任意特征下具有小奇点集的叶状结构
Foliations with small singular set in arbitrary characteristic
浏览论文内容
中文总结 AI 辅助
研究任意特征下光滑代数簇上叶状结构几何性质，给出极小有理和德尔佩佐曲面上正则叶状结构分类，推广卡马乔 - 萨德指标，证明射影空间叶状结构相关结论，建立$p$-闭叶状结构的博特消没定理版本。
中文摘要 AI 辅助
本文研究了在任意特征$p \ge 0$的代数闭域上光滑代数簇上叶状结构的几何性质。我们探讨了正特征下叶状结构的几个特定特征，旨在突出与特征零情况的异同。首先，我们给出了极小有理曲面和德尔佩佐曲面上正则叶状结构的完整分类，确定在正特征下此类叶状结构是$p$-闭的。然而，在特征2的弱德尔佩佐曲面上存在非$p$-闭的正则叶状结构。我们将卡马乔 - 萨德指标及其相关的和公式推广到任意特征，用它来研究具有小奇点集的叶状结构和分布的行为。对于射影空间上的叶状结构，我们证明存在具有足够小奇点集的不变超曲面会迫使叶状结构是$p$-闭的，并对其法丛的次数施加严格的可除性条件。最后，我们在霍奇上同调与周环中建立了$p$-闭叶状结构的博特消没定理的版本，为复几何中的经典消没结果提供了正特征类似物。
英文摘要
This paper investigates the geometry of foliations on smooth algebraic varieties over an algebraically closed field of arbitrary characteristic $p \ge 0$. We address several specific features of foliations in positive characteristic, aiming to highlight both similarities and differences with the characteristic zero case. First, we provide a complete classification of regular foliations on minimal rational and del Pezzo surfaces, establishing that in positive characteristic, most of them are $p$-closed. However, there are regular foliations on Hirzebruch surfaces over arbitrary characteristic $p>0$, and on weak del Pezzo surfaces, in characteristic $p=2$, which are not $p$-closed. We extend the Camacho-Sad index and its associated sum formula to arbitrary characteristic, using it to study the behavior of foliations and distributions with small singular set. For foliations on projective spaces, we prove that the existence of an invariant hypersurface with a sufficiently small singular set forces the foliation to be $p$-closed and imposes strict divisibility conditions on the degree of its normal bundle. Finally, we establish versions of the Bott vanishing theorem in both Hodge cohomology and the Chow ring for $p$-closed foliations, providing a positive characteristic analogue to classical vanishing results in complex geometry.
发表机构
- Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）
- Institut Montpelliérain Alexander Grothendieck, Université de Montpellier（蒙彼利埃亚历山大·格罗滕迪克研究所，蒙彼利埃大学）
- Univ Rennes, CNRS, IRMAR, UMR 6625（雷恩大学，法国国家科学研究中心，IRMAR）
机构由 AI 辅助整理，请以论文原文为准。
