The Arnold conjecture for singular symplectic manifolds

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-04-18 DOI:10.1007/s11784-024-01105-y

Joaquim Brugués, Eva Miranda, Cédric Oms

引用次数: 0

Abstract

In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of \(b^m\)-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number of 1-periodic Hamiltonian orbits for \(b^{2m}\)-symplectic manifolds depending only on the topology of the manifold. Moreover, for \(b^m\)-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions.

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奇异交映流形的阿诺德猜想

在这篇文章中，我们研究了奇异交映流形上的哈密顿动力学，并证明了一大类 \(b^m\)-symplectic 流形的阿诺德猜想。文章引入了新技术，在一些温和条件下将光滑交映形式与原始奇异交映结构联系起来。这些技术使得奇点交映流形的阿诺德猜想在多种情况下都有效。更准确地说，我们证明了 \(b^{2m}\)-symplectic 流形的单周期哈密顿轨道数量的下限，这仅取决于流形的拓扑结构。此外，对于\(b^{m}\)-交错曲面，我们改进了取决于一对（M，Z）拓扑的下界。然后，我们将大胆研究这个奇异领域的浮子同调，最后列出一些开放性问题。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.