Hodge theory of Log Singular Loci

镜像对称是不同类型的几何体之间的深层关系：复杂几何体与辛几何体。1990年前后，物理学家在研究Calabi-Yau几何学时发现了它，从那时起，它就产生了非常有趣的数学结果。尽管如此，许多重要的问题仍然悬而未决，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。迈尔斯通在与这一提议相关的主题的发展中一直是巴蒂雷夫-鲍里索夫、斯特罗明格-尤-扎斯洛和格罗斯-西伯特的大部分工作。这个项目涉及到这两者之间的联系，还结合了关于各种一般类型的镜像对称性的最新结果。在Gross-Siebert方法中，推广了Batyrev-Borisv的镜像结构，使用了对数几何和环面退化。原木结构的奇异性首先是发展中的一个障碍。然而，后来的研究表明，当构造退化的Calabi-Yau几何的形式光滑化时，它们构成一个重要的特征。这个项目涉及到对这一奇异轨迹的霍奇理论方面的分析，将它与更一般几何的镜像对称性联系起来。

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.