Mixed Hodge structures in homotopy theory

本研究计画的重点是研究代数簇的同伦结构，并特别强调其动机公式，使用代数几何与同伦理论的技巧。特别是，我们计划研究的同伦理论不变量的混合霍奇结构的存在性和性质所产生的有理同伦，交上同调和变形理论。我们的目标是为动力有理同伦理论建立合适的代数框架，主要研究微分分次代数的混合Hodge图。这是一个乘法模拟的混合霍奇复杂的德利涅涉及过滤和bifiltered微分代数的理性和复杂的领域，编码的重量和霍奇filtrations到过滤准同构。本项目的发展需要将它们的应用范围扩展到不同的同伦框架，并发展代数簇的同伦理论的具体应用。

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.