维1全纯叶状结构芽的全纯与形式首次积分。在二维情形下，假设解消例外除子的对偶图为树，我们证明该叶状结构存在全纯首次积分当且仅当它的叶在奇点外是闭的，且仅有有限多片叶在该奇点处积累。这将Mattei和Moussu的经典可积性定理推广到了奇异环境空间。我们还证明了关于正规商芽的全纯延拓定理，该商芽存在光滑拟平覆盖及一个一般二维截口的光滑连通提升。此外，我们记录了一个条件性形式延拓命题，其依赖于余法幂的深度假设及对应微分形式障碍模的单射条件。在商-延拓假设下，且在需检测形式限制的约化切锥条件下，高维可积性结果可由其二维对应结果导出。我们给出了一个约化非正规例子，该例子满足上述两个动力学条件但不存在全纯首次积分，表明正规性是必要的。我们的论证结合了奇点解消、和乐技术、形式完备化及正规解析空间上全纯函数的延拓性质。
英文摘要
We study holomorphic and formal first integrals for germs of codimension-one holomorphic foliations on normal complex analytic spaces. In dimension two, under the assumption that the dual graph of the exceptional divisor of a resolution is a tree, we prove that the foliation admits a holomorphic first integral if and only if its leaves are closed outside the singular point and only finitely many leaves accumulate at that point. This extends a classical integrability theorem of Mattei and Moussu to singular ambient spaces. We also prove a holomorphic prolongation theorem for normal quotient germs admitting a smooth quasi-étale cover and a smooth connected lift of a generic two-dimensional section. We record, in addition, a conditional formal prolongation statement under depth assumptions on the conormal powers and an injectivity condition for the corresponding differential-form obstruction modules. Under the quotient-prolongation hypothesis, and with a reduced tangent cone where formal restriction must be detected, the higher-dimensional integrability results follow from their surface counterparts. We give a reduced nonnormal example satisfying both dynamical conditions but admitting no holomorphic first integral, showing that normality is essential. Our arguments combine resolution of singularities, holonomy techniques, formal completion, and extension properties of holomorphic functions on normal analytic spaces.
