Cofibrant models of diagrams: mixed Hodge structures in rational homotopy

TL;DR

This paper develops a homotopy theory framework for diagram categories with variable vertices, applying it to mixed Hodge diagrams to establish functorial mixed Hodge structures on rational homotopy types of complex varieties.

Contribution

It introduces a new homotopy theory approach for diagram categories and applies it to mixed Hodge diagrams, proving a multiplicative Beilinson's theorem and functoriality of mixed Hodge structures.

Findings

Established a model for mixed Hodge diagrams using cofibrant objects.

Proved a multiplicative version of Beilinson's theorem.

Demonstrated functoriality of mixed Hodge structures on rational homotopy types.

Abstract

We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is applied to the category of mixed Hodge diagrams of differential graded algebras. Using Sullivan's minimal models, we prove a multiplicative version of Beilinson's Theorem on mixed Hodge complexes. As a consequence, we obtain functoriality for the mixed Hodge structures on the rational homotopy type of complex algebraic varieties. In this context, the mixed Hodge structures on homotopy groups obtained by Morgan's theory follow from the derived functor of the indecomposables of mixed Hodge diagrams.