Rectification of Deligne's mixed Hodge structures

TL;DR

This paper advances the understanding of mixed Hodge structures by establishing a symmetric monoidal equivalence of categories and constructing algebraic models that compute Deligne's mixed Hodge structures on cohomology.

Contribution

It extends Beilinson's equivalence to a symmetric monoidal setting and constructs presheaves of algebraic structures that compute mixed Hodge structures.

Findings

Established a symmetric monoidal equivalence of quasi-categories.

Constructed presheaves of commutative differential graded algebras.

Provided models for computing mixed Hodge structures on cohomology.

Abstract

We promote Beilinson's triangulated equivalence between the bounded derived category of rational polarizable mixed Hodge structures and the derived category of rational polarizable mixed Hodge complexes to an equivalence of symmetric monoidal quasi-categories. We use this equivalence to construct a presheaf of commutative differential graded algebras in the ind-completion of the category of rational mixed Hodge structures which computes Deligne's mixed Hodge structure on the rational Betti cohomology of $\mathbf{C}$-schemes of finite type. This leads to a preshea--in the quasi-categorical sense--of $\mathbf{E}_{\infty}$-algebras computing integral mixed Hodge structures.