Mixed Hodge structures and formality of symmetric monoidal functors
aa r X i v : . [ m a t h . A T ] M a r MIXED HODGE STRUCTURES AND FORMALITY OF SYMMETRICMONOIDAL FUNCTORS
JOANA CIRICI AND GEOFFROY HOREL
Abstract.
We use mixed Hodge theory to show that the functor of singular chains withrational coefficients is formal as a lax symmetric monoidal functor, when restricted tocomplex schemes whose weight filtration in cohomology satisfies a certain purity prop-erty. This has direct applications to the formality of operads or, more generally, ofalgebraic structures encoded by a colored operad. We also prove a dual statement, withapplications to formality in the context of rational homotopy theory. In the general caseof complex schemes with non-pure weight filtration, we relate the singular chains functorto a functor defined via the first term of the weight spectral sequence. Introduction
There is a long tradition of using Hodge theory as a tool for proving formality results. Thefirst instance of this idea can be found in [DGMS75] where the authors prove that compactKähler manifolds are formal (i.e. the commutative differential graded algebra of differentialforms is quasi-isomorphic to its cohomology). In the introduction of that paper, the authorsexplain that their intuition came from the theory of étale cohomology and the fact thatthe degree n étale cohomology group of a smooth projective variety over a finite field ispure of weight n . This purity is what morally prevents the existence of non-trivial Masseyproducts. In the setting of complex algebraic geometry, Deligne introduced in [Del71, Del74]a filtration on the rational cohomology of every complex algebraic variety X , called the weightfiltration , with the property that each of the successive quotients of this filtration behavesas the cohomology of a smooth projective variety, in the sense that it has a Hodge-typedecomposition. Deligne’s mixed Hodge theory was subsequently promoted to the rationalhomotopy of complex algebraic varieties (see [Mor78], [Hai87], [NA87]). This can then beused to make the intuition of the introduction of [DGMS75] precise. In [Dup16] and [CC17],it is shown that purity of the weight filtration in cohomology implies formality, in the senseof rational homotopy, of the underlying topological space. However, the treatment of thetheory in these references lacks functoriality and is restricted to smooth varieties in the firstpaper and to projective varieties in the second.In another direction, in the paper [GNPR05], the authors elaborate on the method of[DGMS75] and prove that operads (as well as cyclic operads, modular operads, etc.) internalto the category of compact Kähler manifolds are formal. Their strategy is to introduce thefunctor of de Rham currents which is a functor from compact Kähler manifolds to chaincomplexes that is symmetric monoidal and quasi-isomorphic to the singular chain functoras a lax symmetric monoidal functor. Then they show that this functor is formal as a laxsymmetric monoidal functor. Recall that, if C is a symmetric monoidal category and A is an Key words and phrases.
Mixed Hodge structures, formality, operads, rational homotopy.Cirici would like to acknowledge financial support from the German Research Foundation (SPP-1786)and partial support from the Spanish Ministry of Economy and Competitiveness (MTM2016-76453-C2-2-P).Horel acknowledges support from the project ANR-16-CE40-0003 ChroK. abelian symmetric monoidal category, a lax symmetric monoidal functor F : C −→ Ch ∗ ( A ) is said to be formal if it is weakly equivalent to H ∗ ◦ F in the category of lax symmetricmonoidal functors. It is then straightforward to see that such functors send operads in C toformal operads in Ch ∗ ( A ) . The functoriality also immediately gives us that a map of operadsin C is sent to a formal map of operads or that an operad with an action of a group G issent to a formal operad with a G -action. Of course, there is nothing specific about operadsin these statements and they would be equally true for monoids, cyclic operads, modularoperads, or more generally any algebraic structure that can be encoded by a colored operad.The purpose of this paper is to push this idea of formality of symmetric monoidal functorsfrom complex algebraic varieties in several directions in order to prove the most generalpossible theorem of the form “purity implies formality”. Before explaining our results moreprecisely, we need to introduce a bit of terminology.Let X be a complex algebraic variety. Deligne’s weight filtration on the rational n -thcohomology vector space of X is bounded by W − H n ( X, Q ) ⊆ W H n ( X, Q ) ⊆ · · · ⊆ W n H n ( X, Q ) = H n ( X, Q ) . If X is smooth then W n − H n ( X, Q ) = 0 , while if X is projective W n H n ( X, Q ) = H n ( X, Q ) .In particular, if X is a smooth and projective then we have W n − H n ( X, Q ) ⊆ W n H n ( X, Q ) = H n ( X, Q ) . In this case, the weight filtration on H n ( X, Q ) is said to be pure of weight n . More generally,for α a rational number and X a complex algebraic variety, we say that the weight filtrationon H ∗ ( X, Q ) is α -pure if, for all n ≥ , we have Gr Wp H n ( X, Q ) := W p H n ( X, Q ) W p − H n ( X, Q ) = 0 for all p = αn. The bounds on the weight filtration tell us that this makes sense only when ≤ α ≤ .Note as well that if we write α = a/b with ( a, b ) = 1 , α -purity implies that the cohomologyis concentrated in degrees that are divisible by b , and that H bn ( X, Q ) is pure of weight an .Aside from smooth projective varieties, some well-known examples of varieties with -pure weight filtration are: projective V -manifolds, projective varieties whose underlyingtopological space is a Q -homology manifold and the moduli spaces M Dol and M dR appearingin the non-abelian Hodge correspondence. Complements of hyperplane arrangements andcomplements of toric arrangements as well as the moduli spaces M ,n of smooth projectivecurves of genus 0 with n marked points make examples with -pure weight filtration. As weshall see in Section , complements of codimension d subspace arrangements are examplesof smooth schemes whose weight filtration in cohomology is d/ (2 d − -pure.Our main result is Theorem . . We show that, for a non-zero rational number α , thesingular chains functor S ∗ ( − , Q ) : Sch C −→ Ch ∗ ( Q ) is formal as a lax symmetric monoidal functor when restricted to complex schemes whoseweight filtration in cohomology is α -pure. Here Sch C denotes the category of complexschemes, that are reduced, separated and of finite type. This generalizes the main result of[GNPR05] on the formality of S ∗ ( X, Q ) for any operad X on smooth projective varieties,to the case of operads in possibly singular and/or non-compact varieties with pure weightfiltration in cohomology.As direct applications of the above result, we prove formality of the operad of singularchains of some operads in complex schemes, such as the noncommutative analogue of the(framed) little 2-discs operad introduced in [DSV15] and the monoid of self-maps of thecomplex projective line studied by Cazanave in [Caz12] (see Theorems . and . ). We also IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 3 reinterpret in the language of mixed Hodge theory the proofs of the formality of the littledisks operad and Getzler’s gravity operad appearing in [Pet14] and [DH17]. These last tworesults do not fit directly in our framework, since the little disks operad and the gravityoperad do not quite come from operads in algebraic varieties. However, the action of theGrothendieck-Teichmüller group provides a bridge to mixed Hodge theory.In Theorem 8.1 we prove a dual statement of our main result, showing that Sullivan’sfunctor of piece-wise linear forms A ∗ P L : Sch op C −→ Ch ∗ ( Q ) is formal as a lax symmetric monoidal functor when restricted to schemes whose weightfiltration in cohomology is α -pure, where α is a non-zero rational number.This gives functorial formality in the sense of rational homotopy for such schemes, gener-alizing both “purity implies formality” statements appearing in [Dup16] for smooth varietiesand in [CC17] for singular projective varieties. Our generalization is threefold: we allowrational weights, obtain functoriality and we study possibly singular and open varietiessimultaneously.Theorems . and . deal with situations in which the weight filtration is pure. In thegeneral context with mixed weights, it was shown by Morgan [Mor78] for smooth schemesand in [CG14] for possibly singular schemes, that the first term of the multiplicative weightspectral sequence carries all the rational homotopy information of the scheme. In Theorem . we provide the analogous statement for the lax symmetric monoidal functor of singularchains. A dual statement for Sullivan’s functor of piece-wise linear forms is proven in The-orem . , enhancing the results of [Mor78] and [CG14] with functoriality.We now explain the structure of this paper. The first four sections are purely algebraic. InSection we collect the main properties of formal lax symmetric monoidal functors that weuse. In particular, in Theorem 2.3 we recall a recent theorem of rigidification due to Hinichthat says that, over a field of characteristic zero, formality of functors can be checked atthe level of ∞ -functors. We also introduce the notion of α -purity for complexes of bigradedobjects in a symmetric monoidal abelian category and show that, when restricted to α -purecomplexes, the functor defined by forgetting the degree is formal.The connection of this result with mixed Hodge structures is done in Section , where weprove a symmetric monoidal version of Deligne’s weak splitting of mixed Hodge structuresover C . Such splitting is a key tool towards formality. In Section we study lax monoidalfunctors to vector spaces over a field of characteristic zero equipped with a compatiblefiltration. We show, in Theorem . , that the existence of a lax monoidal splitting for suchfunctors is independent of the field. As a consequence, we obtain splittings for the weightfiltration over Q . This result enables us to bypass the theory of descent of formality foroperads of [GNPR05], which assumes the existence of minimal models. Putting the aboveresults together we are able to show that the forgetful functor Ch ∗ (MHS Q ) −→ Ch ∗ ( Q ) induced by sending a rational mixed Hodge structure to its underlying vector space, is formalwhen restricted to those complexes whose mixed Hodge structure in homology is α -pure.In order to obtain a symmetric monoidal functor from the category of complex schemesto an algebraic category encoding mixed Hodge structures, we have to consider more flexibleobjects than complexes of mixed Hodge structures. This is the content of Section , where westudy the category MHC k of mixed Hodge complexes. In Theorem . we explain a promo-tion of Beilinson’s equivalence of categories D b (MHS k ) −→ ho(MHC k ) between the derived JOANA CIRICI AND GEOFFROY HOREL category of mixed Hodge structures and the homotopy category of mixed Hodge complexes,to an equivalence of symmetric monoidal ∞ -categories (see also [Dre15], [BNT15]).The geometric character of this paper comes in Section , where we construct a symmetricmonoidal functor from complex schemes to mixed Hodge complexes. This is done in twosteps. First, for smooth schemes, we dualize Navarro’s construction [NA87] of functorialmixed Hodge complexes to obtain a lax monoidal ∞ -functor D ∗ : N (Sm C ) −→ MHC Q such that its composite with the forgetful functor MHC Q −→ Ch ∗ ( Q ) is naturally weaklyequivalent to S ∗ ( − , Q ) as a lax symmetric monoidal ∞ -functor (see Theorem . ). Note thatin order to obtain monoidality, we move to the world of ∞ -categories, denoted in boldfaceletters. In the second step, we extend this functor from smooth, to singular schemes, bystandard descent arguments.The main results of this paper are stated and proven in Section , where we also explainseveral applications to operad formality. Lastly, Section contains applications to therational homotopy theory of complex schemes. Acknowledgments.
This project was started during a visit of the first author at the Haus-dorff Institute for Mathematics as part of the Junior Trimester Program in Topology. Wewould like to thank the HIM for its support. We would also like to thank Alexander Berglund,Brad Drew, Clément Dupont, Vicenç Navarro, Thomas Nikolaus and Bruno Vallette forhelpful conversations.
Notations.
As a rule, we use boldface letters to denote ∞ -categories and normal letters todenote -categories. For C a -category, we denote by N ( C ) its nerve seen as an ∞ -category.If C is a relative category we also use N ( C ) for the ∞ -categorical localization of C at its weakequivalences.For A an additive category, we will denote by Ch ? ∗ ( A ) the category of (homologicallygraded) chain complexes in A , where ? denotes the boundedness condition: nothing forunbounded, b for bounded below and above and ≥ (resp. ≤ for non-negatively (resp.non-positively) graded complexes. We denote by Ch ? ∗ ( A ) the ∞ -category obtained from Ch ? ∗ ( A ) by inverting the quasi-isomorphisms.2. Formal symmetric monoidal functors
Let ( A , ⊗ , ) be an abelian symmetric monoidal category. The homology functor H ∗ :Ch ∗ ( A ) −→ Q n ∈ Z A is a lax symmetric monoidal functor, via the usual Künneth morphism.In the cases that will interest us, all the objects of A will be flat and the homology functoris in fact strong symmetric monoidal.We recall the following definition from [GNPR05]. Definition 2.1.
Let C be a symmetric monoidal category and F : C −→ Ch ∗ ( A ) a laxsymmetric monoidal functor. Then F is said to be a formal lax symmetric monoidal functorif F and H ∗ ◦ F are weakly equivalent in the category of lax symmetric monoidal functors:there is a string of monoidal natural transformations of lax symmetric monoidal functors F Φ ←−− F −→ · · · ←− F n Φ n −−→ H ∗ ◦ F such that for every object X of C , the morphisms Φ i ( X ) are quasi-isomorphisms. Definition 2.2.
Let C be a symmetric monoidal category and F : N ( C ) → Ch ∗ ( A ) alax symmetric monoidal functor (in the ∞ -categorical sense). We say that F is a formal IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 5 lax symmetric monoidal ∞ -functor if F and H ∗ ◦ F are weakly equivalent as lax monoidalfunctors from N ( C ) to Ch ∗ ( A ) .Clearly a formal lax symmetric monoidal functor C → Ch ∗ ( A ) induces a formal laxsymmetric monoidal ∞ -functor N ( C ) → Ch ∗ ( A ) . The following theorem and its corollarygive a partial converse. Theorem 2.3 (Hinich) . Let k be a field of characteristic . Let C be a small symmetricmonoidal category. Let F and G be two lax symmetric monoidal functors C → Ch ∗ ( k ) . If F and G are weakly equivalent as lax symmetric monoidal ∞ -functors N ( C ) −→ Ch ∗ ( k ) , then F and G are weakly equivalent as lax symmetric monoidal functors.Proof. This theorem is true more generally if C is a colored operad. Indeed recall that anysymmetric monoidal category has an underlying colored operad whose category of algebrasis equivalent to the category of lax monoidal functors out of the original category.Now since we are working in characteristic zero, the operad underlying C is homotopicallysound (following the terminology of [Hin15]). Therefore, [Hin15, Theorem 4.1.1] gives us anequivalence of ∞ -categories N (Alg C (Ch ∗ ( k )) ∼ −→ Alg C ( Ch ∗ ( k )) where we denote by Alg C (resp. Alg C ) the category of lax monoidal functors (resp. the ∞ -category of lax monoidal functors) out of C . Now, the two functors F and G are twoobjects in the source of the above map that become weakly equivalent in the target. Hence,they are already equivalent in the source, which is precisely saying that they are connectedby a zig-zag of weak equivalences of lax monoidal functors. (cid:3) Corollary 2.4.
Let k be a field of characteristic . Let C be a small symmetric monoidalcategory. Let F : C → Ch ∗ ( k ) be a lax symmetric monoidal functor. If F is formal aslax symmetric monoidal ∞ -functor N ( C ) −→ Ch ∗ ( k ) , then F is formal as lax symmetricmonoidal functor.Proof. It suffices to apply Theorem 2.3 to F and H ∗ ◦ F . (cid:3) The following proposition whose proof is trivial is the reason we are interested in formallax monoidal functors.
Proposition 2.5 ([GNPR05], Proposition 2.5.5) . If F : C −→ Ch ∗ ( A ) is a formal laxsymmetric monoidal functor then F sends operads in C to formal operads in Ch ∗ ( A ) . In rational homotopy, there is a criterion of formality in terms of weight decompositionswhich proves to be useful in certain situations (see for example [BMSS98] and [BD78]). Wenext provide an analogous criterion in the setting of symmetric monoidal functors.Denote by gr A the category of graded objects of A . It inherits a symmetric monoidalstructure from that of A , with the tensor product defined by ( A ⊗ B ) n := M p A p ⊗ B p − n . The unit in gr A is given by concentrated in degree zero. The functor U : gr A −→ A obtained by forgetting the degree is symmetric monoidal. The category of graded complexes Ch ∗ ( gr A ) inherits a symmetric monoidal structure via a graded Künneth morphism. Definition 2.6.
Given a rational number α , denote by Ch ∗ ( gr A ) α -pure the full subcategoryof Ch ∗ ( gr A ) given by those graded complexes A = L A pn with α -pure homology : H n ( A ) p = 0 for all p = αn. JOANA CIRICI AND GEOFFROY HOREL
Note that if α = a/b , with a and b coprime, then the above condition implies that H ∗ ( A ) is concentrated in degrees that are divisible by b , and in degree kb , it is pure of weight ka : H kb ( A ) p = 0 for all p = ka. Proposition 2.7.
Let A be an abelian category and α a non-zero rational number. Thefunctor U : Ch ∗ ( gr A ) α -pure −→ Ch ∗ ( A ) defined by forgetting the degree is formal as a laxsymmetric monoidal functor.Proof. We will define a functor τ : Ch ∗ ( gr A ) −→ Ch ∗ ( gr A ) together with natural transfor-mations Φ : U ◦ τ ⇒ U and Ψ : U ◦ τ ⇒ H ◦ U giving rise to weak equivalences when restricted to chain complexes with α -pure homology.Consider the truncation functor τ : Ch ∗ ( gr A ) −→ Ch ∗ ( gr A ) defined by sending a gradedchain complex A = L A pn to the graded complex given by: ( τ A ) pn := A pn n > ⌈ p/α ⌉ Ker( d : A pn → A pn − ) n = ⌈ p/α ⌉ n < ⌈ p/α ⌉ , where ⌈ p/α ⌉ denotes the smallest integer greater than or equal to p/α . Note that for each p , τ ( A ) p ∗ is the chain complex given by the canonical truncation of A p ∗ at ⌈ p/α ⌉ , which satisfies H n ( τ ( A ) p ∗ ) ∼ = H n ( A p ∗ ) for all n ≥ ⌈ p/α ⌉ . To prove that τ is a lax symmetric monoidal functor it suffices to see that τ ( A ) pn ⊗ τ ( B ) qm ⊆ τ ( A ⊗ B ) p + qn + m for all A, B ∈ Ch ∗ ( gr A ) . It suffices to consider three cases:(1) If n > ⌈ p/α ⌉ and m ≥ ⌈ q/α ⌉ then n + m > ⌈ p/α ⌉ + ⌈ q/α ⌉ ≥ ⌈ ( p + q ) /α ⌉ . Thereforewe have τ ( A ⊗ B ) p + qn + m = ( A ⊗ B ) p + qn + m and the above inclusion is trivially satisfied.(2) If n = ⌈ p/α ⌉ and m = ⌈ q/α ⌉ then n + m = ⌈ p/α ⌉ + ⌈ q/α ⌉ ≥ ⌈ ( p + q ) /α ⌉ . Now,if n + m > ⌈ ( p + q ) /α ⌉ then again we have τ ( A ⊗ B ) p + qn + m = ( A ⊗ B ) p + qn + m . If n + m = ⌈ ( p + q ) /α ⌉ then the above inclusion reads Ker( d : A pn → A pn − ) ⊗ Ker( d : B qm → B qm − ) ⊆ Ker( d : ( A ⊗ B ) p + qn + m → ( A ⊗ B ) p + qn + m − ) . This is verified by the Leibniz rule.(3) Lastly, if n < ⌈ p/α ⌉ then τ ( A ) pn = 0 and there is nothing to verify.The projection Ker( d : A pn → A pn − ) ։ H n ( A ) p defines a morphism τ A → H ( A ) by ( τ A ) pn (cid:26) n = ⌈ p/α ⌉ H n ( A ) p n = ⌈ p/α ⌉ . This gives a monoidal natural transformation
Ψ : U ◦ τ ⇒ H ◦ U = U ◦ H . Likewise, theinclusion τ A ֒ → A defines a monoidal natural transformation Φ : U ◦ τ ⇒ U .Let A be a complex of Ch ∗ ( gr A ) α -pure . Then both morphisms Ψ( A ) : τ ◦ U ( A ) → H ◦ U ( A ) and Φ( A ) : U ◦ τ ( A ) → U ( A ) are clearly quasi-isomorphisms. (cid:3) For graded chain complexes whose homology is pure up to a certain degree, we obtain aresult of partial formality as follows.
Definition 2.8.
Let q ≥ be an integer. A morphism of chain complexes f : A → B iscalled q -quasi-isomorphism if the induced morphism in homology H i ( f ) : H i ( A ) → H i ( B ) is an isomorphism for all i ≤ q and an epimorphism for i = q + 1 . IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 7
Definition 2.9.
Let q ≥ be an integer. A functor F : C −→ Ch ∗ ( A ) is a q -formal laxsymmetric monoidal functor if the maps Φ i ( X ) in Definition . are q -quasi-isomorphismfor all ≤ i ≤ n . Proposition 2.10.
Let A be an abelian category. Given a non-zero rational number α andan integer q ≥ , denote by Ch ∗ ( gr A ) α -pure q the full subcategory of Ch ∗ ( gr A ) given by thosegraded complexes A = L A pn whose homology in degrees ≤ q + 1 is α -pure: for all n ≤ q + 1 , H n ( A ) p = 0 for all p = αn. Then the functor U : Ch ∗ ( gr A ) α -pure q −→ Ch ∗ ( A ) defined by forgetting the degree is q -formal.Proof. The proof is parallel to that of Proposition 2.7 by noting that, if H n ( A ) is α -pure for n ≤ q + 1 , then the morphisms Ψ( A ) : τ ◦ U ( A ) → H ◦ U ( A ) and Φ( A ) : U ◦ τ ( A ) → U ( A ) are q -quasi-isomorphisms. (cid:3) Mixed Hodge structures
Denote by FA the category of filtered objects of an abelian symmetric monoidal category ( A , ⊗ , ) . All filtrations will be assumed to be of finite length and exhaustive. With thetensor product W p ( A ⊗ B ) := X i + j = p Im( W i A ⊗ W j B −→ A ⊗ B ) , and the unit given by concentrated in weight zero, FA is a symmetric monoidal category.The functor U fil : gr A −→ FA defined by A = L A p W m A := L q ≤ m A q is symmetricmonoidal. The category of filtered complexes Ch ∗ ( FA ) inherits a symmetric monoidalstructure via a filtered Künneth morphism and we have a symmetric monoidal functor U fil : Ch ∗ ( gr A ) −→ Ch ∗ ( FA ) . Let k ⊂ R be a subfield of the real numbers. Definition 3.1. A mixed Hodge structure on a finite dimensional k -vector space V is givenby a filtration W of V , called the weight filtration , together with a filtration F on V C := V ⊗ C ,called the Hodge filtration , such that for all m ≥ , each k -module Gr Wm V := W m V /W m − V carries a pure Hodge structure of weight m given by the filtration induced by F on Gr Wm V ⊗ C ,that is, there is a direct sum decomposition Gr mW V ⊗ C = M p + q = m V p,q where V p,q = F p ( Gr Wm V ⊗ C ) ∩ F q ( Gr Wm V ⊗ C ) = V q,p . Morphisms of mixed Hodge structures are given by morphisms f : V → V ′ of k -modulescompatible with filtrations: f ( W m V ) ⊂ W m V ′ and f ( F p V C ) ⊂ F p V ′ C . Denote by MHS k thecategory of mixed Hodge structures over k . It is an abelian category by [Del71, Theorem2.3.5]. Remark 3.2.
Given mixed Hodge structures V and V ′ , then V ⊗ V ′ is a mixed Hodgestructure with the filtered tensor product. This makes MHS k into a symmetric monoidalcategory. Also, Hom ( V, V ′ ) is a mixed Hodge structure, with the weight filtration given by W p Hom ( V, V ′ ) := { f : V → V ′ ; f ( W q V ) ⊂ W q + p V ′ , ∀ q } and the Hodge filtration defined in the same way. In particular, the dual of a mixed Hodgestructure is again a mixed Hodge structure. JOANA CIRICI AND GEOFFROY HOREL k ⊂ K be a field extension. The functors Π K : MHS k −→ Vect K and Π W K : MHS k −→ F Vect K defined by sending a mixed Hodge structure ( V, W, F ) to V K := V k ⊗ K and ( V K , W ) respec-tively, are symmetric monoidal functors.Deligne introduced a global decomposition of V C := V ⊗ C into subspaces I p,q , for anymixed Hodge structure ( V, W, F ) which generalizes the decomposition of pure Hodge struc-tures of a given weight. In this case, one has a congruence I p,q ≡ I q,p modulo W p + q − . Westudy this decomposition in the context of symmetric monoidal functors. Lemma 3.3 (Deligne’s splitting) . The functor Π W C admits a factorization MHS k G / / Π W C ●●●●●●●●●●●●●●●●● gr Vect C U fil (cid:15) (cid:15) F Vect C into symmetric monoidal functors. In particular, there is an isomorphism of functors U fil ◦ gr ◦ Π W C ∼ = Π W C , where gr : F Vect C −→ gr Vect C is the graded functor given by gr ( V C , W ) p = Gr Wp V C .Proof. Let ( V, W, F ) be a mixed Hodge structure. By [Del71, 1.2.11] (see also [GS75, Lemma1.12]), there is a direct sum decomposition V C = L I p,q ( V ) where I p,q ( V ) = ( F p W p + q V C ) ∩ F q W p + q V C + X i> F q − i W p + q − − i V C ! . This decomposition is functorial for morphisms of mixed Hodge structures and satisfies W m V C = M p + q ≤ m I p,q ( V ) . Define G by letting G ( V, W, F ) n := L p + q = n I p,q ( V ) for any mixed Hodge structure. Since f ( I p,q ( V )) ⊂ I p,q ( V ′ ) for every morphism f : ( V, W, F ) → ( V ′ , W, F ) of mixed Hodgestructures, G is symmetric monoidal. The functor U fil : gr Vect −→ F
Vect is the symmetricmonoidal functor given by M n V n ( V, W ) , with W m V := M n ≤ m V n . Therefore we have U fil ◦ G = Π W C . In order to prove the isomorphism U fil ◦ gr ◦ Π W C ∼ = Π W C it suffices to note that there is an isomorphism of functors gr ◦ U fil ∼ = Id . (cid:3) Descent of splittings of lax monoidal functors
In this section, we study lax monoidal functors to vector spaces over a field of character-istic zero k equipped with a compatible filtration. More precisely, we are interested in laxmonoidal maps C −→ F
Vect k . Our goal is to prove that the existence of a lax monoidalsplitting for such a functor, i.e. of a lift of this map to C −→ gr Vect k , does not depend onthe field k . Our proof follows similar arguments to those appearing in [CG14, Section 2.4],see also [GNPR05] and [Sul77]. A main advantage of our approach with respect to these IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 9 references is that, in proving descent at the level of functors, we avoid the use of minimalmodels (and thus restrictions to, for instance, operads with trivial arity 0).It will be a bit more convenient to study a more general situation where C is allowedto be a colored operad instead of a symmetric monoidal category. Indeed recall that anysymmetric monoidal category can be seen as an operad whose colors are the objects of C andwhere a multimorphism from ( c , . . . , c n ) to d is just a morphism in C from c ⊗ . . . ⊗ c n to d .Then, given another symmetric monoidal category D , there is an equivalence of categoriesbetween the category of lax monoidal functors from C to D and the category of C -algebrasin the symmetric monoidal category D .We fix ( V, W ) a map of colored operads C −→ F
Vect k such that for each color c of C , thevector space V ( c ) is finite dimensional. We denote by Aut W ( V ) the set of its automorphismsin the category of C -algebras in F Vect k and by Aut ( Gr W V ) the set of automorphisms of Gr W V in the category of C -algebras in gr Vect k . We have a morphism gr : Aut W ( V ) → Aut ( Gr W V ) .Let k → R be a commutative k -algebra. The correspondence R Aut W ( V )( R ) := Aut W ( V ⊗ k R ) defines a functor Aut W ( V ) : Alg k −→ Gps from the category
Alg k of commutative k -algebras, to the category Gps of groups. Clearly, we have
Aut W ( V )( k ) = Aut W ( V ) . Wedefine in a similar fashion a functor Aut ( Gr W V ) from Alg k to Gps .We recall the following properties:
Proposition 4.1.
Let ( V, W ) be as above.(1) Aut W ( V ) is a pro-algebraic matrix group over k .(2) Aut W ( V ) is a pro-algebraic affine group scheme over k represented by Aut W ( V ) .(3) The grading morphism gr defines a morphism gr : Aut W ( V ) → Aut ( Gr W V ) ofpro-algebraic affine group schemes.(4) The kernel N := Ker (cid:0) gr : Aut W ( V ) → Aut ( Gr W V ) (cid:1) is a pro-unipotent algebraicaffine group scheme over k .Proof. We can write C as a filtered colimit of suboperads with finitely many objects. Thenthe category of algebras is just the limit of the category of algebras for each of these subop-erads. Hence, in order to prove this proposition, it is enough to show it when C has finitelymany objects and when we remove the prefix pro everywhere.Let N be such that the vector space ⊕ c ∈C V ( c ) can be linearly embedded in k N . Then Aut W ( V ) is the closed subgroup of GL N ( k ) defined by the polynomial equations that expressthe data of a lax monoidal natural filtration preserving automorphism. Thus Aut W ( V ) is analgebraic matrix group. Moreover, Aut W ( V ) is obviously the algebraic affine group schemerepresented by Aut W ( V ) . Hence (1) and (2) are satisfied.For every commutative k -algebra R , the map Aut W ( V )( R ) = Aut W ( V ⊗ k R ) −→ Aut ( Gr W V ⊗ k R ) = Aut ( Gr W V )( R ) is a morphism of groups which is natural in R . Thus (3) follows.Since by (2) both groups Aut W ( V ) and Aut ( grV ) are algebraic and k has character-istic zero, the kernel N is represented by an algebraic matrix group defined over k (see[Bor91, Corollary 15.4]). Therefore to prove (4) it suffices to verify that all elementsin N ( k ) are unipotent. We see that it is enough to show that for any f in N ( k ) andany c ∈ C , the restriction f ( c ) to V ( c ) is unipotent. Consider the Jordan decomposition f = f s · f u into semi-simple and unipotent parts. We want to show that f s ( c ) = 1 for all c . By [Bor91, Theorem 4.4] we have f s ( c ) , f u ( c ) ∈ Aut W ( V ( c ))( k ) . Since grf ( c ) = 1 andan algebraic group morphism preserves semi-simple and unipotent parts, we deduce that gr ( f s ( c )) = gr ( f u ( c )) = 1 . Let V ( c ) = Ker( f s ( c ) − I ) and decompose V ( c ) into f s ( c ) -invariant subspaces V = V ( c ) ⊕ V ′ ( c ) . Therefore we have grV ( c ) = grV ( c ) ⊕ grV ′ ( c ) .Since grV ( c ) contains nothing but the eigenspaces of eigenvalue , we have grV ′ ( c ) = 0 ,and so V ′ ( c ) = 0 . Therefore f s ( c ) = 1 and f ( c ) is unipotent. (cid:3) Lemma 4.2.
Let ( V, W ) be as above. The following are equivalent:(1) The pair ( V, W ) admits a lax monoidal splitting: W p V ∼ = L q ≤ p Gr Wq V .(2) The morphism gr : Aut W ( V ) → Aut ( Gr W V ) is surjective.(3) There exists α ∈ k ∗ which is not a root of unity together with an automorphism Φ ∈ Aut W ( V ) such that gr (Φ) = ψ α is the grading automorphism of Gr W V associatedwith α , defined by ψ α ( a ) = α p a, for a ∈ Gr Wp V. Proof.
The implications (1) ⇒ (2) ⇒ (3) are trivial. We show that (3) implies (1). Let Φ ∈ Aut W ( V ) be such that gr Φ = ψ α . Consider a Jordan decomposition Φ = Φ s · Φ u .Note that the Jordan decomposition exists even for a pro-algebraic affine group scheme,it suffices to do it levelwise. Moreover, we have the property that for each object c of C ,the restrictions (Φ u ( c ) , Φ v ( c )) to V ( c ) form a Jordan decomposition of Φ( c ) = Φ | V ( c ) . By[Bor91, Theorem 4.4] we have that Φ s ( c ) , Φ u ( c ) ∈ Aut W ( V ( c )) and there is a decompositionof the form V ( c ) = V ′ ( c ) ⊕ U ( c ) , where V ′ ( c ) = M V p ( c ) with V p ( c ) := Ker(Φ s ( c ) − α p I ) and U ( c ) is the complementary subspace corresponding to the remaining factors of thecharacteristic polynomial of Φ s ( c ) . As in the proof of Proposition 4.1 (4) one concludes that U ( c ) = 0 .In order to show that W p V = L i ≤ p V p it suffices to prove it objectwise. Let c be anobject of C . For x ∈ V p ( c ) , let q be the smallest integer such that x ∈ W q V ( c ) . Then x defines a class x + W q +1 V ( c ) ∈ grV ( c ) , and ψ α ( x + W q − V ( c )) = α q x + W q − V ( c ) = Φ( x ) + W q − V ( c ) = α p x + W q − V ( c ) . Then ( α q − α p ) x ∈ W q − V ( c ) . Since x / ∈ W q − V ( c ) we have q = p , hence x ∈ W p V . (cid:3) We may now state and prove the main theorem of this section.
Theorem 4.3.
Let ( V, W ) be a map of colored operad C −→ F
Vect k such that for eachcolor c of C , the vector space V ( c ) is finite dimensional. Let k ⊂ K be a field extension.Then V admits a lax monoidal splitting if and only if V K := V ⊗ k K : C −→
Vect K admits alax monoidal splitting.Proof. We may assume without loss of generality that K is algebraically closed. If V K admitsa splitting, the map Aut W ( V )( K ) −→ Aut ( gr ( V ))( K ) is surjective by Lemma . . From [Wat79, Section 18.1] there is an exact sequence of groups −→ N ( k ) −→ Aut W ( V )( k ) −→ Aut ( grV )( k ) −→ H ( K / k , N ) −→ . . . where N is pro-unipotent by Proposition . . Since k has characteristic zero the group H ( K / k , N ) is trivial (see [Wat79, Example 18.2.e]). This gives the exact sequence → N ( k ) −→ Aut W ( V ) −→ Aut ( gr ( V )) −→ . Hence V admits a splitting by Lemma . . (cid:3) IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 11
From this theorem we deduce that Deligne’s splitting holds over Q . We have the followinglemma. Lemma 4.4 (Deligne’s splitting over Q ) . The forgetful functor Π W Q : MHS Q −→ F Vect Q given by ( V, W, F ) ( V, W ) admits a factorization MHS Q G / / Π W Q ●●●●●●●●●●●●●●●●● gr Vect Q U fil (cid:15) (cid:15) F Vect Q into lax symmetric monoidal functors. In particular, there is an isomorphism of functors U fil ◦ gr ◦ Π W Q ∼ = Π W Q , where gr : F Vect Q −→ gr Vect Q is the graded functor given by gr ( V Q , W ) p = Gr Wp V Q .Proof. We apply Theorem 4.3 to the lax monoidal functor Π W Q using the fact that Π W Q ⊗ Q C admits a splitting by Lemma 3.3. (cid:3) Remark 4.5.
We want to emphasize that Theorem 4.3 does not say that the splitting ofthe previous lemma recovers the splitting of Lemma 3.3 after tensoring with C . In fact,it can probably be shown that such a splitting cannot exist. Nevertheless, the existenceof Deligne’s splitting over C abstractly forces the existence of a similar splitting over Q which is all this Lemma is saying. Note as well that these are not splittings of mixed Hodgestructures, but only of the weight filtration. They are also referred to as weak splittings of mixed Hodge structures (see for example [PS08, Section 3.1]). As is well-known, mixedHodge structures do not split in general.The above splitting over Q yields the following “purity implies formality” statement inthe abstract setting of functors defined from the category of complexes of mixed Hodgestructures. Given a rational number α , denote by Ch ∗ (MHS Q ) α - pure the full subcate-gory of Ch ∗ (MHS Q ) of complexes with pure weight α homology: an object ( K, W, F ) in Ch ∗ (MHS Q ) α - pure is such that Gr pW H n ( K ) = 0 for all p = αn. Corollary 4.6.
The restriction of the functor Π Q : Ch ∗ (MHS Q ) −→ Ch ∗ ( Q ) to the category Ch ∗ (MHS Q ) α - pure is formal for any non-zero rational number α .Proof. This follows from Proposition 2.7 together with Lemma 4.4. (cid:3) Mixed Hodge complexes
We next recall the notion of mixed Hodge complex introduced by Deligne in [Del74] inits chain complex version (with homological degree). Let k ⊂ R be a subfield of the realnumbers. Definition 5.1. A mixed Hodge complex over k is given by a filtered bounded chain complex ( K k , W ) over k , a bifiltered chain complex ( K C , W, F ) over C , together with a finite stringof filtered quasi-isomorphisms of filtered complexes of C -modules ( K k , W ) ⊗ C α −→ ( K , W ) α ←− · · · α l − −−−→ ( K l − , W ) α l −→ ( K C , W ) . We call l the length of the mixed Hodge complex. The following axioms must be satisfied:( MH ) The homology H ∗ ( K k ) is of finite type. ( MH ) The differential of Gr pW K C is strictly compatible with F .( MH ) The filtration on H n ( Gr pW K C ) induced by F makes H n ( Gr pW K k ) into a pure Hodgestructure of weight p + n .Such a mixed Hodge complex will be denoted by K = { ( K k , W ) , ( K C , W, F ) } , omittingthe data of the comparison morphisms α i .Axiom ( MH ) implies that for all n ≥ the triple ( H n ( K k ) , Dec
W, F ) is a mixed Hodgestructure over k , where Dec W denotes Deligne’s décalage of the weight filtration (see [Del71,Definition 1.3.3]).Morphisms of mixed Hodge complexes are given by levelwise bifiltered morphisms ofcomplexes making the corresponding diagrams commute. Denote by MHC k the category ofmixed Hodge complexes of a certain fixed length, which we omit in the notation. The tensorproduct of mixed Hodge complexes is again a mixed Hodge complex (see [PS08, Lemma3.20]). This makes MHC k into a symmetric monoidal category, with a filtered variant of theKünneth formula. Definition 5.2.
A morphism f : K → L in MHC k is said to be a weak equivalence if H ∗ ( f k ) is an isomorphism of k -vector spaces.Since the category of mixed Hodge structures is abelian, the homology of every complexof mixed Hodge structures is a graded mixed Hodge structure. We have a functor T : Ch b ∗ (MHS k ) −→ MHC k given on objects by ( K, W, F )
7→ { ( K, T W ) , ( K ⊗ C , T W, F ) } , where T W is the shiftedfiltration ( T W ) p K n := W p + n K n . The comparison morphisms α i are the identity. Also, T is the identity on morphisms. This functor clearly preserves weak equivalences. Lemma 5.3.
The shift functor T : Ch b ∗ (MHS k ) −→ MHC k is symmetric monoidal.Proof. It suffices to note that given filtered complexes ( K, W ) and ( L, W ) , we have: T ( W ⊗ W ) p ( K ⊗ L ) n = ( T W ⊗ T W ) p ( K ⊗ L ) n . (cid:3) Beilinson [Be˘ı86] gave an equivalence of categories between the derived category of mixedHodge structures and the homotopy category of a shifted version of mixed Hodge complexes.We will require a finer version of Beilinson’s equivalence, in terms of symmetric monoidal ∞ -categories. Denote by MHC k the ∞ -category obtained by inverting weak equivalencesof mixed Hodge complexes, omitting the length in the notation. As explained in [Dre15,Theorem 2.7.], this object is canonically a symmetric monoidal stable ∞ -category. Note thatin loc. cit., mixed Hodge complexes have fixed length 2 and are polarized. The results of[Dre15] as well as Beilinson’s equivalence, are equally valid for the category of mixed Hodgecomplexes of an arbitrary fixed length. Theorem 5.4.
The shift functor induces an equivalence Ch b ∗ (MHS k ) −→ MHC k of sym-metric monoidal ∞ -categories.Proof. A proof in the polarizable setting appears in [Dre15]. Also, in [BNT15], a similarstatement is proven for a shifted version of mixed Hodge complexes. We sketch a proof inour setting.We first observe as in Lemma 2.6 of [BNT15] that both ∞ -categories are stable and thatthe shift functor is exact. The stability of MHC k follows from the observation that this ∞ -category is the Verdier quotient at the acyclic complexes of the ∞ -category of mixedHodge complexes with the homotopy equivalences inverted. This last ∞ -category underliesa dg-category that can easily be seen to be stable. The stability of Ch b ∗ (MHS k ) follows in a IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 13 similar way. Since a complex of mixed Hodge structures is acyclic if and only if the underlyingcomplex of k -modules is acyclic, and T is the identity on the underlying complexes of k -modules, it follows that T is exact. Therefore, in order to prove that T is an equivalence of ∞ -categories, it suffices to show that it induces an equivalence of homotopy categories D b (MHS k ) −→ ho(MHC k ) . In [Be˘ı86, Lemma 3.11], it is proven that the shift functor T : Ch b ∗ (MHS p k ) −→ MHC p k induces an equivalence at the level of homotopy categories. Here the superindex p indicatesthat the mixed Hodge objects are polarized. But in fact the result remains true if we removethe polarization (see also [CG16, Theorem 4.10] for a proof of this last fact). The fact that T can be given the structure of a symmetric monoidal ∞ -functor follows from the work ofDrew in [Dre15]. (cid:3) Logarithmic de Rham currents
The goal of this section is to construct a symmetric monoidal functor from schemes over C to mixed Hodge complexes which computes the correct mixed Hodge structure after passingto homology. The construction for smooth schemes is relatively straightforward. It sufficesto take a functorial mixed Hodge complex model for the cochains as constructed for instancein [NA87] and dualize it. The monoidality of that functor is slightly tricky as one has tomove to the world of ∞ -categories for it to exist. Once one has constructed this functor forsmooth schemes, it can be extended to more general schemes by standard descent arguments.We denote by Sch C the category of complex schemes that are reduced, separated andof finite type and we denote by Sm C the subcategory of smooth schemes. Both of thesecategories are essentially small (i.e. there is a set of isomorphisms classes of objects) andsymmetric monoidal under the cartesian product.We will make use of the following very simple observation. Proposition 6.1.
Let C and D be two categories with finite products seen as symmetricmonoidal categories with respect to the product. Then any functor F : C −→ D has apreferred oplax monoidal structure.Proof.
We need to construct comparison morphisms F ( c × c ′ ) −→ F ( c ) × F ( c ′ ) . By definitionof the product, there is a unique such functor whose composition with the first projection isthe map F ( c × c ′ ) −→ F ( c ) induced by the first projection c × c ′ −→ c and whose compositionwith the second projection is the map F ( c × c ′ ) −→ F ( c ′ ) induced by the second projection c × c ′ −→ c ′ . Similarly, one has a unique map F ( ∗ ) −→ ∗ . One checks easily that these twomaps give F the structure of an oplax monoidal functor. (cid:3) For smooth schemes.
In this section, we construct a lax monoidal functor D ∗ : N (Sm C ) −→ MHC Q such that its composition with the forgetful functor MHC Q −→ Ch ∗ ( Q ) is naturally weaklyequivalent to S ∗ ( − , Q ) as a lax symmetric monoidal functor.Let X be a smooth projective complex scheme and j : U ֒ → X an open subscheme suchthat D := X − U is a normal crossings divisor. Denote by A ∗ X the analytic de Rham complexof the underlying real analytic variety of X and let A ∗ X (log D ) denote the subsheaf of j ∗ A ∗ U of logarithmic forms in D . This sheaf may be naturally endowed with weight and Hodgefiltrations W and F (see 8.6 of [NA87]). Furthermore, Proposition 8.4 of loc. cit. gives astring of quasi-isomorphisms of sheaves of filtered cdga’s: ( R TW j ∗ Q U , τ ) ⊗ C ∼ −→ ( R TW j ∗ A ∗ U , τ ) ∼ ←− ( A ∗ X ( log D ) , τ ) ∼ −→ ( A ∗ X ( log D ) , W ) , where τ is the canonical filtration.In this diagram, R TW j ∗ : Sh( U, Ch ≤ ∗ ( Q )) −→ Sh( X, Ch ≤ ∗ ( Q )) is the functor defined by R TW j ∗ := s TW ◦ j ∗ ◦ G + where G • : Sh( X, Ch ≤ ∗ ( Q )) −→ ∆Sh( X, Ch ≤ ∗ ( Q )) is the Godement canonical cosimplicial resolution functor and s TW : ∆Sh( X, Ch ≤ ∗ ( Q )) −→ Sh( X, Ch ≤ ∗ ( Q )) is the Thom-Whitney simple functor introduced by Navarro in Section 2 of loc. cit. Bothfunctors are symmetric monoidal and hence R TW j ∗ is a symmetric monoidal functor (see[RR16, Section 3.2]). The above string of quasi-isomorphisms gives a commutative algebraobject in (cohomological) mixed Hodge complexes after taking global sections. Specifically,the composition R TW Γ( X, − ) := s TW ◦ Γ( X, − ) ◦ G + gives a derived global sections functor R TW Γ( X, − ) : Sh( X, Ch ≤ ∗ ( Q )) −→ Ch ≤ ∗ ( Q ) which again is symmetric monoidal. There is also a filtered version of this functor definedvia the filtered Thom-Whitney simple (see Section 6 of [NA87]). Theorem 8.15 of loc. cit.asserts that by applying the (bi)filtered versions of R TW Γ( X, − ) to each of the pieces ofthe above string of quasi-isomorphisms, one obtains a commutative algebra object in mixedHodge complexes H dg ( X, U ) whose cohomology gives Deligne’s mixed Hodge structure on H ∗ ( U, Q ) and such that H dg ( X, U ) Q = R TW Γ( X, R TW j ∗ Q U ) is naturally quasi-isomorphic to S ∗ ( U, C ) (as a cochain complex). This construction isfunctorial for morphisms of pairs f : ( X, U ) → ( X ′ , U ′ ) . The definition of H dg ( f ) followsas in the additive setting (see [Hub95, Lemma 6.1.2] for details), by replacing the classicaladditive total simple functor with the Thom-Whitney simple functor.Now in order to get rid of the dependence on the compactification, we define for U asmooth scheme over C , a mixed Hodge complex D ∗ ( U ) by the formula D ∗ ( U ) := colim ( X,U ) H dg ( X, U ) where the colimit is taken over the category of pairs ( X, U ) where X is smooth and properscheme containing U as an open subscheme, and X − U is a normal crossing divisor. By the-orems of Hironaka and Nagata, the category of such pairs is a non-empty filtered category.Note that we have to be slightly careful here as the category of mixed Hodge complexes doesnot have all filtered colimits. However, we can form this colimit in the category of pairs ( K Q , W ) , ( K C , W, F ) having the structure required in Definition 5.1 but not necessarily sat-isfying the axioms MH , MH and MH . Since taking filtered colimit is an exact functor, wededuce from the classical isomorphism between sheaf cohomology and singular cohomologythat there is a quasi-isomorphism D ∗ ( U ) Q → S ∗ ( U, Q ) This shows that the cohomology of D ∗ ( U ) is of finite type and hence, that D ∗ ( U ) satisfiesaxiom MH . The other axioms are similarly easily seen to be satisfied. Moreover, filtered IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 15 colimits preserve commutative algebra structures, therefore the functor D ∗ is a functor from Sm op C to commutative algebras in MHC Q .Since the coproduct in commutative algebras is the tensor product, we deduce from thedual of Proposition 6.1 that D ∗ is canonically a lax symmetric monoidal functor from Sm op C to MHC Q . But since the comparison map D ∗ ( U ) Q ⊗ Q D ∗ ( V ) Q −→ D ∗ ( U × V ) Q is a quasi-isomorphism, this functor extends to a symmetric monoidal ∞ -functor D ∗ : N (Sm C ) op −→ MHC Q Remark 6.2.
A similar construction for real mixed Hodge complexes is done in [BT15,Section 3.1]. There is also a similar construction in [Dre15] that includes polarizations.Now, the category
MHC Q is equipped with a duality functor. It sends a mixed Hodgecomplex { ( K Q , W ) , ( K C , W, F ) } to the linear duals { ( K ∨ Q , W ∨ ) , ( K ∨ C , W ∨ , F ∨ ) } where thedual of a filtered complex is defined as in 3.2. One checks easily that this dual object satisfiesthe axioms of a mixed Hodge complex. Moreover, the duality functor MHC op Q −→ MHC Q islax monoidal and the canonical map K ∨ ⊗ L ∨ −→ ( K ⊗ L ) ∨ is a weak equivalence. It follows that the duality functor induces a symmetric monoidal ∞ -functor MHC op Q −→ MHC Q Composing it with D ∗ , we get a symmetric monoidal ∞ -functor D ∗ : N (Sm C ) −→ MHC Q Remark 6.3.
One should note that D ∗ comes from a lax symmetric monoidal functor from Sm op C to MHC Q . On the other hand, D ∗ is induced by a strict functor which is neither laxnor oplax. Its monoidal structure only exists at the ∞ -categorical level.To conclude this construction, it remains to compare the functor D ∗ ( − ) Q with the singularchains functor. These two functors are naturally quasi-isomorphic as shown in [NA87] butwe will need that they are quasi-isomorphic as symmetric monoidal ∞ -functors. We denoteby S ∗ ( − , R ) the singular chain complex functor from the category of topological spaces tothe category of chain complexes over a commutative ring R . The functor S ∗ ( − , R ) is laxmonoidal. Moreover, the natural map S ∗ ( X, R ) ⊗ S ∗ ( Y, R ) → S ∗ ( X × Y, R ) is a quasi-isomorphism. This implies that S ∗ ( − , R ) induces a symmetric monoidal ∞ -functorfrom the category of topological spaces to the ∞ -category Ch ∗ ( R ) of chain complexes over R . We still use the symbol S ∗ ( − , R ) to denote this ∞ -functor. Theorem 6.4.
The functors D ∗ ( − ) Q and S ∗ ( − , Q ) are weakly equivalent as symmetricmonoidal ∞ -functors from N (Sm C ) to Ch ∗ ( Q ) .Proof. We introduce the category
Man of smooth manifolds. We consider the ∞ -category PSh (Man) of presheaves of spaces on the ∞ -category N (Man) . This is a symmetricmonoidal ∞ -category under the product. We can consider the reflective subcategory T spanned by presheaves G satisfying the following two conditions:(1) Given a hypercover U • → M of a manifold M , the induced map G ( M ) → lim ∆ G ( U • ) is an equivalence. (2) For any manifold M , the map G ( M ) → G ( M × R ) induced by the projection M × R → M is an equivalence.The presheaves satisfying these conditions are stable under product, hence the ∞ -category T inherits the structure of a symmetric monoidal locally presentable ∞ -category. It has auniversal property that we now describe.Given another symmetric monoidal locally presentable ∞ -category D , we denote by Fun L, ⊗ ( T , D ) the ∞ -category of colimit preserving symmetric monoidal functors T → D .Then, we claim that the restriction map Fun L, ⊗ ( T , D ) → Fun ⊗ ( N Man , D ) is fully faithful and that its essential image is the full subcategory of Fun ⊗ (Man , D ) spannedby the functors F that satisfy the following two properties:(1) Given a hypercover U • → M of a manifold M , the map colim ∆ op F ( U • ) → F ( M ) is an equivalence.(2) For any manifold M , the map F ( M × R ) → F ( M ) induced by the projection M × R → M is an equivalence.This statement can be deduced from the theory of localizations of symmetric monoidal ∞ -categories (see [Hin16, Section 3]).In particular, there exists an essentially unique symmetric monoidal and colimit preserv-ing functor from T to S (the ∞ -category of spaces) that is determined by the fact that itsends a manifold M to the simplicial set Sing( M ) . This functor is an equivalence of ∞ -categories. This is a floklore result. A proof of a model category version of this fact can befound in [Dug01, Proposition 8.3.].The ∞ -category S is the unit of the symmetric monoidal ∞ -category of presentable ∞ -categories. It follows that it has a commutative algebra structure (which corresponds to thesymmetric monoidal structure coming from the cartesian product) and that it is the initialsymmetric monoidal presentable ∞ -category. Since T is equivalent to S as a symmetricmonoidal presentable ∞ -category, we deduce that, up to equivalence, there is a uniquefunctor T −→ Ch ∗ ( Q ) that is symmetric monoidal and colimit preserving. But, usingthe universal property of T , we easily see that S ∗ ( − , Q ) and D ∗ ( − ) Q can be extended tosymmetric monoidal and colimits preserving functors from T to Ch ∗ ( Q ) . It follows thatthey must be equivalent. (cid:3) For schemes.
In this subsection, we extend the construction of the previous subsectionto the category of schemes.We have the site (Sch C ) pro of schemes over C with the proper topology and the site (Sm C ) pro which is the restriction of this site to the category of smooth schemes (see [Bla16,Section 3.5] for the definition of the proper topology). Proposition 6.5 (Blanc) . Let C be a symmetric monoidal presentable ∞ -category. Wedenote by Fun ⊗ pro, (Sch C , C ) the ∞ -category of symmetric monoidal functors from Sch C to C whose underlying functor satisfies descent with respect to proper hypercovers. Similarly, wedenote by Fun ⊗ pro (Sm C , C ) the ∞ -category of symmetric monoidal functors from Sm C to C whose underlying functor satisfies descent with respect to proper hypercovers. The restrictionfunctor Fun ⊗ pro (Sch C , C ) −→ Fun ⊗ pro (Sm C , C ) is an equivalence. IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 17
Proof.
We have the categories
Fun(Sch op C , sSet) and Fun(Sm op C , sSet) of presheaves of sim-plicial sets over Sch C and Sm C respectively. These categories are related by an adjunction π ∗ : Fun(Sm op C , sSet) ⇆ Fun(Sch op C , sSet) : π ∗ where the right adjoint is just the restriction. Both sides if this adjunction have a symmetricmonoidal structure by taking objectwise product. The functor π ∗ is obviously symmetricmonoidal. We can equip both sides with the local model structure. We obtain a Quillenadjunction π ∗ : Fun(Sm op C , sSet) ⇆ Fun(Sch op C , sSet) : π ∗ between symmetric monoidal model categories. In which the right adjoint is a symmetricmonoidal functor. In [Bla16, Proposition 3.22], it is proved that this is a Quillen equiv-alence. The local model structure on the category Fun(Sm op C , sSet) presents the ∞ -toposof hypercomplete sheaves over the proper site on Sm C and similarly for the local modelstructure on Fun(Sch op C , sSet) . Therefore, this Quillen equivalence implies that these two ∞ -topoi are equivalent. Moreover, as in the proof of 6.4, these topoi, seen as symmet-ric monoidal presentable ∞ -categories under the cartesian product, represent the functor C Fun ⊗ pro (Sm C , C ) (resp. C Fun ⊗ pro (Sm C , C ) ). The result immediately follows. (cid:3) Theorem 6.6.
Up to weak equivalences, there is a unique symmetric monoidal functor D ∗ : N (Sch C ) −→ MHC Q which satisfies descent with respect to proper hypercovers and whose restriction to Sm C isequivalent to the functor D ∗ constructed in the previous subsection.There is also a unique symmetric monoidal functor D ∗ : N (Sch C ) op −→ MHC Q which satisfies descent with respect to proper hypercovers and whose restriction to Sm C isequivalent to the functor D ∗ constructed in the previous subsection.Proof. Let
Ind(
MHC Q ) be the Ind-category of the ∞ -category of mixed Hodge complexes.This is a stable presentable ∞ -category. We first prove that the composite D ∗ : N (Sm C ) −→ MHC Q −→ Ind(
MHC Q ) satisfies descent with respect to proper hypercovers. Let Y be a smooth scheme and X • → Y be a hypercover for the proper topology. We wish to prove that the map α : colim ∆ op D ∗ ( X • ) −→ D ∗ ( Y ) is an equivalence in Ind(
MHC Q ) . By [Bla16, Proposition 3.24] and the fact that takingsingular chains commutes with homotopy colimits in spaces, we see that the map β : colim ∆ op S ∗ ( X • , Q ) −→ S ∗ ( Y, Q ) is an equivalence. On the other hand, writing Ch ∗ ( Q ) ω for the ∞ -category of chain com-plexes whose homology is finite dimensional, the forgetful functor U : Ind( MHC Q ) −→ Ind( Ch ∗ ( Q ) ω ) ≃ Ch ∗ ( Q ) preserves colimits and by Theorem 6.4, the composite U ◦ D ∗ is weakly equivalent to S ∗ ( − , Q ) .Therefore, the map β is weakly equivalent to the map U ( α ) in particular, we deduce that thesource of α is in MHC Q (as opposed to Ind(
MHC Q ) ). And since the functor U : MHC Q → Ch ∗ ( C ) is conservative, it follows that α is an equivalence as desired.Hence, by Proposition 6.5, there is a unique extension of D ∗ to a symmetric monoidalfunctor N (Sch C ) −→ MHC Q that has proper descent. Moreover, as we proved above, if Y is an object of Sch C and X • −→ Y is a proper hypercover by smooth schemes, then colim ∆ op D ∗ ( X • , Q ) has finitely generated homology. It follows that this unique extensionof D ∗ to Sch C lands in MHC Q ⊂ Ind(
MHC Q ) .For the case of D ∗ , since dualization induces a symmetric monoidal equivalence of ∞ -categories MHC op Q ≃ MHC Q , we see that we have no other choice but to define D ∗ as thecomposite N (Sch) op ( D ∗ ) op −−−−→ MHC op Q ( − ) ∨ −−−→ MHC Q and this will be the unique symmetric monoidal functor D ∗ : N (Sch C ) op −→ MHC Q which satisfies descent with respect to proper hypercovers and whose restriction to Sm C isequivalent to the functor D ∗ constructed in the previous subsection. (cid:3) Proposition 6.7. (1) There is a weak equivalence D ∗ ( − ) Q ≃ S ∗ ( − , Q ) in the categoryof symmetric monoidal ∞ -functors N (Sch C ) −→ Ch ∗ ( Q ) .(2) There is a weak equivalence A ∗ P L ( − ) ≃ D ∗ ( − ) Q ≃ S ∗ ( − , Q ) in the category ofsymmetric monoidal ∞ -functors N (Sch C ) op −→ Ch ∗ ( Q ) .Proof. We prove the first claim. By construction D ∗ ( − ) Q is a symmetric monoidal functorthat satisfies proper descent. By [Bla16, Proposition 3.24], the same is true for S ∗ ( − , Q ) .Since these two functors are moreover weakly equivalent when restricted to Sm C , they areequivalent by Proposition 6.5.The linear dual functor is strong monoidal when restricted to chain complexes whosehomology is of finite type. Moreover, both S ∗ ( − , Q ) and D ∗ ( − ) Q land in the ∞ -category ofsuch chain complexes. Therefore, the equivalence S ∗ ( − , Q ) ≃ D ∗ ( − ) Q follows from the firstpart. The equivalence A ∗ P L ( − ) ≃ S ∗ ( − , Q ) is classical. (cid:3) Formality of the singular chains functor
In this section, we prove the main results of the paper on the formality of the singularchains functor. We also explain some applications to operad formality.
Definition 7.1.
Let X be a complex scheme and let α be a rational number. We say thatthe weight filtration on H ∗ ( X, Q ) is α -pure if for all n ≥ we have Gr Wp H n ( X, Q ) = 0 for all p = αn. Remark 7.2.
Note that since the weight filtration on H n ( − , Q ) has weights in the interval [0 , n ] ∩ Z , the above definition makes sense only for α ∈ [0 , ∩ Q . For α = 1 we recoverthe purity property shared by the cohomology of smooth projective varieties. A very simpleexample of a variety whose filtration is α -pure, with α not integer, is given by C \ { } . Itscohomology is concentrated in degree and weight , so its weight filtration is / -pure.We refer to Proposition . in the following section for more elaborate examples.Here is our main theorem. Theorem 7.3.
Let α be a non-zero rational number. The singular chains functor S ∗ ( − , Q ) : Sch C −→ Ch ∗ ( Q ) is formal as a lax symmetric monoidal functor when restricted to schemes whose weightfiltration in cohomology is α -pure. IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 19
Proof.
By Corollay 2.4, it suffices to prove that this functor is formal as an ∞ -lax monoidalfunctor. By Proposition 6.7, it is equivalent to prove that D ∗ ( − ) Q is formal. We denote by ¯ D ∗ the composite of D ∗ with a symmetric monoidal inverse of the equivalence of Theorem5.4. Because of that theorem, D ∗ ( − ) Q is weakly equivalent to Π Q ◦ ¯ D ∗ . The restriction of ¯ D ∗ to Sch α - pure C lands in Ch ∗ (MHS Q ) α - pure , the full subcategory of Ch ∗ (MHS Q ) spannedby chain complexes whose homology is α -pure. By Corollary 4.6, the ∞ -functor Π Q from Ch ∗ (MHS Q ) α - pure to Ch ∗ ( Q ) is formal and hence so is Π Q ◦ ¯ D ∗ . (cid:3) We now list a few applications of this result.7.1.
Noncommutative little disks operad.
The authors of [DSV15] introduce two non-symmetric topological operads A s S and A s S ⋊ S . In each arity, these operads are givenby a product of copies of C − { } and the operad maps can be checked to be algebraic maps.It follows that the operads A s S and A s S ⋊ S are operads in the category Sm C and thatthe weight filtration on their cohomology is -pure. Therefore, by 7.3 we have the followingresult. Theorem 7.4.
The operads S ∗ ( A s S , Q ) and S ∗ ( A s S ⋊ S , Q ) are formal. Remark 7.5.
The fact that the operad S ∗ ( A s S , Q ) is formal is proved in [DSV15, Propo-sition 7] by a more elementary method and it is true even with integral coefficients. Theother formality result was however unknown to the authors of [DSV15].7.2. Self-maps of the projective line.
We denote by F d the algebraic variety of degree d algebraic maps from P C to itself that send the point ∞ to the point . Explicitly, a pointin F d is a pair ( f, g ) of degree d monic polynomials without any common roots. Sending amonic polynomials to its set coefficients, we may see the variety F d as a Zariski open subsetof A d C . See [Hor16, Section 5] for more details. Proposition 7.6.
The weight filtration on H ∗ ( F d , Q ) is -pure.Proof. The variety F d is denoted Poly d, in [FW16, Definition 1.1.]. It is explained inStep 4 of the proof of Theorem 1.2. in that paper that the variety F d is the quotient ofthe complement of a hyperplane arrangement H in A d C by the group Σ d × Σ d acting bypermuting the coordinates. A transfer argument then shows that H k ( F d , Q ) is a subspace of H k ( A d C − H, Q ) . Moreover this inclusion is a morphism of mixed Hodge structures. Sincethe mixed Hodge structure of H k ( A d C − H, Q ) is well-known to be pure of weight k (byProposition 8.2 or by [Kim94]), the desired result follows. (cid:3) In [Caz12, Proposition 3.1.], Cazanave shows that the scheme F d F d has the structure ofa graded monoid in Sm C . The structure of a graded monoid can be encoded by a coloredoperad. Thus the following follows from 7.3. Theorem 7.7.
The graded monoid in chain complexes L d S ∗ ( F d , Q ) is formal. The little disks operad.
In [Pet14], Petersen shows that the operad of little disks D isformal. The method of proof is to use the action of a certain group GT( Q ) on S ∗ ( PAB Q , Q ) which follows from work of Drinfeld’s. Here the operad PAB Q is rationally equivalent to D and GT( Q ) is the group of Q -points of the pro-algebraic Grothendieck-Teichmüller group.We can reinterpret this proof using the language of mixed Hodge structures. Indeed, thegroup GT receives a map from the group Gal(MT( Z )) , the Galois group of the Tannakiancategory of mixed Tate motives over Z (see [And04, 25.9.2.2]). Moreover there is a map Gal(MHTS Q ) → Gal(MT( Z )) from the Tannakian Galois group of the abelian category of mixed Hodge Tate structures (the full subcategory of MHS Q generated under extensions bythe Tate twists Q ( n ) for all n ) which is Tannaka dual to the tensor functor MT( Z ) −→ MHTS Q sending a mixed Tate motive to its Hodge realization. This map of Galois group allowsus to view S ∗ ( PAB Q , Q ) as an operad in Ch ∗ (MHS Q ) which moreover has a -pure weightfiltration (as follows from the computation in [Pet14]). Therefore by Corollary 4.6, theoperad S ∗ ( PAB Q , Q ) is formal and hence also S ∗ ( D , Q ) .7.4. The gravity operad.
In [DH17], Dupont and the second author prove the formalityof the gravity operad of Getzler. It is an operad structure on the collection of gradedvector spaces { H ∗− ( M ,n +1 ) , n ∈ N } . It can be defined as the homotopy fixed points ofthe circle action on S ∗ ( D , Q ) . The method of proof in [DH17] can also be interpreted interms of mixed Hodge structures. Indeed, a model G rav W ′ of gravity is constructed in 2.7of loc. cit. This model comes with an action of GT( Q ) and a GT( Q ) -equivariant map ι : G rav W ′ −→ S ∗ ( PAB Q , Q ) which is injective on homology. As in the previous subsection,this action of GT( Q ) lets us interpret G rav W ′ as an operad in Ch ∗ (MHS Q ) . Moreover,the injectivity of ι implies that G rav W ′ also has a -pure weight filtration. Therefore byCorollary 4.6, we deduce the formality of G rav W ′ . In fact, we obtain the stronger resultthat the map ι : G rav W ′ −→ S ∗ ( PAB Q , Q ) is formal as a map of operads (i.e. it is connected to the induced map in homology by azig-zag of maps of operads).7.5. E -formality. The above results deal with objects whose weight filtration is pure. Ingeneral, for mixed weights, the singular chains functor is not formal, but it is E -formal aswe now explain.The r -stage of the spectral sequence associated to a filtered complex is an r -bigradedcomplex with differential of bidegree ( − r, r − . By taking its total degree and consideringthe column filtration we obtain a filtered complex. Denote by E r : Ch ∗ ( F Q ) −→ Ch ∗ ( F Q ) the resulting symmetric monoidal ∞ -functor. Denote by ˜Π W Q : MHC Q −→ Ch ∗ ( F Q ) the forgetful functor defined by sending a mixed Hodge complex to its rational componenttogether with the weight filtration. Note that, since the weight spectral sequence of a mixedHodge complex degenerates at the second stage, the homology of E ◦ ˜Π W Q gives the weightfiltration on the homology of mixed Hodge complexes. We have: Theorem 7.8.
Denote by S fil ∗ : N (Sch C ) −→ Ch ∗ ( Q ) the composite functor N (Sch C ) D ∗ −−→ MHC Q ˜Π W Q −−→ Ch ∗ ( F Q ) . There is an equivalence of symmetric monoidal ∞ -functors E ◦ S fil ∗ ≃ S fil ∗ . IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 21
Proof.
It suffices to prove an equivalence ˜Π W Q ≃ E ◦ ˜Π W Q . We have a commutative diagramof symmetric monoidal ∞ -functors. Ch ∗ (MHS Q ) Π W Q (cid:15) (cid:15) T / / MHC Q ˜Π W Q (cid:15) (cid:15) Ch ∗ ( F Q ) T / / E (cid:15) (cid:15) Ch ∗ ( F Q ) E (cid:15) (cid:15) Ch ∗ ( F Q ) T / / Ch ∗ ( gr Q ) The commutativity of the top square follows from the definition of T . We prove that thebottom square commutes. Recall that T ( K, W ) is the filtered complex ( K, T W ) defined by T W p K n := W p + n K n . It satisfies d ( T W p K p ) ⊂ T W p +1 K n − . In particular, the induceddifferential on Gr T W K is trivial. Therefore we have: E − p,q ( K, T W ) ∼ = H q − p ( Gr pT W K ) ∼ = Gr pT W K q − p = Gr qW K q − p = E − q, q − p ( K, W ) . This proves that the above diagram commutes.Since T is an equivalence of ∞ -categories, it is enough to prove that E ◦ ˜Π W Q ◦ T isequivalent to ˜Π W Q ◦T . By the commutation of the above diagram it suffices to prove that thereis an equivalence E ◦ Π W Q ∼ = Π W Q . This follows from Lemma 4.4, since E = U fil ◦ gr . (cid:3) Rational homotopy of schemes and formality
For X a space, we denote by A ∗ P L ( X ) , Sullivan’s algebra of piecewise linear differentialforms. This is a commutative dg-algebra over Q that captures the rational homotopy typeof X . A contravariant version of Theorem . gives: Theorem 8.1.
Let α be a non-zero rational number. The functor A ∗ P L : Sch op C −→ Ch ∗ ( Q ) is formal as a lax symmetric monoidal functor when restricted to schemes whose weightfiltration in cohomology is α -pure.Proof. The proof is the same as the proof of Theorem . using D ∗ instead of D ∗ and usingthe fact that D ∗ ( − ) Q is quasi-isomorphic to A ∗ P L as a lax monoidal functor (see [NA87,Théorème 5.5]). (cid:3)
Recall that a topological space X is said to be formal if there is a string of quasi-isomorphisms of commutative dg-algebras from A ∗ P L ( X ) to H ∗ ( X, Q ) , where H ∗ ( X, Q ) isconsidered as a commutative dg-algebra with trivial differential. Likewise, a continuous mapof topological spaces f : X −→ Y is formal if there is a string of homotopy commutativediagrams of morphisms A ∗ P L ( Y ) f ∗ (cid:15) (cid:15) ∗ o o (cid:15) (cid:15) · · · o o / / ∗ (cid:15) (cid:15) / / H ∗ ( Y, Q ) H ∗ ( f ) (cid:15) (cid:15) A ∗ P L ( X ) ∗ o o · · · o o / / ∗ / / H ∗ ( X, Q ) where the horizontal arrows are quasi-isomorphisms. Note that if f : X → Y is a map oftopological spaces and X and Y are both formal spaces, then it is not always true that f isa formal map. Also, in general, the composition of formal morphisms is not formal. Theorem . gives functorial formality for schemes with pure weight filtration in coho-mology, generalizing both “purity implies formality” statements appearing in [Dup16] forsmooth varieties and in [CC17] for singular projective varieties. We also get a result ofpartial formality as done in these references, via Proposition 2.10. Our generalization isthreefold, as explained in the following three subsections.8.1. Rational weights.
To our knowledge, in the existing references where α -purity of theweight filtration is discussed, only the cases α = 1 and α = 2 are considered, whereas weobtain formality for varieties with α -pure cohomology, for α an arbitrary non-zero rationalnumber. This gives a whole new family of formal spaces. For instance, we have: Proposition 8.2.
Let H = { H , . . . , H k } be a set of linear subspaces of C n such that for allproper subset S ⊂ { , . . . , k } , the intersection H S := ∩ i ∈ S H i is of codimension d | S | . Thenthe mixed Hodge structure on H ∗ ( C n − ∪ i H i , Q ) is pure of weight d/ (2 d − .Proof. We proceed by induction on k . This is easy to do for k = 1 . Now, we consider thevariety X = C n − ∪ k − i H i , It contains an open subvariety U = C n − ∪ ki H i and its closedcomplement Z = H k − ∪ k − i H i ∩ H k which has codimension d . Therefore the purity longexact sequence on cohomology groups has the form . . . −→ H r − d ( Z )( − d ) −→ H r ( X ) −→ H r ( U ) −→ H r +1 − d ( Z )( − d ) −→ . . . By the induction hypothesis, the Hodge structure on H r +1 − d ( Z )( − d ) and on H r ( X ) arepure of weight dr/ (2 d − and hence it is also the case for H r ( U ) as desired. (cid:3) Remark 8.3.
This proposition is well-known for d = 1 and is proved for instance in [Kim94].8.2. Functoriality.
Every morphism of smooth complex projective schemes is formal. How-ever, if f : X → Y is an algebraic morphism of complex schemes (possibly singular and/ornon-projective), and both X and Y are formal, the morphism f need not be formal. Example 8.4.
Consider the algebraic Hopf fibration f : C \ { } −→ P C defined by ( x , x ) [ x : x ] . Both spaces C \ { } ≃ S and P C ≃ S are formal. As is well-known, the morphism induced by f in cohomology is trivial, while its homotopy type is not.Therefore f is not formal. Note in fact, that P C has -pure weight filtration while C \ { } has / -pure weight filtration.Theorem . tells us that if f : X −→ Y is a morphism of algebraic varieties and both X and Y have α -pure cohomology, with α a non-zero rational number (the same α for X and Y ), then f is a formal morphism. This generalizes the formality of holomorphic morphismsbetween compact Kähler manifolds of [DGMS75] and enhances the results of [Dup16] and[CC17] by providing them with functoriality. In fact, we have: Proposition 8.5.
Let f : X −→ Y be a morphism of complex schemes inducing a non-trivialmorphism f ∗ : A ∗ P L ( Y ) −→ A ∗ P L ( X ) . Assume that the weight filtration on the cohomologyof X (resp. Y ) is α -pure (resp. β -pure). Then:(1) If α = β then H ∗ ( f ) = 0 .(2) f is formal if and only if α = β .Proof. Assume that α = β . Since morphisms of mixed Hodge structures are strictly com-patible with the weight filtration, to show that H ∗ ( f ) is trivial it suffices to show that themorphism Gr Wp H n ( Y, Q ) −→ Gr Wp H n ( X, Q ) is trivial for all p ∈ Z and all n > . This follows from the purity conditions. Now, since f ∗ is assumed to be non-trivial and H ∗ ( f ) = 0 , it follows that f cannot be formal. To end theproof, note that when α = β , Theorem . ensures that f is formal. (cid:3) IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 23
Non-projective singular schemes.
The formality of non-projective singular complexvarieties with pure Hodge structure seems to be a new result.
Example 8.6.
Let X be an irreducible singular projective variety of dimension n with -pure weight filtration in cohomology (for instance, a V -manifold). Let p ∈ X be a smoothpoint of X . Then the complement X − p has -pure weight filtration in cohomology. Indeed,using a Mayer-Vietoris long exact sequence argument, one can show that for k ≤ n − wehave H k ( X − p ) ∼ = H k ( X ) and H n ( X − p ) = 0 . Therefore the inclusion X − p ֒ → X isformal.8.4. E -formality. We also have a contravariant version of Theorem . . Theorem 8.7.
Denote by A ∗ fil : N (Sch C ) op −→ Ch ∗ ( F Q ) the composite functor N (Sch C ) op D ∗ −−→ MHC Q ˜Π W Q −−→ Ch ∗ ( F Q ) . Then(1) The lax symmetric monoidal ∞ -functors A ∗ fil and E ◦ A ∗ fil are weakly equivalent.(2) Let U : Ch ∗ ( F Q ) −→ Ch ∗ ( Q ) denote the forgetful functor. The lax symmetricmonoidal ∞ -functor U ◦ E ◦ A ∗ fil : N (Sch C ) op → Ch ∗ ( Q ) is weakly equivalent toSullivan’s functor A ∗ P L of piece-wise linear forms.(3) The lax symmetric monoidal functor U ◦ E ◦ A ∗ fil : Sm op C → Ch ∗ ( Q ) is weaklyequivalent to Sullivan’s functor A ∗ P L of piece-wise linear forms.Proof.
The first part is proven as Theorem 7.8 replacing D ∗ by D ∗ . The second partfollows from the first part and the fact that A ∗ P L ( − ) is naturally weakly equivalent to D ∗ ( − ) Q ≃ U ◦ A ∗ fil (Proposition 6.7). The third part follows from the second part andTheorem 2.3, using the fact that both functors are ordinary lax monoidal functors whenrestricted to smooth schemes. (cid:3) Remark 8.8.
In [Mor78] it is proven that the complex homotopy type of every smoothcomplex scheme is E -formal. This is extended to possibly singular schemes and their mor-phisms in [CG14]. Then, a descent argument is used to prove that for nilpotent spaces (withfinite type minimal models), this result descends to the rational homotopy type. Theorem . enhances the contents of [CG14] in two ways: first, since descent is done at the level offunctors, we obtain E -formality over Q for any complex scheme, without nilpotency con-ditions (the only property needed is finite type cohomology). Second, the functorial natureof our statement makes E -formality at the rational level, compatible with composition ofmorphisms.8.5. Formality of Hopf cooperads.
Our main theorem takes two dual forms, one covari-ant and one contravariant. The covariant theorem yields formality for algebraic structures(like monoids, operads, etc.), the contravariant theorem yields formality for coalgebraicstructure (like the comonoid structure coming from the diagonal X → X × X for any va-riety X ). One might wonder if there is a way to do both at the same time. For example,if M is a topological monoid, then H ∗ ( M, Q ) is a Hopf algebra where the multiplicationcomes from the diagonal of M and the comultiplication comes from the multiplication of M .One may ask wether S ∗ ( M, Q ) is formal as a Hopf algebra. This question is not well-posedbecause S ∗ ( M, Q ) is not a Hopf algebra on the nose. The problem is that there does notseem to exist a model for singular chains or cochains that is strong monoidal : the stan-dard singular chain functor S ∗ ( − , Q ) is lax monoidal and Sullivan’s functor A ∗ P L is oplaxmonoidal functor from
Top to Ch ∗ ( Q ) op . Nevertheless, the functor A ∗ P L is strong monoidal “up to homotopy”. It follows that, if M is a topological monoid, A ∗ P L ( M ) has the structure of a cdga with a comultiplication upto homotopy and it makes sense to ask if it has formality as such an object. In order toformulate this more precisely, we introduce the notion of an algebraic theory. The followingis inspired by Section 3 of [LV14]. Definition 8.9.
An algebraic theory is a small category T with finite products. For C acategory with finite products, a T -algebra in C is a finite product preserving functor T −→ C .There exist algebraic theories for which the T -algebras are monoids, groups, rings, oper-ads, cyclic operads, modular operads etc. Remark 8.10.
Definitions of algebraic theories in the literature are usually more restrictive.This definition will be sufficient for our purposes.
Definition 8.11.
Let T be an algebraic theory. Let k be a field. Then a dg Hopf T -coalgebra over k is a finite coproduct preserving functor from T op to the category of cdga’sover k . Remark 8.12.
Recall that the coproduct in the category of cdga’s is the tensor product. Itfollows that a dg Hopf T -algebra for T the algebraic theory of monoids is a dg Hopf algebrawhose multiplication is commutative. A dg Hopf T -algebra for T the theory of operads iswhat is usually called a dg Hopf cooperad in the literature. Definition 8.13.
Let T be an algebraic theory and C a category with products and with anotion of weak equivalences. A weak T -algebra in C is a functor F : T −→ C such that foreach pair ( s, t ) of objects of T , the canonical map F ( t × s ) −→ F ( t ) × F ( s ) is a weak equivalence. A weak T -algebra in the opposite category of CDGA k is called aweak dg Hopf T -coalgebra.Observe that if X : T −→ Top is a T -algebra in topogical spaces (or even a weak T -algebra), then A ∗ P L ( X ) is a weak dg Hopf T -coalgebra. Our main theorem for Hopf T -coalgebras is the following. Theorem 8.14.
Let α be a rational number different from zero. Let X : T −→ Sch C be a T -algebra such that for all t ∈ T , the weight filtration on the cohomology of X ( t ) is α -pure.Then A ∗ P L ( X ) is formal as a weak dg Hopf T -coalgebra.Proof. Being a weak T -coalgebra is a property of a functor T op → CDGA k that is invariantunder quasi-isomorphism. Thus the result follows immediately from Theorem 8.1. (cid:3) It should be noted that knowing that A ∗ P L ( X ) is formal as a dg Hopf T -coalgebra im-plies that the data of H ∗ ( X, Q ) is enough to reconstruct X as a T -algebra up to rationalequivalence. Indeed, recall the Sullivan spatial realization functor h−i : CDGA k −→ Top
Applying this functor to a weak dg Hopf T -coalgebra yields a weak T -algebra in rationalspaces. Specializing to A ∗ P L ( X ) where X is a T -algebra in spaces, we get a rational modelfor X in the sense that the map X −→ hA ∗ P L ( X ) i is a rational weak equivalence of weak T -algebras whose target is objectwise rational. Itshould also be noted that for reasonable algebraic theories T (including in particular thetheory for monoids, commutative monoids, operads, cyclic operads), the homotopy theory IXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC MONOIDAL FUNCTORS 25 of T -algebras in spaces is equivalent to that of weak T -algebras by the main theorem of[Ber06]. In particular our weak T -algebra hA ∗ P L ( X ) i can be stricitified to a strict T -algebrathat models the rationalization of X . If A ∗ P L ( X ) is formal, one also get a rational model for X by applying the spatial realization to the strict Hopf T -coalgebra H ∗ ( X, Q ) . Thus therational homotopy type of X as a T -algebra is a formal consequence of H ∗ ( X, Q ) as a Hopf T -coalgebra. Example 8.15.
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