Mixed Hodge structures and formality of symmetric monoidal functors

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.

29 pages

References in corpus (2)

• The Beilinson regulator is a map of ring spectra
• Rectification of Deligne's mixed Hodge structures

Cited by in corpus (7)

• Toric varieties of Loday's associahedra and noncommutative cohomological field theories
• Etale cohomology, purity and formality with torsion coefficients
• On the formality of the little disks operad in positive characteristic
• Derived categories of character sheaves
• Hodge-de Rham numbers of almost complex 4-manifolds
• Getzler-Kapranov complexes and moduli stacks of curves
• Formality of little disks and algebraic geometry