Mixed Hodge structures in homotopy theory

本课题主要利用代数几何和同伦理论的方法研究代数变异的同伦结构，重点研究它们的动机公式。特别地，我们计划研究由理性同伦、交上同伦和变形理论引起的同伦理论不变量上的混合Hodge结构的存在性和性质。我们的目的是为动机理性同伦理论建立合适的代数框架。本课题研究的主要代数对象是微分梯度代数的混合Hodge图。这些是Deligne混合Hodge复形的乘法模拟，涉及有理域和复域上的过滤和二分微分代数，将权重和Hodge滤波编码为过滤的拟同构。本课题的发展需要将它们的应用范围扩展到不同的同伦框架，并对代数变异的同伦理论进行具体的应用。

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.