Mixed Hodge structures in homotopy theory

同伦理论中的混合 Hodge 结构

• 批准号: 269490478
• 负责人: Dr. Joana Cirici
• 金额: --
• 依托单位: Forschungsgebiet Algebraische Geometrie und Arithmetik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2017-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/269490478?language=en
• 关键词: Mixed; Hodge; structures; homotopy; theory

This research project focuses on the study of homotopical structures of algebraic varieties, with a special emphasis on their motivic formulation, using techniques of algebraic geometry and homotopy theory. In particular, we plan to study the existence and properties of mixed Hodge structures on homotopy theoretic invariants arising from rational homotopy, intersection cohomology and deformation theory. We aim to develop suitable algebraic frameworks for motivic rational homotopy theory.The main algebraic objects under consideration in the present research project are mixed Hodge diagrams of differential graded algebras. These are a multiplicative analogue of the mixed Hodge complexes of Deligne involving filtered and bifiltered differential algebras over the rational and complex fields that encode the weight and Hodge filtrations up to filtered quasi-isomorphisms. The development of the present project requires to extend their scope of application to different homotopical frameworks, and to develop concrete applications to the homotopy theory of algebraic varieties.

本研究利用代数几何和同伦理论的方法，对代数簇的同伦结构进行了研究，重点研究了代数簇的基元表示。特别地，我们计划研究由有理同伦、交上同调和形变理论产生的同伦理论不变量的混合Hodge结构的存在性和性质。我们的目标是为Motivic有理同伦理论建立合适的代数框架。本研究的主要代数对象是微分分次代数的混合Hodge图。这些是Deligne的混合Hodge复形的乘法模拟，涉及有理和复域上的过滤和双滤微分代数，这些域编码了权和Hodge滤子直到滤子拟同构。本项目的发展要求将它们的应用范围扩展到不同的同伦框架，并开发代数簇的同伦理论的具体应用。