Degenerations of Calabi-Yau Manifolds and Related Geometries

Calabi-Yau流形的退化及相关几何。Calabi-Yau流形形成了一个中心几何类，在其他数学领域和数学物理中有着大量的联系和应用。迄今为止，关于特别有趣的三维几何形状的各种结构问题都无法回答，例如变形类型的数量是否有限，是否所有变形类型都由极值转换连接，或者镜像对称适用于何种程度。本提案的目标是发展研究这些问题的方法。该项目的基本方法是最大程度地退化Calabi-Yau几何。在过去的十年中，控制这种退化的方法主要是由Gross和Siebert开发的。它们采用对数几何和热带几何。简并的动机源于数学物理。1990年左右，弦理论学家发现了镜像对称，这是不同的卡拉比-丘几何之间的一种深刻关系，具有退化特征。镜像对称将一个Calabi-Yau流形的复几何与另一个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。提出的项目旨在将这种关系扩展到高属曲线和相应的复杂数据。为此，应将Costello-Li和Barannikov-Kontsevich的现有方法转化为对数几何，然后进行增强。此外，热带方法应通过研究热带变形加以扩展。这些直接关系到莫里森的猜想，即镜像对称与极端过渡是相容的。一个极值转换连接了两个不同的Calabi-Yau流形，Reid推测所有的三维Calabi-Yau流形都是通过这样的转换连接起来的。我们寻求在这些猜想方面取得进展。最后，将分析相关结构，如相对于除数的同调镜像对称和基于光谱曲线的非紧化Calabi-Yau流形，因为这扩展了所开发技术的应用范围。

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.