Hodge theory of Log Singular Loci

对数奇异位点的 Hodge 理论

• 批准号: 241231364
• 负责人: Professor Dr. Helge Ruddat
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 无数据
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/241231364?language=en
• 关键词: Hodge; theory; Log; Singular; Loci

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.

镜像对称是不同类型几何之间的深层关系：复几何与辛几何。它是由卡-丘几何物理学家在1990年左右发现的，此后产生了非常有趣的数学结果。还有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。里程碑在发展的主题相关的这一建议一直是最所有的工作Batyrev-Borisov，Strominger-Yau-Zaslow和格罗斯-Siebert。这个项目涉及到这些之间的联系，也包括最近的结果镜像对称的品种一般类型。在Gross-Siebert方法中，一个推广Batyrev-Borisov的镜像构造使用对数几何和复曲面退化。原木结构的奇异性首先成为发展的障碍。然而，后来证明，它们构成了一个重要的功能时，一个正式的光滑退化的卡-丘几何构造。这个项目涉及霍奇理论方面的分析，这个奇异的轨迹连接它与镜像对称更一般的几何形状。

项目成果

Perverse curves and mirror symmetry

反常曲线和镜像对称

• DOI: 10.1090/jag/666
• 发表时间: 2014
• 期刊: Journal of Algebraic Geometry
• 影响因子: 1.8
• 作者: Ruddat