Degenerations of Calabi-Yau Manifolds and Related Geometries

Calabi-Yau流形和相关几何的退化。Calabi-Yau流形形成了一个中心几何类，与其他数学领域和数学物理有大量的联系和应用。关于特别有趣的三维几何形状的各种结构问题至今无法回答，例如，变形类型的数量是否有限，所有变形类型是否通过极值过渡连接，或者镜像对称适用于何种程度。该项目的目标是发展研究这些问题的方法，其基本途径是卡-丘几何的最大退化。在过去的十年中，控制这种退化的方法特别是由Gross和Siebert开发的。这些采用对数和热带几何。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，这是不同卡-丘几何之间以简并为特征的深刻关系。镜像对称将一个卡-丘流形的复几何与另一个卡-丘流形的辛几何联系起来。在辛侧的结构数据由全纯曲线支配，在复侧的结构数据由复结构的变化支配。拟议的项目旨在将这种关系扩展到更高属的曲线和相应的复杂数据。为此目的，Costello-Li和Barannikov-Kontsevich的现有方法应被转化为对数几何，然后被增强。此外，还应通过研究热带变形来扩展热带方法。这些直接关系到莫里森的猜想指出，镜像对称是兼容的极值过渡。一个极值变迁连接了两个不同的Calabi-Yau流形，Reid证明了所有的三维Calabi-Yau流形都是由这样的变迁连接的。我们力求在实现这些目标方面取得进展。最后，相关的结构，如同调镜像对称相对于一个除数和非紧的卡-丘流形的基础上的频谱曲线应进行分析，因为这扩展了开发的技术的应用范围。

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.