Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations

Q3 Mathematics

Arnold Mathematical Journal Pub Date : 2021-10-15 DOI:10.1007/s40598-021-00192-w

Xianghong Gong, Laurent Stolovitch

引用次数: 4

Abstract

We consider an embedded n-dimensional compact complex manifold in \(n+d\) dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of \(C_n\) in \(M_{n+d}\) is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in \(M_{n+d}\) having \(C_n\) as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors condition arising from solutions to their cohomological equations.

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嵌入紧复流形的邻域等价与高余维叶

我们在（n+d）维复流形中考虑一个嵌入的n维紧致复流形。作为Grauert形式原理程序的一部分，我们对邻域的全纯分类感兴趣。我们将给出确保\（M_｛n+d｝\）中\（C_n\）的邻域是其正规丛的零部分的邻域的双全纯的条件。这扩展了Arnold关于曲面中复杂环面邻域的结果。我们还证明了\（C_n\）为紧叶的\（M_{n+d}\）中全纯叶理的存在性，将Ueda的理论推广到高余维情形。这两个问题都表现为一类线性化问题，涉及由其上同调方程的解引起的小因子条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Arnold Mathematical Journal Mathematics-Mathematics (all)

CiteScore

1.50

自引率

0.00%

发文量

期刊介绍： The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.