Degenerations of Calabi-Yau Manifolds and Related Geometries

Calabi-Yau 流形的退化及相关几何形状

• 批准号: 272561367
• 负责人: Professor Dr. Helge Ruddat
• 金额: --
• 依托单位: Dipartimento di Mathematica
• 依托单位国家: 德国
• 项目类别: Independent Junior Research Groups
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/272561367?language=en
• 关键词: Degenerations; Calabi; Yau; Manifolds; Related

Degenerations of Calabi-Yau Manifolds and Related Geometries.Calabi-Yau manifolds form a central geometric class with a plethora of connections and applications to other mathematical areas and mathematical physics. Various structural questions about the particularly interesting three-dimenensional such geometries could not be answered to date, e.g. whether the number of deformation types is finite, whether all deformation types are connected by extremal transitions or to which extent mirror symmetry applies. The goal of this proposal is the development of methodology to study these questions.The basic approach of the project is the maximal degeneration of the Calabi-Yau geometry. In the past decade, methods controlling such degenerations were developed in particular by Gross and Siebert. These employ logarithmic and tropical geometry. The motivation for degenerating stems from mathematical physics. Around 1990, string theorists discovered mirror symmetry, a deep relationship between different Calabi-Yau geometries featuring degenerations. Mirror symmetry relates the complex geometry of one Calabi-Yau manifold to the symplectic geometry of another Calabi-Yau manifold. The structural data on the symplectic side are governed by holomorphic curves, that on the complex side by variations of the complex structure. The proposed project aims at extending this relationship to curves of higher genus and the corresponding complex data. For this purpose, existing methods by Costello-Li and Barannikov-Kontsevich shall be translated into logarithmic geometry and then be enhanced. Also tropical methods shall be extended by studying tropical deformations. These directly relate to Morrison's conjecture stating that mirror symmetry is compatible with extremal transitions. An extremal transitions connects two different Calabi-Yau manifolds and Reid conjectured that all three-dimensional Calabi-Yau manifolds are connected by such transitions. We seek to make progress towards these conjectures. Finally, related structures like homological mirror symmetry relative to a divisor and non-compact Calabi-Yau manifolds based on a spectral curve shall be analysed as this extends the scope of the applications for the developed techniques.

Calabi-Yau流形及其相关几何的退化。Calabi-Yau流形形成了一个中心几何类，它与其他数学领域和数学物理有着大量的联系和应用。关于这种特别有趣的三维几何的各种结构问题到目前为止还无法回答，例如变形类型的数量是否有限，是否所有的变形类型都通过极端跃迁联系在一起，或者镜像对称在多大程度上适用。这个方案的目标是发展研究这些问题的方法论。该项目的基本方法是最大限度地简并Calabi-Yau几何。在过去的十年里，格罗斯和西伯特开发了控制这种退化的方法。它们使用对数几何和热带几何。退化的动机源于数学物理。大约在1990年，弦理论家发现了镜像对称性，这是不同的Calabi-Yau几何之间以简并为特征的深层次关系。镜像对称将一个Calabi-Yau流形的复杂几何与另一个Calabi-Yau流形的辛几何联系起来。辛侧的结构数据由全纯曲线控制，复侧的结构数据由复结构的变化控制。建议的项目旨在将这种关系扩展到更高亏格的曲线和相应的复杂数据。为此，Costello-Li和Barannikov-Kontsevich的现有方法应转化为对数几何，然后加以加强。此外，热带方法还应通过研究热带变形来推广。这与莫里森关于镜像对称性与极值跃迁相容的猜想有直接关系。极值跃迁连接了两个不同的Calabi-Yau流形，Reid猜想所有三维的Calabi-Yau流形都是通过这种极值跃迁连接的。我们寻求在这些猜想方面取得进展。最后，将分析相关的结构，如相对于除数的同调镜像对称性和基于谱曲线的非紧致Calabi-Yau流形，因为这扩展了所开发技术的应用范围。