# Mixed Hodge structures and formality of symmetric monoidal functors

## TL;DR

This paper demonstrates that the singular chains functor with rational coefficients is formal as a lax symmetric monoidal functor for certain complex schemes, using mixed Hodge theory, with implications for operad formality and rational homotopy theory.

## Contribution

It establishes formality of the singular chains functor via mixed Hodge theory for schemes with pure weight filtration, extending to non-pure cases through spectral sequences.

## Key findings

- Proves formality of the singular chains functor for pure weight schemes.
- Links the functor to the weight spectral sequence in non-pure cases.
- Provides applications to operad and rational homotopy formality.

## Abstract

We use mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex schemes whose weight filtration in cohomology satisfies a certain purity property. This has direct applications to the formality of operads or, more generally, of algebraic structures encoded by a colored operad. We also prove a dual statement, with applications to formality in the context of rational homotopy theory. In the general case of complex schemes with non-pure weight filtration, we relate the singular chains functor to a functor defined via the first term of the weight spectral sequence.