AI 中文总结
受Ekedahl-Shepherd-Barron-Taylor猜想启发，作者证明几乎所有素数p-闭的余秩1叶状结构的有限和乐定理，完整证明相关代数可积性命题，还建立局部解析p-闭叶状结构的全纯可积性并验证二维联络论等价命题。
AI 中文摘要
受Ekedahl-Shepherd-Barron-Taylor猜想的启发，我们研究特征零下叶状结构整体与局部可积性的算术判据。主要结果为：对几乎所有素数p-闭的余秩1叶状结构的不变素除子正则部分，存在有限和乐定理。由此，我们完整证明了Ekedahl-Shepherd-Barron-Taylor提出的关于余秩1叶状结构的命题：光滑射影簇上几乎所有素数p-闭的叶状结构，只要存在紧叶，就是代数可积的。我们还对正规复簇上的叶状结构芽引入了p-闭性的局部解析概念，并提出了将其与亚纯及全纯首次积分关联的局部猜想。我们证明了光滑曲面及klt曲面芽上具有典范奇点的解析p-闭叶状结构的全纯可积性。将这些局部结果与有限和乐定理结合，我们得到了某些解析p-闭非临界曲面芽及任意维解析p-闭简单奇点的全纯首次积分。最后，我们基于Bott部分联络的平坦亚纯延拓，提出了与局部全纯可积性对应的联络论等价命题，并在二维情形验证了该命题。
英文摘要
Motivated by the Ekedahl-Shepherd-Barron-Taylor conjecture, we study arithmetic criteria for the global and local integrability of foliations in characteristic zero. Our main result is a finite-holonomy theorem for the regular parts of invariant prime divisors of corank one foliations which are \(p\)-closed for almost all primes. As a consequence, we give a complete proof for corank one foliations of the statement proposed by Ekedahl-Shepherd-Barron-Taylor: a foliation on a smooth projective variety which is \(p\)-closed for almost all primes is algebraically integrable whenever it admits a compact leaf. We also introduce a local analytic notion of \(p\)-closedness for foliation germs on normal complex varieties and formulate local conjectures relating it to meromorphic and holomorphic first integrals. We establish holomorphic integrability for analytically \(p\)-closed foliations with canonical singularities on smooth surfaces and on klt surface germs. Combining these local results with the finite-holonomy theorem, we obtain holomorphic first integrals for certain analytically \(p\)-closed non-dicritical surface germs and for analytically \(p\)-closed simple singularities in arbitrary dimension. Finally, we formulate a connection-theoretic counterpart to local holomorphic integrability in terms of flat meromorphic extensions of the Bott partial connection, and verify it in dimension two.
Comments37 pages
