Homotopy Theory of Mixed Hodge Complexes

TL;DR

This paper establishes a Cartan-Eilenberg structure on the category of mixed Hodge complexes, facilitating homotopy calculations and providing a conceptual framework for classical results in mixed Hodge theory.

Contribution

It introduces a Cartan-Eilenberg structure for mixed Hodge complexes, enabling better homotopy category computations and unifying existing results.

Findings

Provides a new conceptual framework for mixed Hodge complexes.

Simplifies calculations in the homotopy category.

Connects classical results through this new structure.

Abstract

We show that the category of mixed Hodge complexes admits a Cartan-Eilenberg structure, a notion introduced in [GNPR10] leading to a good calculation of the homotopy category in terms of (co)fibrant objects. This result provides a conceptual framework from which Beilinson's [Bei86] and Carlson's [Car80] results on mixed Hodge complexes and extensions of mixed Hodge structures follow easily.