Hodge theory of Log Singular Loci

镜像对称是不同几何类型之间的深层关系：复杂几何与辛几何。它是1990年左右由物理学家在卡拉比-丘几何中发现的，从那时起就产生了非常有趣的数学结果。但仍有许多重要的问题有待解决，例如同调镜像对称猜想的证明以及镜像对称如何扩展到更一般的几何。与这一建议相关的主题发展的里程碑是Batyrev-Borisov， strominger - you - zaslow和Gross-Siebert的大部分工作。这个项目关注的是这些之间的联系，也包括最近关于各种一般类型的镜像对称的结果。在Gross-Siebert方法中，一种推广Batyrev-Borisov方法的镜像结构，使用对数几何和环退化。原木结构的奇异性最初是开发中的一个障碍。然而，后来证明了它们在构造退化的Calabi-Yau几何的形式平滑时是一个重要的特征。该项目涉及对霍奇理论方面的分析，该奇异轨迹将其与更一般几何的镜像对称联系起来。

Mirror Symmetry is a deep relationship between different types of geometries: complex vs. symplectic geometry. It was discovered by physicists for Calabi-Yau geometries around 1990 and has since then led to very interesting mathematical results. Still many important questions remain open, e.g. a proof of the homological mirror symmetry conjecture and how mirror symmetry extends to more general geometries. Mile stones in the development of the subject relevant for this proposal have been most of all the work of Batyrev-Borisov, Strominger-Yau-Zaslow and Gross-Siebert. This project concerns a connection between these incorporating also very recent results on mirror symmetry for varieties of general type. In the Gross-Siebert approach, a mirror construction generalizing Batyrev-Borisov's, uses logarithmic geometry and toric degenerations. The singularities of the log structure had first been a road block in the development. Later however, it was shown that they constitute an important feature when a formal smoothing of a degenerate Calabi-Yau geometry was constructed. This project concerns an analysis of the Hodge theoretic aspects of this singular locus linking it with mirror symmetry for more general geometries.