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.

Deformations of representations of fundamental groups of complex varieties

“…So we recover the main result of [Lef17]: in this case O ρ is quadratic. And we recover some form of purity implies formality (see [CH17]): the purity of weights implies some partial formality of L hence it behaves as in the compact case. (4) If X ⊂ X (with X compact Kähler) and ρ is the monodromy of a polarized VHS over X extendable to X, then H 1 (L) has again weights 1, 2 and H 2 (L) has weights 2, 3, 4.…”

Formality of little disks and algebraic geometry

“…) is a natural "analytic geometry" home for the mixed Hodge structure on the operad of chains C * LD given by Tamarkin's construction [45] (see also [9]): in particular, both the Hodge and the weight filtration are clearly visible in this picture. Moreover, the complex of sheaves Ω * ( LD log,top ) has naturally a rational lattice (lifting the rational structure on Ω * (FLC)), giving naturally a de Rham lattice C * dR (LD).…”

“…The canonical splittings for C * (LD, C) and C * (FLD, C) constructed here are particularly interesting compared to previously known splittings, because they are the first explicitly constructed splittings which are compatible with a rational structure on C * (LD, C); namely, the de Rham rational structure. This structure (or rather its dual, C * dR ) was first constructed as part of a mixed Hodge structure in the paper [9], where it was observed to follow from the Grothendieck-Teichmüller action discovered in [45] (the corresponding action on the framed operad FLD follows from [42]). Another, log geometric, interpretation for this rational structure was given in [50].…”

“…Its algebro-geometric provenance also gives it automatic compatibility with a number of structures. In particular this splitting is rational (and indeed definable integrally) for the "de Rham" rational structure on the operad FLD (the de Rham structure on the complex C * (FLD) as well as on C * (LD) is part of a larger known derived mixed Hodge structure on these dg operads: see for exampe [9] for a definition in terms of the Grothendieck-Teichmüller group, or [50] for a log algebro-geometric definition.) No explicit splitting that has previously been constructed was known to be rational in any lattice, though rational splittings have been known to exist by certain standard lifting theorems.…”

“…In particular 1-pure schemes are schemes with Gr k W (H * ) purely in cohomological degree k, and include all smooth proper varieties. See also[9].…”