On Hodge polynomials for nonalgebraic complex manifolds

• 成果类型: Article
• 署名作者: Katzarkov, Ludmil; Lee, Kyoung-Seog; Lupercio, Ernesto; Meersseman, Laurent
• 署名单位: University of Miami; HSE University (National Research University Higher School of Economics); Bulgarian Academy of Sciences; Pohang University of Science & Technology (POSTECH); CINVESTAV - Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional; Centre National de la Recherche Scientifique (CNRS); Universite d'Angers
• 刊物名称: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• ISSN/ISSBN: 0027-13584
• DOI: 10.1073/pnas.2415722122
• 发表日期: 2025-04-10
• 关键词: lvmb manifolds; compact; VARIETIES; ring

摘要

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-K & auml;hler 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.