Solving the Kodaira-Spencer problem using Harmonic Analysis on torus bundles.

使用环面束的调和分析解决 Kodaira-Spencer 问题。

• 批准号: 2105841
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2105841
• 关键词: Solving; Kodaira; Spencer; problem; using

The goal of this research is to answer in full the Kodaira-Spencer problem, which appeared as Problem 20 in Hirzebruch's 1954 problem list. The problem concerns the extension of the Hodge numbers from complex manifolds to almost-complex manifolds. Specifically, by considering the case of the Kodaira-Thurston manifold it will be demonstrated that the property of metric-invariance which applies in the complex setting is lost when moving to the almost-complex setting. Furthermore, it can be shown that by varying the almost-complex structure the Hodge numbers can be made arbitrarily large. The main technique used to arrive at these results is derived from the Weil-Brezin transform and will be further generalised so as to apply on any torus bundle over S1. The main beneficiaries of the proposed research will be research mathematicians working in differential geometry, symplectic geometry and complex geometry.

这项研究的目标是完整回答 Kodaira-Spencer 问题，该问题在 Hirzebruch 的 1954 年问题列表中作为问题 20 出现。该问题涉及霍奇数从复流形到近复流形的扩展。具体来说，通过考虑 Kodaira-Thurston 流形的情况，将证明适用于复杂设置的度量不变性在移动到几乎复杂设置时会丢失。此外，可以证明，通过改变几乎复杂的结构，霍奇数可以变得任意大。用于获得这些结果的主要技术源自 Weil-Brezin 变换，并将进一步推广，以便应用于 S1 上的任何环面丛。拟议研究的主要受益者将是从事微分几何、辛几何和复几何研究的数学家。