非代数复流形的Hodge多项式

IF 9.1 1区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES
Ludmil Katzarkov, Kyoung-Seog Lee, Ernesto Lupercio, Laurent Meersseman
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引用次数: 0

摘要

霍奇理论是研究代数变异的复杂几何和拓扑的关键:它提供了对其结构的基本见解。霍奇分解定理在变异的几何和它们的上同调群之间建立了深刻的联系,有助于理解它们的潜在性质。此外,霍奇理论在镜像对称领域的初始阶段至关重要,它揭示了看似不同的代数变体之间的深刻联系。它还揭示了代数循环和动机的研究,代数几何的关键对象。本文探讨Hodge多项式及其性质,特别关注non-Kähler复流形。我们研究了这种流形的不同范围,包括(拟-)Hopf,(拟-)Calabi-Eckmann和LVM流形,以及一类包含代数变体和上述特殊情况的可定义复流形。我们的研究在这个更广泛的背景下建立了霍奇多项式的动机性质的保存。通过明确的计算和彻底的分析,这项工作有助于更深入地了解复杂的流形几何超出了代数变化的领域。本研究的结果在复杂流形发挥重要作用的数学和物理的各个领域具有潜在的应用前景。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On Hodge polynomials for nonalgebraic complex manifolds
Hodge theory is pivotal in studying algebraic varieties’ intricate geometry and topology: it provides essential insights into their structure. The Hodge decomposition theorem establishes a profound link between the geometry of varieties and their cohomology groups, helping to understand their underlying properties. Moreover, Hodge theory was crucial at the inception of the field of mirror symmetry, revealing deep connections among seemingly disparate algebraic varieties. It also sheds light on studying algebraic cycles and motives, crucial objects in algebraic geometry. This article explores Hodge polynomials and their properties, specifically focusing on non-Kähler complex manifolds. We investigate a diverse range of such manifolds, including (quasi-)Hopf, (quasi-)Calabi–Eckmann, and LVM manifolds, alongside a class of definable complex manifolds encompassing both algebraic varieties and the aforementioned special cases. Our research establishes the preservation of the motivic nature of Hodge polynomials inside this broader context. Through explicit calculations and thorough analyses, this work contributes to a deeper understanding of complex manifold geometry beyond the realm of algebraic varieties. The outcomes of this study have potential applications in various areas of mathematics and physics where complex manifolds play a significant role.
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来源期刊
CiteScore
19.00
自引率
0.90%
发文量
3575
审稿时长
2.5 months
期刊介绍: The Proceedings of the National Academy of Sciences (PNAS), a peer-reviewed journal of the National Academy of Sciences (NAS), serves as an authoritative source for high-impact, original research across the biological, physical, and social sciences. With a global scope, the journal welcomes submissions from researchers worldwide, making it an inclusive platform for advancing scientific knowledge.
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