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Real non-abelian mixed Hodge structures for quasi-projective varieties: formality and splitting

We define and construct mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x-i), giving new results even for simply connected varieties. The mixed Hodge structures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these structures all become real variations of mixed Hodge structure.


Introduction
The main aims of this paper are to construct mixed Hodge structures on the real relative Malcev homotopy types of open complex varieties, and to investigate how far these can be recovered from the structures on cohomology groups of local systems, and in particular the Gysin spectral sequence. In this respect, this paper is a sequel to [Pri4], which considers the same question for proper complex varieties.
In [Mor], Morgan established the existence of natural mixed Hodge structures on the minimal model of the rational homotopy type of a smooth variety X, and used this to define natural mixed Hodge structures on the rational homotopy groups π * (X ⊗ Q) of X. This construction was extended to singular varieties by Hain in [Hai].
For non-nilpotent topological spaces, the rational homotopy type is too crude an invariant to recover much information, so schematic homotopy types were introduced in [Toë], based on ideas from [Gro]. [Pri2] showed how to recover the groups π n (X) ⊗ Z R from schematic homotopy types for very general topological spaces, and also introduced the intermediate notion of relative Malcev homotopy type, simultaneously generalising both rational and schematic homotopy types.
In [Pri4], the notions of mixed Hodge and mixed twistor structures on real relative Malcev homotopy types were introduced, and were constructed for homotopy types of compact Kähler manifolds. [Pri4] also introduced an important class of MHS or MTSthose which are SL 2 -split or S-split. These split on tensoring with the ring S := R[x] whose Hodge filtration on S ⊗ R C is given by powers of (x − i). It was then shown in [Pri4] that any S-split MHS or MTS on relative Malcev homotopy types gives rise to MHS or MTS on the relative Malcev homotopy groups, with the latter also being S-split. Adapting [DGMS] gave rise to an S-splitting of the MHS and MTS for homotopy types of compact Kähler manifolds, and hence S-split MHS/MTS on the homotopy groups. This paper is broken into two main parts: we first adapt [Pri4] to construct MHS/MTS for relative Malcev homotopy types of quasi-projective varieties in §2, but only when the monodromy around the divisor is trivial. A more general case (unitary monodromy around the divisor) is addressed in §3. Whereas the S-splittings of [Pri4] were realised concretely using the principle of two types, the second part of the paper ( § §4-5) establishes abstract existence results for S-splittings of general mixed Hodge and mixed twistor structures. These latter results are then needed to construct mixed Hodge and mixed twistor structures on relative Malcev homotopy groups of quasi-projective varieties ( §5.4).
The structure of the paper is as follows: §1 recalls several basic results from [Pri4] concerning non-abelian filtrations, mixed Hodge structures and mixed twistor structures. These are adapted slightly here to specialise to non-positively weighted homotopy types.
§2 deals with the Malcev homotopy type (Y, y) ρ,Mal of a quasi-projective variety Y = X − D with respect to a Zariski-dense representation ρ : π 1 (X, y) → R(R). For the local system O(R) on X corresponding to the regular representation O(R), the construction of MHS and MTS is based on the complex A • (X, O(R)) D , defined by modifying the O(R)-valued de Rham complex by allowing logarithmic singularities around the divisor.
When Y is smooth, Theorem 2.21 establishes a non-positively weighted MTS on (Y, y) ρ,Mal , with the associated graded object gr W (Y, y) ρ,Mal corresponding to the Requivariant DGA ( is the differential on the E 2 sheet of the Leray spectral sequence for j : Y → X, and H a−b (X, R b j * j −1 O(R)) has weight a + b. Theorem 2.22 shows that if R-representations underlie variations of Hodge structure, then the MTS above extends to a non-positively weighted MHS on (Y, y) ρ,Mal . Theorem 2.30 gives the corresponding results for singular quasi-projective varieties Y , with gr W (Y, y) ρ,Mal now characterised in terms of cohomology of a smooth simplicial resolution of Y . In §3, these results are extended to Zariski-dense representations ρ : π 1 (Y, y) → R(R) with unitary monodromy around local components of the divisor. The construction of MHS and MTS in these cases is much trickier than for trivial monodromy. The idea behind Theorem 3.16, inspired by [Mor], is to construct the Hodge filtration on the complexified homotopy type, and then to use homotopy limits of diagrams to glue this to the real form. When R-representations underlie variations of Hodge structure on Y , this gives a non-positively weighted MHS on (Y, y) ρ,Mal , with gr W (Y, y) ρ,Mal corresponding to the R-equivariant DGA ( regarded as a Hodge structure via the VHS structure on O(R). For more general R, Theorem 3.19 gives a non-positively weighted MTS, with the construction based on homotopy gluing over an affine cover of the analytic space P 1 (C). Simplicial resolutions then extend these results to singular varieties in Theorems 3.21 and Theorem 3.22. §3.5 discusses possible extensions to more general monodromy. §4 is concerned with splittings of MHS and MTS on finite-dimensional vector spaces. Every mixed Hodge structure V splits on tensoring with the ring S defined above, giving an S-linear isomorphism V ⊗ S ∼ = (gr W V ) ⊗ S preserving the Hodge filtration F . Differentiating with respect to V , this gives a map β : (gr W V ) → (gr W V ) ⊗ Ω(S/R) from which V can be recovered. Theorem 4.8 shows that the S-splitting can be chosen canonically, corresponding to imposing certain restrictions on β, and this gives an equivalence of categories. In Remark 4.11, β is explicitly related to the complex splitting of [Del4]. Theorem 4.15 then gives the corresponding results for mixed twistor structures.
The main result in §5 is Theorem 5.16, which shows that every non-positively weighted MHS or MTS on a real relative Malcev homotopy type admits a strictification, in the sense that it is represented by an R-equivariant DGA in ind-MHS or ind-MTS. Corollary 5.22 In this section, we summarise several results from [Pri4] concerning non-abelian generalisations of real mixed Hodge and mixed twistor structures.
Lemma 1.1. There is an equivalence of categories between flat quasi-coherent G mequivariant sheaves on A 1 , and exhaustive (i.e. V = n F n V ) filtered vector spaces, where G m acts on A 1 via the standard embedding G m ֒→ A 1 .
Proof. This is [Pri4,Lemma 1.6]. Given a filtered vector space V , the equivalence sets M to be the Rees module ξ(V, F ) := F n V , with G m -action given by setting F n V to be weight n, and the k[t]-module structure determined by letting t be the inclusion F n V ֒→ F n+1 V .
1.1. Mixed Hodge and mixed twistor structures. Definition 1.2. Define C to be the real affine scheme C/R A 1 obtained from A 1 C by restriction of scalars, so for any real algebra A, C(A) = A 1 C (A ⊗ R C) ∼ = A ⊗ R C. Choosing i ∈ C gives an isomorphism C ∼ = A 2 R , and we let C * be the quasi-affine scheme C − {0}. Define S to be the real algebraic group C/R G m obtained as in [Del1,2.1.2] from G m,C by restriction of scalars. Note that there is a canonical inclusion G m ֒→ S, and that S acts on C and C * by inverse multiplication, i.e.
Remark 1.3. Fix an isomorphism C ∼ = A 2 , with co-ordinates u, v on C so that the isomorphism C(R) ∼ = C is given by (u, v) → u + iv. Thus the algebra O(C) associated to C is the polynomial ring R [u, v]. S is isomorphic to the scheme A 2 R − {(u, v) : u 2 + v 2 = 0}. On C C , we have alternative co-ordinates w = u + iv andw = u − iv, which give the standard isomorphism S C ∼ = G m,C × G m,C . Note that on C the co-ordinates w andw are of types (−1, 0) and (0, −1) respectively. Definition 1.4. Given an S-representation V , the inclusion G m ֒→ S (given by v = 0 in the co-ordinates above) gives a grading on V , which we denote by Equivalently, W n (V ⊗ C) is the sum of elements of type (p, q) for p + q = n.
Lemma 1.5. The category of flat S-equivariant quasi-coherent sheaves on C * is equivalent to the category of pairs (V, F ), where V is a real vector space and F an exhaustive decreasing filtration on V ⊗ R C.
Proof. This is contained in [Pri4,Corollary 1.8]. The construction is given by first forming the complex Rees module with respect to F andF . This has a C[w,w-module structure, and an S-action as a submodule of V C [w, w −1 ,w,w −1 ]. We then form ξ(V, F) ⊂ ξ(V C , F,F ) to consist of real elements. This a flat S-equivariant R[u, v]-module, so defines a flat S-equivariant quasicoherent sheaf on C, and hence pulls back to one on C * . Definition 1.6. Given an affine scheme X over R, we define an algebraic mixed Hodge structure X MHS on X to consist of the following data: (1) an G m × S-equivariant affine morphism X MHS → A 1 × C * , (2) a real affine scheme grX MHS equipped with an S-action, an isomorphism X ∼ = X MHS × (A 1 ×C * ),(1,1) Spec R, (4) a G m × S-equivariant isomorphism grX MHS × C * ∼ = X MHS × A 1 ,0 Spec R, where G m acts on grX MHS via the inclusion G m ֒→ S. This is called the opposedness isomorphism.
Definition 1.7. Define a (real) quasi-MHS to be a real vector space V , equipped with an exhaustive increasing filtration W on V , and an exhaustive decreasing filtration F on V ⊗ C.
We adopt the convention that a (real) MHS is a finite-dimensional quasi-MHS on which W is Hausdorff, satisfying the opposedness condition Define a (real) ind-MHS to be a filtered direct limit of MHS. Say that an ind-MHS is bounded below if W N V = 0 for N ≪ 0.
Lemma 1.8. The category of flat G m ×S-equivariant quasi-coherent sheaves M on A 1 ×C * is equivalent to the category of quasi-MHS.
Under this equivalence, bounded below ind-MHS (V, W, F ) correspond to flat algebraic mixed Hodge structures M on V whose weights with respect to the G m × 1-action are bounded below.
A real splitting of the Hodge filtration is equivalent to giving a (real) Hodge structure on V (i.e. an S-action).
Proof. This is [Pri4,Proposition 1.40]. The construction is given by combining Lemmas 1.1 and 1.5.

Mixed twistor structures.
Definition 1.9. Adapting [Sim1] §1 from complex to real structures, say that a twistor structure on a real vector space V consists of a vector bundle E on P 1 R , with an isomorphism V ∼ = E 1 , the fibre of E over 1 ∈ P 1 . Lemma 1.10. The category of finite flat algebraic twistor filtrations on real vector spaces is equivalent to the category of twistor structures.
Proof. This is [Pri4,Proposition 1.8]. The flat algebraic twistor filtration is a flat G mequivariant quasi-coherent sheaf M on C * , with M | 1 = V . Taking the quotient by the right G m -action, M corresponds to a flat quasi-coherent sheaf M Gm on [C * /G m ]. Now, [C * /G m ] ∼ = [(A 2 − {0})/G m ] = P 1 , so Lemma 1.1 implies that M Gm corresponds to a flat quasi-coherent sheaf E on P 1 . Note that E 1 = (M | Gm ) Gm ∼ = M 1 ∼ = V , as required.
Definition 1.11. Let C * → C * be theétale covering of C * given by cutting out the divisor {u − iv = 0} from C * ⊗ R C, for co-ordinates u, v as in Definition 1.3.
Note that C * ∼ = A 1 C × G m,C , with the isomorphism given by sending (u, v) to (u + iv, u − iv). Definition 1.12. Adapting [Sim1] §1 from complex to real structures, say that a (real) mixed twistor structure (real MTS) on a real vector space V consists of a finite locally free sheaf E on P 1 R , equipped with an exhaustive Hausdorff increasing filtration by locally free subsheaves W i E , such that for all i the graded bundle gr W i E is semistable of slope i (i.e. a direct sum of copies of O P 1 (i)). We also require an isomorphism V ∼ = E 1 , the fibre of E over 1 ∈ P 1 .
Define a quasi-MTS on V to be a flat quasi-coherent sheaf E on P 1 R , equipped with an exhaustive increasing filtration by quasi-coherent subsheaves W i E , together with an isomorphism V ∼ = E 1 . Define an ind-MTS to be a filtered direct limit of real MTS, and say that an ind-MTS E on V is bounded below if W N E = 0 for N ≪ 0.
Definition 1.13. Given an affine scheme X over R, we define an algebraic mixed twistor structure X MTS on X to consist of the following data: ( (2) a real affine scheme grX MTS equipped with a G m -action, This is called the opposedness isomorphism.
Lemma 1.14. The category of flat G m ×G m -equivariant quasi-coherent sheaves on A 1 ×C * is equivalent to the category of quasi-MTS. Under this equivalence, bounded below ind-MTS on V correspond to flat algebraic mixed twistor structures ξ(V, MTS) on V whose weights with respect to the G m × 1-action are bounded below.
For an M -representation A in algebras, we define DG Z Alg A (M ) (resp. DGAlg A (M )) to be the comma category A ↓ DG Z Alg(M ) (resp. A ↓ DGAlg(M )).
Denote the opposite categories by dg Z Aff A (M ) and dgAff A (M ). Given an object A ∈ DG Z Alg(M ) * , write Spec A ∈ dg Z Aff(M ) * for the corresponding object of the opposite category. For each of the categories C above, let Ho(C) be the category obtained by formally inverting quasi-isomorphisms.
Definition 1.16. Given a reductive pro-algebraic monoid M , and an M -representation Y in schemes, define DG Z Alg Y (M ) to be the category of M -equivariant quasi-coherent Z-graded graded-commutative cochain algebras on Y . Define a weak equivalence in this category to be a map giving isomorphisms on cohomology sheaves (over Y ), and define Ho(DG Z Alg Y (M )) to be the homotopy category obtained by localising at weak equivalences. Define the categories dg Z Aff Y (M ), Ho(dg Z Aff Y (M )) to be the respective opposite categories.
When Y is affine, define DGAlg Y (M ) ⊂ DG Z Alg Y (M ) to consist of non-negatively graded cochain graded-commutative R-algebras on Y , with dgAff Y (M ), Ho(DGAlg Y (M )) and Ho(dg Z Aff Y (M )) defined similarly.
Definition 1.17. Given a reductive pro-algebraic monoid K acting on a reductive proalgebraic monoid M and on a scheme Y , define dg Z Aff Y (M ) * (K) to be the category Definition 1.18. Recall from [Pri4,Proposition 3.34] that for a reductive pro-algebraic group R, the relative Malcev homotopy type (X, x) R,Mal of a pointed manifold (X, x) where O(R) is the local system on X corresponding to the left action of π 1 (X, x) on O(R).

Hodge and twistor structures.
Definition 1.19. Define the real algebraic group S 1 to be the circle group, whose Avalued points are given by {(a, b) ∈ A 2 : a 2 + b 2 = 1}. Note that S 1 ֒→ S, and that S/G m ∼ = S 1 . This latter S-action gives S 1 a split Hodge filtration.
The following definitions and results are taken from [Pri4,§4]. Fix a real reductive pro-algebraic group R, a pointed connected topological space (X, x), and a Zariski-dense morphism ρ : π 1 (X, x) → R(R).
Definition 1.20. An algebraic Hodge filtration on a pointed Malcev homotopy type (X, x) ρ,Mal consists of the following data: (1) an algebraic action of S 1 on R, (2) an object (X, x) ρ,Mal F ∈ Ho(dg Z Aff C * (R) * (S)), where the S-action on R is defined via the isomorphism S/G m ∼ = S 1 , while the R ⋊ S-action on R combines multiplication by R with conjugation by S.
Note thatS-representations correspond via the morphism S →S to real Hodge structures of non-negative weights. In the co-ordinates of Remark 1.3, The following adapts [Pri4,Definition 4.4] to non-positive weights, replacing G m and S with Mat 1 andS respectively. Definition 1.23. A non-positively weighted algebraic mixed Hodge structure (X, x) R,Mal MHS on a pointed Malcev homotopy type (X, x) R,Mal consists of the following data: (1) an algebraic action of S 1 on R, (2) an object , where S acts on R via the S 1 -action, using the canonical isomorphism an isomorphism (called the opposedness isomorphism) , for the canonical map θ : Mat 1 ×S →S given by combining the inclusion Mat 1 ֒→S with the inclusion S ֒→S.
Definition 1.24. An non-positively weighted algebraic mixed twistor structure (X, x) R,Mal MTS on a pointed Malcev homotopy type (X, x) R,Mal consists of the following data: (1) an object (2) an object gr(X, x) R,Mal MTS ∈ Ho(dg Z Aff(R) * (Mat 1 )), an isomorphism (called the opposedness isomorphism) , for the canonical map θ : Mat 1 × G m → Mat 1 given by combining the identity on Mat 1 with the inclusion G m ֒→ Mat 1 .

Algebraic MHS/MTS for quasi-projective varieties I
Fix a smooth compact Kähler manifold X, a divisor D locally of normal crossings, and set Y := X − D. Let j : Y → X be the inclusion morphism.
Definition 2.1. Denote the sheaf of real C ∞ n-forms on X by A n X , and let A • X be the resulting complex (the real sheaf de Rham complex on X). Let These inherit filtrations, given by Remark 2.5. The filtration J essentially corresponds to the weight filtration W of [Del1,3.1.5]. However, the true weight filtration on cohomology, and hence on homotopy types, is given by the décalage Dec J (as in [Del1,Theorem 3.2.5] or [Mor]). Since Dec J gives the correct notion of weights, not only for mixed Hodge structures but also for Frobenius eigenvalues in the ℓ-adic case of [Pri5], we reserve the terminology "weight filtration" for W := Dec J.
2.1. The Hodge and twistor filtrations. If we write J for the complex structure on A • (X), then there is a differential d c := J −1 dJ on the underlying graded algebra A * (X). Note that dd c + d c d = 0.
Definition 2.6. There is an action of S on A * (X), which we will denote by a → λ ⋄ a, for λ ∈ C × = S(R). For a ∈ (A * (X) ⊗ C) pq , the action is given by It follows from [Sim2] Theorem 1 that there exists a harmonic metric on every semisimple real local system V on X. We then decompose the associated connection D : A 0 X (V) → A 1 X (V) as D = d + + ϑ into antisymmetric and symmetric parts, and let D c := i ⋄ d + − i ⋄ ϑ. Note that this decomposition is independent of the choice of metric, since the pluriharmonic metric is unique up to global automorphisms Γ(X, Aut(V)).
Definition 2.7. Given a semisimple real local system V on X, define the sheavesÃ • , uD + vD c ), for co-ordinates u, v as in §1.1. We denote the differential byD := uD + vD c .
Define the quasi-coherent sheafÃ • (X, V) D of cochain complexes on C bỹ . Note that the ⋄ action on A gives an action of G m ⊂ S onÃ • X (V) D over C. Definition 2.8. Given a semisimple local system V and an element t ∈ S 1 , define the semisimple local system t ⊛ V as follows. Decompose the connection D : A 0 X (V) → A 1 X (V) as D = d + + ϑ into antisymmetric and symmetric parts, and set ). Definition 2.9. Assume that we have a semisimple local system V, equipped with a discrete (resp. algebraic) action of S 1 on A 0 for v ∈ A 0 X (V) and t ⊛ D as above. Then define a discrete S(R) = C × -action (resp. an algebraic S-action) for a ∈ A X , f ∈ O(C) and v ∈ V. This gives an action on the global sectionsÃ • X (V) D over C. Note thatD(λ b) = λ (Db), so this is indeed an action on cochain complexes.
In the above definition, observe that the discrete action of S 1 on A 0 X (V) is algebraic if and only if V is a weight 0 real variation of Hodge structure.
Definition 2.10. Given a Zariski-dense representation ρ : π 1 (X, jy) → R(R), for R a proreductive pro-algebraic group, define an algebraic twistor filtration on the relative Malcev homotopy type (Y, y) R,Mal by (Y, y) R,Mal where O(R) is the local system of Definition 1.18, which is necessarily a sum of finite-dimensional semisimple local systems, and G m ⊂ S acts via the action of Definition 2.9.
A Zariski-dense representation ρ : π 1 (X, jy) → R(R) is equivalent to a morphism ̟ 1 (X, jy) red ։ R of pro-algebraic groups, where ̟ 1 (X, jy) red is the reductive quotient of the real pro-algebraic fundamental group ̟ 1 (X, jy). [Sim2] effectively gives a discrete S 1 -action on ̟ 1 (X, jy) red , corresponding (as in [Pri4,Lemma 5.7]) to the ⊛ action on semisimple local systems from Definition 2.8. This S 1 -action thus descends to R if and only if O(R) satisfies the conditions of Definition 2.9. Moreover, the S 1 -action is algebraic on R if and only if O(R) becomes a weight 0 variation of Hodge structures under the ⊛ action, by [Pri4,Proposition 5.12].
Definition 2.11. Take a Zariski-dense representation ρ : π 1 (X, jy) → R(R), for R a proreductive pro-algebraic group to which the S 1 -action on ̟ 1 (X, jy) red descends and acts algebraically. Then define an algebraic Hodge filtration on the relative Malcev homotopy type (Y, y) R,Mal by (Y, y) R,Mal in Ho(dg Z Aff C * (R) * (S)), where the S-action is given by the action of Definition 2.9.
If the S 1 action descends to R but is not algebraic, we still have the following: Proposition 2.12. The algebraic twistor filtration (Y, y) R,Mal T of Definition 2.10 is equipped with a (S 1 ) δ -action (i.e. a discrete S 1 -action) with the properties that (1) the S 1 -action and G m -actions commute, Proof. This is the same as the proof of [Pri4,Proposition 6.3]. The action comes from Definition 2.9, with t ∈ (S 1 ) δ acting on A *

Higher direct images and residues.
Definition 2.13. Let D m ⊂ X denote the union of all m-fold intersections of local components of the divisor D ⊂ X, and set D (m) to be its normalisation. Write ν m : D (m) → X for the composition of the normalisation map with the embedding of D m , and set C (m) := ν −1 m D m+1 . As in [Tim2,1.2], observe that D m − D m+1 is a smooth quasi-projective variety, isomorphic to D (m) − C (m) . Moreover, D (m) is a smooth projective variety, with C (m) a normal crossings divisor.
Definition 2.14. Recall from [Del1] Definition 2.1.13 that for n ∈ Z, Z(n) is the lattice (2πi) n Z, equipped with the pure Hodge structure of type (−n, −n). Given an abelian group A, write A(n) := A ⊗ Z Z(n).
Definition 2.15. On D (m) , define ε m by the property that ε m (m) is the integral local system of orientations of D m in X. Thus ε n is the local system ε n Z defined in [Del1,3.1.4].
That Res m is a quasi-isomorphism follows immediately from Lemmas 2.3 and 2.16.
Proposition 2.18. If the S 1 -action on ̟ 1 (X, jy) red descends to an algebraic action on R, then for the algebraic Hodge filtration (Y, y) R,Mal H ) on C * defines a pure ind-Hodge structure of weight a + b, corresponding to the S-action on Proof. We need to show that H a (gr J bÃ • (X, O(R)) D )| C * corresponds to a pure ind-Hodge structure of weight a + b, or equivalently a sum of vector bundles of slope a + b. We are therefore led to study the complex gr J bÃ • , and the right-hand side is just As in [Pri4, §1.1.2], we have anétale pushout C * = C * ∪ S C S of affine schemes, so to give an isomorphism F → G of quasi-coherent sheaves on C * is the same as giving an isomorphism f : making use of the fact that ε b already contains a factor of i b (coming from Z(−b)). Therefore As in [Pri4,Theorem 5.14], inclusion of harmonic forms gives an S-equivariant isomorphism on C * defines a pure ind-twistor structure of weight a + b, corresponding to the canonical G m -action on Proof. The proof of Proposition 2.18 carries over, replacing S-equivariance with G mequivariance, and Theorem 5.14 with Theorem 6.1.
Proposition 2.20. If the S 1 -action on ̟ 1 (X, jy) red descends to R, then the associated discrete S 1 -action of Proposition 2.12 on H a (gr J b O(Y, y) R,Mal T ) corresponds to the action of S 1 ⊂ S (see Definition 2.9) on Proof. The proof of Proposition 2.18 carries over, replacing S-equivariance with discrete S-equivariance.
Theorem 2.21. There is a canonical non-positively weighted mixed twistor structure (Y, y) R,Mal MTS on (Y, y) R,Mal , in the sense of Definition 1.24.
For the Rees algebra construction ξ of Lemma 1.1, we then set O(Y, y) R,Mal noting that this is flat and that (X, is the differential in the E 1 sheet of the spectral sequence associated to the filtration J. Combining Lemmas 2.16 and 2.17, it follows that this is the same as the differential In order to show that this defines a mixed twistor structure, it only remains to establish opposedness.
Spec R, and properties of Rees modules mean that this is just given by where the Mat 1 -action assigns gr Dec J n the weight n. By [Del1,Proposition 1.3.4], décalage has the formal property that the canonical map is a quasi-isomorphism. Since the right-hand side is just is of weight 2a − n + 2(n − a) = n for the G m action, and of weight n for the Mat 1 -action, being gr Dec J n . Theorem 2.22. If the local system on X associated to any R-representation underlies a polarisable variation of Hodge structure, then there is a canonical non-positively weighted mixed Hodge structure (Y, y) R,Mal MHS on (Y, y) R,Mal , in the sense of Definition 1.23. Proof. We adapt the proof of Theorem 2.21, replacing Proposition 2.19 with Proposition 2.18. The first condition is equivalent to saying that the S 1 action descends to R and is algebraic, by [Pri4,Proposition 5.12]. We therefore set for (Y, y) R,Mal H as in Definition 2.11, and let Hodge structure, and hence anSrepresentation.
Proposition 2.23. If the discrete S 1 -action on ̟ 1 (X, jy) red descends to R, then there are natural (S 1 ) δ -actions on (Y, y) R,Mal MTS and gr(Y, y) R,Mal MTS , compatible with the opposedness isomorphism, and with −1 ∈ S 1 acting as −1 ∈ G m .
Proof. This is a direct consequence of Proposition 2.12 and Proposition 2.20, since the Rees module construction transfers the discrete S 1 -action.

Singular and simplicial varieties.
Proposition 2.24. If Y is any separated complex scheme of finite type, there exists a simplicial smooth proper complex variety X • , a simplicial divisor D • ⊂ X • with normal crossings, and a map ( Proof. The results in [Del2,§8.2] and [SD,Propositions 5.1.7 and 5.3.4], adapted as in [Pri4,Corollary 9.3], give the equivalence required.
Definition 2.25. Given a simplicial diagram X • of smooth proper varieties and a point x ∈ X 0 , define the fundamental group ̟ 1 (|X • |, x) norm to be the quotient of ̟ 1 (|X • |, x) by the normal subgroup generated by the image of R u ̟ 1 (X 0 , x). We call its representations normally semisimple local systems on |X • | -these correspond to local systems V (on the connected component of |X| containing Then define ̟ 1 (|X • |, x) norm,red to be the reductive quotient of ̟ 1 (|X • |, x) norm . Its representations are semisimple and normally semisimple local systems on the connected component of |X| containing x.
Definition 2.26. If X • → X is any resolution as in Proposition 2.24, with x 0 ∈ X 0 mapping to x ∈ X, we denote the corresponding pro-algebraic group by ̟ 1 (X, x) norm := ̟ 1 (|X • |, x 0 ) norm , noting that this is independent of the choices X • and x 0 , by [Pri4,Lemma 9.5].
Proposition 2.27. If X is a proper complex variety with a smooth proper resolution a : X • → X, then normally semisimple local systems on X • correspond to local systems on X which become semisimple on pulling back to the normalisation π : X norm → X of X.
Proposition 2.28. If X • is a simplicial diagram of compact Kähler manifolds, then there is a discrete action of the circle group S 1 on ̟ 1 (|X • |, x) norm , such that the composition This also holds if we replace X • with any proper complex variety X.
Definition 2.29. Recall that the Thom-Sullivan (or Thom-Whitney) functor Th from cosimplicial DG algebras to DG algebras is defined as follows. Let Ω(|∆ n |) be the DG algebra of rational polynomial forms on the n-simplex, so for t i of degree 0. The usual face and degeneracy maps for simplices yield ∂ i : Ω(|∆ n |) → Ω(|∆ n−1 |) and σ i : Ω(|∆ n |) → Ω(|∆ n−1 |), giving a simplicial DGA. Given a cosimplicial DG algebra A •• (with the first index denoting cosimplicial structure and the second, DG) , we then set Now, let X • be a simplicial smooth proper complex variety, and D • ⊂ X • a simplicial divisor with normal crossings. Set Y • = X • − D • , assume that |Y • | is connected, and pick a point y ∈ |Y • |. Let j : |Y • | → |X • | be the natural inclusion map.
Using Proposition 2.24, the following gives mixed twistor or mixed Hodge structures on relative Malcev homotopy types of arbitrary complex varieties.
Theorem 2.30. If R is any quotient of ̟ 1 (|X • |, jy) norm,red (resp. any quotient to which the (S 1 ) δ -action of Proposition 2.28 descends and acts algebraically), then there is an algebraic mixed twistor structure (resp. mixed Hodge structure) on pulling back along row 1 : SL 2 → C * , whose pullback over 0 ∈ A 1 is given by the opposedness isomorphism.
Proof. We adapt the proof of [Pri4,Theorem 9.12]. Define the cosimplicial DGÃ observing that functoriality ensures that the cosimplicial and DGA structures are compatible. This has an augmentation (jy) * : We then define the mixed Hodge structure to be the object of dg Z Aff A 1 ×C * (Mat 1 ×R⋊S) given by MTS is defined similarly, replacing S with G m . The graded object is given by MTS given by replacingS with Mat 1 . For any DGA B, we may regard B as a cosimplicial DGA (with constant cosimplicial structure), and then Th (B) = B. In particular, Th (O(R)) = O(R), so we have a basepoint Spec Th ((jy) * ) : MTS . The proofs of Theorems 2.22 and 2.21 now carry over.

Algebraic MHS/MTS for quasi-projective varieties II -non-trivial monodromy
In this section, we assume that X is a smooth projective complex variety, with Y = X − D (for D still a divisor locally of normal crossings). The hypothesis in Theorems 2.21 and 2.22 that R be a quotient of ̟ 1 (X, jy) is unnecessarily strong, and corresponds to allowing only those semisimple local systems on Y with trivial monodromy around the divisor. By [Moc1], every semisimple local system on Y carries an essentially unique tame imaginary pluriharmonic metric, so it is conceivable that Theorem 2.21 could hold for any reductive quotient R of ̟ 1 (Y, y).
However, Simpson's discrete S 1 -action on ̟ 1 (X, jy) red does not extend to the whole of ̟ 1 (Y, y) red , but only to a quotient ν ̟ 1 (Y, y) red . This is because given a tame pure imaginary Higgs form θ and λ ∈ S 1 , the Higgs form λθ is only pure imaginary if either λ = ±1 or θ is nilpotent. The group ν ̟ 1 (Y, y) red is characterised by the property that its representations are semisimple local systems whose associated Higgs form has nilpotent residues. This is equivalent to saying that ν ̟ 1 (Y, y) red -representations are semisimple local systems on Y for which the monodromy around any component of D has unitary eigenvalues. Thus the greatest generality in which Proposition 2.23 could possibly hold is for any Denote the maximal quotient of ν ̟ 1 (Y, y) red on which the S 1 -action is algebraic by VHS ̟ 1 (Y, y). Arguing as in [Pri4,Proposition 5.12], representations of VHS ̟ 1 (Y, y) correspond to real local systems underlying variations of Hodge structure on Y , and representations of VHS ̟ 1 (Y, y) ⋊ S 1 correspond to weight 0 real VHS. The greatest generality in which Theorem 2.22 could hold is for any Definition 3.1. Given a semisimple real local system V on Y , use Mochizuki's tame imaginary pluriharmonic metric to decompose the associated connection D : Note that these definitions are independent of the choice of pluriharmonic metric, since the metric is unique up to global automorphisms Γ(X, Aut(V)).
3.1. Constructing mixed Hodge structures. We now outline a strategy for adapting Theorem 2.22 to more general R.
Proposition 3.2. Let R be a quotient of VHS ̟ 1 (Y, y) to which the S 1 -action descends, and assume we have the following data.
• For each weight 0 real VHS V on Y corresponding to an R ⋊ S 1 -representation, an S-equivariant R-linear graded subsheaf T n (V) for all n ≤ r, compatible with the tensor structures, and closed under the operations D and D c .
where the Hodge filtration F is defined in the usual way in terms of the S-action, and assume that Then there is a non-negatively weighted mixed Hodge structure (Y, y) R,Mal MHS , with where H a (X, R b j * O(R)) naturally becomes a pure Hodge structure of weight a + 2b, and is the differential from the E 2 sheet of the Leray spectral sequence for j.
Proof. We proceed along similar lines to [Mor]. To construct the Hodge filtration, we first Explicitly, a homotopy fibre product Equivalently, we could replace F → D with a surjection. That such surjections exist and give well-defined homotopy fibre products up to quasi-isomorphism follows from the observation in [Pri4,Proposition 3.45] that the homotopy category of quasi-coherent DGAs on a quasi-affine scheme can be realised as the homotopy category of a right proper model category.
Observe that for co-ordinates u, v on C as in Definition 1.3, If we let C • (X, −) denote either the cosimplicialČech or Godement resolution on X, then the Thom-Sullivan functor Th of Definition 2.29 gives us a composition Th •C • (X, −) from sheaves of DG algebras on X to DG algebras. We denote this by RΓ(X, −), since it gives a canonical choice for derived global sections. We then define the Hodge filtration by ) as an object of Ho(DG Z Alg C * (R) * (S)). Note that condition (1) above ensures that the pullback of (Y, y) R,Mal To define the mixed Hodge structure, we first note that condition (2) above implies that is a filtered quasi-isomorphism of complexes, where τ denotes the good truncation filtration. We then define O(Y, y) R,Mal MHS to be the homotopy limit of the diagram which can be expressed as an iterated homotopy fibre product of the form Here, ξ denotes the Rees algebra construction as in Lemma 1.1. The basepoint jy ∈ X gives an augmentation of this DG algebra, so we have defined an object of Conditions (2) and (1) above ensure that the second and third maps in the diagram above are both quasi-isomorphisms, with the second map becoming an isomorphism on pulling back along 1 ∈ A 1 (corresponding to forgetting the filtrations). The latter observation means that we do indeed have Setting gr(Y, y) R,Mal MHS as in the statement above, it only remains to establish opposedness. Now, the pullback of ξ(M, W ) along 0 ∈ A 1 is just gr W M . Moreover, [Del1,Proposition 1.3.4] shows that for any filtered complex (M, J), the map is a quasi-isomorphism, where d J 1 is the differential in the E 1 sheet of the spectral sequence associated to J. Thus the structure sheaf G of (Y, y) R,Mal where d 2 denotes the differential on the E 2 sheet of the spectral sequence associated to a bigraded complex.
The second and third maps in the diagram above are isomorphisms, so we can write G as the homotopy fibre product of ) has the structure of an S-representation of weight a + 2b -denote this by E ab , and set E := ( a,b E ab , d 2 ). Then we can apply Lemma 1.8 to rewrite G as 3.2. Constructing mixed twistor structures. Proposition 3.2 does not easily adapt to mixed twistor structures, since an S-equivariant morphism M → N of quasi-coherent sheaves on S is an isomorphism if and only if the fibres M 1 → N 1 are isomorphisms of vector spaces, but the same is not true of a G m -equivariant morphism of quasi-coherent sheaves on S. Our solution is to introduce holomorphic properties, the key idea being that for t the co-ordinate on S 1 , the connection t ⊛ D : does not define a local system of O(S 1 )-modules, essentially because iterated integration takes us outside O(S 1 ). However, as observed in [Sim1,end of §3], t ⊛ D defines a holomorphic family of local systems on X, parametrised by S 1 (C) = C × . Definition 3.3. Given a smooth complex affine variety Z, define O(Z) hol to be the ring of holomorphic functions f : Z(C) → C. Given a smooth real affine variety Z, define O(Z) hol to be the ring of Gal(C/R)-equivariant holomorphic functions f : Z(C) → C.
In particular, O(S 1 ) hol is the ring of functions f : C × → C for which f (z) = f (z −1 ), or equivalently convergent Laurent series n∈Z a n t n for whichā n = a −n .
Definition 3.5. Define P := C * /G m andP := C * /G m . As in Definition 1.19, we have S 1 = S/G m , and hence a canonical inclusion S 1 ֒→ P (given by cutting out the divisor C , the latter isomorphism using the co-ordinate t. The canonical mapP → P is given by t → (1+t : i−it), and the map S 1 C →P by (a, b) → a+ib. Also note that theétale pushout C * = C * ∪ S C S corresponds to anétale pushout where S 1 C ∼ = G m,C is given by the subscheme t = 0 in A 1 C . Note that the Gal(C/R)-action on C[t, t −1 ] given by the real form S 1 is determined by the condition that the non-trivial element σ ∈ Gal(C/R) maps t to t −1 .
Definition 3.7. Given a polarised scheme (Z, O Z (1)) (where Z need not be projective), and a sheaf F of O Z -modules, define Γ(F ) := n∈Z Γ(Z, F (n)). This is regarded as a G m -representation, with Γ(Z, F (n)) of weight n.
). This is the sheaf on Y which sends any open subset U ⊂ Y to the ring of consisting of those smooth functions f : U × P 1 (C) → C which are holomorphic along P 1 (C). Thus for any y ∈ U , f (y, −) is a global holomorphic function on P 1 (C), so is constant. Therefore Γ(P (C),Ȃ 0 Y (V)) = A 0 Y ⊗ C. For general n, a similar argument using finite-dimensionality of Γ(P 1 (C), O(n) hol ) shows that Since the differential in both cases is given by uD + vD c , this establishes the isomorphism of complexes.
Definition 3.9. On the schemes S 1 andP , define the sheaf O(1) by pulling back O P (1) from P . Thus the corresponding module A(1) on Spec A is given by Proposition 3.10. Let R be a quotient of ̟ 1 (Y, y) red , and assume that we have the following data.
• For each finite rank local real system V on Y corresponding to an R-representation, a flat A 0 X -submodule graded subsheaf closed under the operations D and D c . This must be functorial in V, with • An increasing non-negative filtration J of T * (V) with J r T n (V) = T n (V) for all n ≤ r, compatible with the tensor structure, and closed under the operations D and D c .
to be the complex on X × P (C) whose underlying sheaf is is a finite locally free O hol P -module of slope a + 2b. Then there is a non-negatively weighted mixed twistor structure (Y, y) R,Mal MTS , with where H a (X, R b j * O(R)) is assigned the weight a + 2b, and d 2 : H a (X, R b j * O(R)) → H a+2 (X, R b−1 j * O(R)) is the differential from the E 2 sheet of the Leray spectral sequence for j.
to be the homotopy fibre product as an object of Ho(DG Z Alg C * (R) * (G m )), and let O(Y, y) R,Mal MTS be the homotopy limit of the diagram ξ(RΓ(X, Γ(P (C),T • (O(R)))), Dec RΓ(J)) ξ(RΓ(X, Γ(S 1 (C),T • (O(R)))), Dec RΓ(J)) as an object of Ho(DG Z Alg A 1 ×C * (R) * (Mat 1 × G m )). Here, we are extending T • to indlocal systems by setting T , and similarly forT • . Given a Gal(C/R)-equivariant sheaf F of O hol P -modules on X × P (C), the group cohomology complex gives a Gal(C/R)-equivariant cosimplicial sheaf C • (Gal(C/R), F ) on X × P (C) -this is a resolution of F , with C 0 (Gal(C/R), F ) = F ⊕ σ * F . Applying the Thom-Whitney functor Th , this means that . This allows us to consider the Gal(C/R)-equivariant sheaf B • T of O hol P -DGAs on P (C) given by the fibre product of Note that since the second map is surjective, this fibre product is in fact a homotopy fibre product. In particular, gives a functor from Zariski sheaves o O hol P -modules to O(C)-modules, and we consider the functor Γ(P (C), −)| C * to quasi-coherent sheaves on C * . There is a right derived functor RΓ(P (C), −); by [Ser], the map Given a morphism f : Z → P C of polarised varieties, with Z affine, and a quasi-coherent Zariski sheaf of O hol Z -modules on Z, note that There are convergent spectral sequences H a (P (C), H b (B • T )(n)) =⇒ H a+b (P (C), B • T (n)) for all n, and Condition (3) above ensures that H b (B • T ) is a direct sum of coherent sheaves. Since H i RΓ(P (C), B • T ) = n∈Z H i (P (C), B • T (n)), this means that the map Γ(P (C), In particular, (1:0) C) Gal(C/R) )) is a quasi-isomorphism, and note that right-hand side is just , and hence quasi-isomorphic to RΓ(X, j * A • Y (O(R))) by condition (1) above. This proves that (Y, y) R,Mal Spec R follows along exactly the same lines as in Proposition 3.2, so it only remains to establish opposedness.
Arguing as in the proof of Proposition 3.2, we see that the structure sheaf G of gr(Y, y) R,Mal as a (Mat 1 × R × G m )-equivariant sheaf of DGAs over C * . Set grB a,b MHS to be the sheaf on P (C) given by the fibre product of the diagram condition (1) ensures that the first map is injective, so grB a,b MHS is quasi-isomorphic to the kernel of By condition (3), this is a holomorphic vector bundle on P (C) of slope a + 2b. Now, we just observe that for any holomorphic vector bundle F of slope m, the map Γ(P (C), F (−m)) → 1 * F , given by taking the fibre at 1 ∈ P (R), is an isomorphism of complex vector spaces, and that the maps are isomorphisms for n ≥ 0. This gives an isomorphism making use of condition (1) to show that H a (X, R b j * O(R)) ⊗ C is the fibre of grB a,b MHS at 1 ∈ P (R). This completes the proof of opposedness.
Proposition 3.11. Let R be a quotient of ν ̟ 1 (Y, y) red to which the discrete S 1 -action descends, assume that the conditions of Proposition 3.10 hold, and assume in addition that for all λ ∈ C × , the map λ⋄ : MTS and gr(Y, y) R,Mal MTS , compatible with the opposedness isomorphism, and with the action of −1 ∈ S 1 coinciding with that of −1 ∈ G m .
Proof. The proof of Proposition 2.23 carries over, substituting Proposition 3.10 for Theorem 2.21.
3.3. Unitary monodromy. In this section, we will consider only semisimple local systems V on Y with unitary monodromy around the local components of D (i.e. semisimple monodromy with unitary eigenvalues), Definition 3.12. For V as above, let M (V) ⊂ j * A 0 (V)⊗C consist of locally L 2 -integrable functions for the Poincaré metric, holomorphic in the sense that they lie in ker∂, where D = ∂ +∂ + θ +θ.
Then set where O X denotes the sheaf of holomorphic functions on X.
The crucial observation which we now make is that A * X (V) D is closed under the operations D and D c . Closure under∂ is automatic, and closure under ∂ follows because Mochizuki's metric is tame, so ∂ : Since V has unitary monodromy around the local components of D, the Higgs form θ is holomorphic, which ensures that A * X (V) D is closed under both θ andθ. We can thus write A • X (V) D for the complex given by A * X (V) D with differential D. Lemma 3.13. For all m ≥ 0, there is a morphism , compatible with both D and D c , for D (m) , C (m) as in Definition 2.13 .
Proof. As in [Tim2,1.4], Res m is given in level q by the composition where the final map is given by orthogonal projection. The proof of [Tim2, Lemma 1.5] then adapts to show that Res m is compatible with both D and D c .
Note that (j * V ⊗ ε m )| D m −D m+1 inherits a pluriharmonic metric from V, so is necessarily a semisimple local system on the quasi-projective variety D m − D m+1 = D (m) − C (m) .
Since θ is holomorphic, note that the operators θ andθ are bounded, so also act on L * (2) (V) ⊗ C.

Mixed Hodge structures.
Theorem 3.16. If R is a quotient of VHS ̟ 1 (Y, y) for which the representation π 1 (Y, y) → R(R) has unitary monodromy around the local components of D, then there is a canonical non-positively weighted mixed Hodge structure (Y, y) R,Mal MHS on (Y, y) R,Mal , in the sense of Definition 1.23. The associated split MHS is given by with H a (X, R b j * O(R)) a pure ind-Hodge structure of weight a + 2b.
Proof. We apply Proposition 3.2, taking T * (V) := A * X (V) D , equipped with its filtration J. The first condition to check is compatibility with tensor operations. This follows because, although a product of arbitrary L 2 functions is not L 2 , a product of meromorphic L 2 functions is so. Next , this means that it suffices to establish the quasiisomorphism for m = 0 (replacing X with D (m) for the higher cases). The proof of [Tim1, Theorem D.2(a)] adapts to this generality, showing that j It only remains to show that for all a, b, the groups H a+b (X, F p gr J b A • X (V) D ) define a Hodge filtration on H a (X, R b j * V) C , giving a pure ind-Hodge structure of weight a + 2b. This is essentially [Tim2, Proposition 6.4]: the quasi-isomorphism induced above by Res m is in fact a filtered quasi-isomorphism, provided we set ε m to be of type (m, m). By applying a twist, we can therefore reduce to the case b = 0 (replacing X with D (b) for the higher cases), so we wish to show that the groups H a (X, F p J 0 A • X (V) D ) define a Hodge filtration on H a (X, j * V) of weight a.
The proof of [Tim1,Proposition D.4] adapts to give this result, by identifying H * (X, j * V) with L 2 cohomology, which in turn is identified with the space of harmonic forms. We have a bicomplex (Γ(X, L * (2) (V) ⊗ C), D ′ , D ′′ ) satisfying the principle of two types, with

Mixed twistor structures.
Definition 3.17. Given a smooth complex variety Z, let L * (2) (V)O hol Z be the sheaf on X × Z(C) consisting of holomorphic families of L 2 distributions on X, parametrised by Z(C). Explicitly, given a local co-ordinate z on Z(C), the space Γ(U × {|z| < R}, L n (2) (V)O hol P ) consists of power series m≥0 a m z n with a m ∈ Γ(U, L * (2) (V)) ⊗ C, such that for all K ⊂ U compact and all r < R, the sum m≥0 a m 2,K r m converges, where − 2,K denotes the L 2 norm on K.
Definition 3.18. SetL n (2) (X, V) to be the complex of O hol P -modules on P (C) given by with differential uD + vD c . Note that locally on P (C), elements ofL n (2) (X, V) can be characterised as convergent power series with coefficients in L n (2) (X, V) ⊗ C.
Theorem 3.19. If π 1 (Y, y) → R(R) is Zariski-dense, with unitary monodromy around the local components of D, then there is a canonical non-positively weighted mixed twistor structure (Y, y) R,Mal MTS on (Y, y) R,Mal , in the sense of Definition 1.24. The associated split MTS is given by with H a (X, R b j * O(R)) of weight a + 2b.
Proof. We verify the conditions of Proposition 3.10, setting to be T * (V) =: A * X (V) D , with its filtration J defined above. This gives the complex This leads us to study the restriction to S 1 (C) ⊂ P (C), where we can divide T pq (V) by (u + iv) p (u − iv) q , giving for t ∈ C × ∼ = S 1 (C). There is a similar expression forT • (V)| S 1 (C) . Now, as observed in [Sim1, end of §3], t −1 ⊛ D defines a holomorphic family K (V) of local systems on Y , parametrised by S 1 (C) = C × . Beware that for non-unitary points λ ∈ C × , the canonical metric is not pluriharmonic on the fibre K (V) λ , since λ −1 θ + λθ is not Hermitian. The proof of Theorem 3.16 (essentially [Tim2, Proposition 1.7] and [Tim1, Theorem D.2(a)]) still adapts to verify conditions (1) and (2) from Proposition 3.10, replacing V with K (V), so that for instance is a quasi-isomorphism.
It remains to verify condition (3) from Proposition 3.10: we need to show that for all a, b ≥ 0, the Gal(C/R)-equivariant sheaf ker(H a (X, gr J bT is a finite locally free O hol P -module of slope a + 2b. Arguing as in the proof of Theorem 3.16, we may apply a twist to reduce to the case b = 0 (replacing X with D (b) for the higher cases), so we wish to show that is a holomorphic vector bundle on P (C) of slope a. We do this by considering the graded sheaf L * (2) (V) of L 2 -integrable distributions from Definition 3.15, and observe that [Tim1,Proposition D.4] adapts to show that On restricting toP (C) ⊂ P (C), Definition 3.5 gives the co-ordinate t onP (C) as t = u+iv u−iv , and dividing T n (V) by (u − iv) n gives an isomorphism Thus we also wish to show that is a quasi-isomorphism. Condition (1) from Proposition 3.10 combines with the quasiisomorphism above to show that we have a quasi-isomorphism on S 1 (C) ⊂P (C), so cohomology of the quotient is supported on 0 ∈P (C). Studying the fibre over this point, it thus suffices to show that (2) (V) ⊗ C, D ′′ ) is a quasi-isomorphism, which also follows by adapting [Tim1,Proposition D.4].
Combining the quasi-isomorphisms above gives an isomorphism ). The Green's operator G behaves well in holomorphic families, so gives a decomposition (2) (X, V). Since these expressions are Gal(C/R)-equivariant, it suffices to work onP (C). Dividing T n (V) by (u − iv) n gives giving us a direct sum decomposition with the principle of two types (as in [Sim2] Lemmas 2.1 and 2.2) showing that (tD ′ + D ′′ ) : Im ((D ′′ ) * ) → Im (tD ′ + D ′′ ) is an isomorphism. We have therefore shown that E a ∼ = H a (X, V)⊗ R O hol P (a), which is indeed of slope a. Proposition 3.20. Assume that a Zariski-dense representation π 1 (Y, y) → R(R) has unitary monodromy around the local components of D, and that the discrete S 1 -action on ν ̟ 1 (Y, y) red descends to R. Then there are natural (S 1 ) δ -actions on (Y, y) R,Mal MTS and gr(Y, y) R,Mal MTS , compatible with the opposedness isomorphism, and with the action of −1 ∈ S 1 coinciding with that of −1 ∈ G m .
Proof. We just observe that the construction T * (V) = A * X (V) D of Theorem 3.19 satisfies the conditions of Proposition 3.11, being closed under the -action of C × .
3.4. Singular and simplicial varieties. Fix a smooth proper simplicial complex variety X • , and a simplicial divisor D • ⊂ X • with normal crossings. Set Y • := X • − D • , with a point y ∈ Y 0 , and write j : Y • → X • for the embedding. Note that Proposition 2.24 shows that for any separated complex scheme Y of finite type, there exists such a simplicial variety Y • with an augmentation a : Y • → Y for which |Y • | → Y is a weak equivalence.
Theorem 3.21. Take ρ : π 1 (|Y • |, y) → R(R) Zariski-dense with R pro-reductive, and assume that for every local system V on |Y • | corresponding to an R-representation, the local system a −1 0 V on Y 0 is semisimple, with unitary monodromy around the local components of D 0 . Then there is a canonical non-positively weighted mixed twistor structure (|Y • |, y) R,Mal MTS on (|Y • |, y) R,Mal , in the sense of Definition 1.24.
The associated split MTS is given by with H p (X n , R q j * a −1 n O(R)) of weight p + 2q. Here, H p (X • , a −1 V) denotes the cosimplicial vector space n → H p (X n , a −1 n V), and Th is the Thom-Whitney functor of Definition 2.29. Proof. Our first observation is that the pullback of a holomorphic pluriharmonic metric is holomorphic, so for any local system V corresponding to an R-representation, the local system a −1 n V on Y n is semisimple for all n, with unitary monodromy around the local components of D n . We may therefore form objects , and gr(Y n , (σ 0 ) n y) R,Mal MTS ∈ dg Z Aff(R) * (Mat 1 ) as in the proof of Theorem 3.19, together with opposedness quasi-isomorphisms.
These constructions are functorial, giving cosimplicial DGAs . We now apply the Thom-Whitney functor, giving an algebraic MTS with gr(|Y • |, y) R,Mal Theorem 3.22. Take ρ : π 1 (|Y • |, y) → R(R) Zariski-dense with R pro-reductive, and assume that for every local system V on |Y • | corresponding to an R-representation, the local system a −1 0 V underlies a variation of Hodge structure with unitary monodromy around the local components of D 0 . Then there is a canonical non-positively weighted mixed Hodge structure (Y, y) R,Mal MHS on (Y, y) R,Mal , in the sense of Definition 1.23. The associated split MTS is given by with H p (X n , R q j * a −1 n O(R)) a pure ind-Hodge structure of weight p + 2q. Proof. The proof of Theorem 3.21 carries over, replacing Theorem 3.19 with Theorem 3.16, and observing that variations of Hodge structure are preserved by pullback.
Definition 3.23. Define ν ̟ 1 (|Y • |, y) norm to be the quotient of ̟ 1 (|Y • |, y) norm characterised as follows. Representations of ν ̟ 1 (|Y • |, y) norm correspond to local systems V on |Y • | for which a −1 0 V is a semisimple local system on Y 0 whose monodromy around local components of D 0 has unitary eigenvalues.
Proposition 3.25. Take a pro-reductive S 1 -equivariant quotient R of ν ̟ 1 (|Y • |, x) norm , and assume that for every local system V on |Y • | corresponding to an R-representation, the local system a −1 0 V has unitary monodromy around the local components of D 0 . Then there are natural (S 1 ) δ -actions on (|Y • |, y) R,Mal MTS and gr(|Y • |, y) R,Mal MTS , compatible with the opposedness isomorphism, and with the action of −1 ∈ S 1 coinciding with that of −1 ∈ G m .
Proof. This just follows from the observation that the S 1 -action of Proposition 3.20 is functorial, hence compatible with the construction of Theorem 3.21.
3.5. More general monodromy. It is natural to ask whether the hypotheses of Theorems 3.16 and 3.19 are optimal, or whether algebraic mixed Hodge and mixed twistor structures can be defined more widely. The analogous results to Theorem 3.16 for ℓadic pro-algebraic homotopy types in [Pri5] holds in full generality (i.e. for any Galoisequivariant quotient R of ̟ 1 (Y, y) red ). However the proofs of Theorems 3.16 and 3.19 clearly do not extend to non-unitary monodromy, since if θ is not holomorphic, thenθ does not act on A * X (V) D . Thus any proof adapting those theorems would have to take some modification of A * X (V) D closed under the operatorθ. A serious obstruction to considering non-semisimple monodromy around the divisor is that the principle of two types plays a crucial rôle in the proofs of Theorems 3.16 and 3.19, and for quasi-projective varieties this is only proved for L 2 cohomology. The map H * (X, j * V) → H * (2) (X, V) is only an isomorphism either for X a curve or for semisimple monodromy, so L • (2) (V) will no longer have the properties we require. There is not even any prospect of modifying the filtrations in Propositions 3.2 or 3.10 so that J 0 H * (Y, V) := H * (2) (X, V), because L 2 cohomology does not carry a cup product a priori (and nor does intersection cohomology). This means that there is little prospect of applying the decomposition theorems of [Sab] and [Moc2], except possibly in the case of curves.
If the groups H n (X, j * V) all carry natural MTS or MHS, then the other terms in the Leray spectral sequence should inherit MHS or MTS via the isomorphisms ∨ is a local system on D m − D m+1 -this will hopefully inherit a tame pluriharmonic metric from V by taking residues.
It is worth noting that even for non-semisimple monodromy, the weight filtration on homotopy types should just be the one associated to the Leray spectral sequence. Although the monodromy filtration is often involved in such weight calculations, [Del3] shows that for V pure of weight 0 on Y , we still expect j * V to be pure of weight 0 on X. It is only at generic (not closed) points of X that the monodromy filtration affects purity.
Adapting L 2 techniques to the case of non-semisimple monodromy around the divisor would have to involve some complex of Fréchet spaces to replace L • (2) (X, V), with the properties that it calculates H * (X, j * V) and is still amenable to Hodge theory. When monodromy around D is trivial, a suitable complex is A • (X, j * V), since j * V is a local system. In general, one possibility is a modification of Foth's complex B • (V) from [Fot], based on bounded forms. Another possibility might be the complex given by p∈(0,∞) L • (p) (X, V), i.e. the complex consisting of distributions which are L p for all p < ∞. Beware that these are not the same as bounded forms -p-norms are all defined, but the limit lim p→∞ f p might be infinite (as happens for log | log |z||).
Rather than using Fréchet space techniques directly, another approach to defining the MHS or MTS we need (including for V with non-semisimple monodromy) might be via Saito's mixed Hodge modules or Sabbah's mixed twistor modules. Since H n (X, j * V) ∼ = IH n (X, V) for curves X, fibring by families of curves then opens the possibility of putting MHS or MTS on H n (X, j * V) for general X. Again, the main difficulty would lie in defining the cup products needed to construct DGAs. , for any real algebra B, any λ ∈ (B ⊗ R C) × and any A ∈ SL 2 (B). Let row 1 : SL 2 → C * be the S-equivariant map given by projection onto the first row.
Lemma 4.2. The S-equivariant quasi-coherent ringed sheaf row 1 * O SL 2 on C * is flat, corresponding under Lemma 1.5 to the real algebra Proof. This is [Pri4,Lemma 1.18] .  Write row 2 : SL 2 → A 2 for projection onto the second row, so row ♯ 2 O(A 2 ) is a subring of O(SL 2 ). This subring is equivariant for the S-action on SL 2 from Definition 4.1.
Theorem 4.8. The categories MHS and SHS are equivalent. This equivalence is additive, and compatible with tensor products and duals.
Proof. Given (V, β) ∈ SHS as above, define a weight filtration on V by W r V = i≤r W i V , where W * is the weight decomposition associated to the S-action (as in Definition 1.4). Since β is S-equivariant and row ♯ 2 O(A 2 )(−1) is of strictly positive weights, we have β : In particular, (W r V, β| Wr V ) ∈ SHS for all r.
We now form V ⊗ O(SL 2 ), then look at the S-equivariant derivation N β : V ⊗ O(SL 2 ) → V ⊗ O(SL 2 )(−1) given by N β = id ⊗ N + β ⊗ id. Since ker N = O(C), this map is O(C)-linear; by Lemma 4.2, it corresponds under Lemma 1.5 to a real derivation Therefore M (V, β) := ker(N β ) ⊂ V ⊗ S is a real vector space, equipped with an increasing filtration W , and a decreasing filtration F on M (V, β) ⊗ C. We need to show that M (V, β) is a mixed Hodge structure.
Since N : S → S(−1) is surjective, the observation above that gr W N β = (id ⊗ N ) implies that N β must also be surjective (as the filtration W is bounded), so is a exact sequence; this implies that the functor M is exact.
Since gr W r (V, β) = (W r V, 0), we get that M (gr W r (V, β)) = W r V . As M is exact, gr W r M (V, β) = M (gr W r (V, β)), so we have shown that gr W r M ((V, β)) is a pure weight r Hodge structure, and hence that M (V, β) ∈ MHS. Thus we have an exact functor it is straightforward to check that this is compatible with tensor products and duals.
We need to check that M is an equivalence of categories. First, observe that for any S-representation V , we have an object (V, 0) ∈ SHS with M (V ) = V .

Write
Ext 1 SHS ((U, α), (V, β)) := coker (β * − α * : Hom S (U, V ) This gives a an exact sequence →Ext 1 SHS ((U, α), (V, β)) → 0. Note that Ext 1 SHS ((U, α), (V, β)) does indeed parametrise extensions of (U, α) by (V, β): given an exact sequence we may choose an S-equivariant section s of W ։ U , so W ∼ = U ⊕ V . The obstruction to this being a morphism in SHS is o(s) := s * γ − α ∈ Hom S (U, V ⊗ O(C)), and another choice of section differs from s by some The crucial observation on which the construction hinges is that the map . This implies that when β = 0, the maps ρ i are isomorphisms. Since each object (V, β) ∈ SHS is an Artinian extension of S-representations, we deduce that the maps ρ i must be isomorphisms for all such objects.
Taking i = 1 gives that Ext 1 SHS ((U, α), (V, β)) → Ext 1 H (M (U, α), M (V, β)) is an isomorphism; we deduce that every extension in MHS lifts uniquely to an extension in SHS, so M : SHS → MHS is essentially surjective. Taking i = 0 shows that M is full and faithful.
Remark 4.9. Note that the Tannakian fundamental group (in the sense of [DMOS]) of the category SHS is Π(SHS) = S ⋉ Fr(W >0 (row ♯ 2 O(A 2 )(−1))) ∨ , where Fr(V ) denotes the free pro-unipotent group generated by the pro-finite-dimensional vector space V . In other words, SHS is canonically equivalent to the category of finitedimensional Π(SHS)-representations. Likewise, ind(SHS) is equivalent to the category of all Π(SHS)-representations.
The categories SHS and MHS both have vector space-valued forgetful functors. Tannakian formalism shows that the functor SHS → MHS, together with a choice of natural isomorphism between the respective forgetful functors, gives a morphism Π(MHS) → Π(SHS). The choice of natural isomorphism amounts to choosing a Levi decomposition for Π(MHS), or equivalently a functorial isomorphism V ∼ = gr W V of vector spaces for V ∈ MHS.
A canonical choice b 0 of such an isomorphism is given by composing the embedding b : M (V, β) ֒→ V ⊗ S with the map p 0 : S → R given by x → 0. This allows us to put a new MHS on V , with Hodge filtration b 0 (F ) and the same weight filtration as V , so b 0 : M (V, β) → (V, W, b 0 (F )) is an isomorphism of MHS. To describe this new MHS, first observe that S(−1) ∼ = Ω(S/R) = Sdx, and that for β : V → V ⊗ Ω(S/R), we get an isomorphism exp(− x 0 β) : V → M (V, β), which is precisely b −1 0 . Since the map p i : S → C given by x → i preserves F , it follows that the map This raises the question of which F -preserving maps β : . Using the explicit description from the proof of [Pri4, Lemma 1.18], we see that this amounts to the restriction that Remark 4.11. In [Del4], Deligne established a characterisation of real MHS in terms of S-representations equipped with additional structure. For any λ ∈ C, we have a map p λ : S → C given by x → λ, and b −1 β). Comparing the filtrations b 0 (F ) and b 0 (F ) on V , we are led to consider This maps V to V , and has the properties thatd = d −1 and This is precisely the data of an M-representation in the sense of [Del4, Proposition 2.1], so corresponds to a MHS. Explicitly, we first find the unique operator d 1/2 satisfying the properties for d above, then define the mixed Hodge structure M (V, d) to have underlying vector space V , with the same weight filtration, and with F p M (V, d) := d 1/2 (F p V ).
For our choice of d as above, we then have an isomorphism , so a is an isomorphism of MHS. We have therefore shown directly how our category SHS is equivalent to Deligne's category of M-representations by sending the pair (V, β) to (V, exp( i −i β)). This also gives a canonical isomorphism M ∼ = Π(SHS), once we specify the associated isomorphism a • b −1 0 : V → V on fibre functors. This isomorphism can be understood in terms of identifying the generating elements of [Del4, Construction 1.6] with explicit elements of For an explicit quasi-inverse functor from M-representations to SHS, take a pair (V, d). Since d is unipotent, δ := log d : V C → V C is well-defined, and decomposes into types as δ = a,b<0 δ ab . We now just set for co-ordinates x, y on A 2 . 4.2. Splittings of mixed twistor structures. The following lemma ensures that a mixed twistor structure can be regarded as an Artinian extension of G m -representations.
Lemma 4.12. If E and F are pure twistor structures of weights m and n respectively, then Proof. By hypothesis, E = gr W m E and F = gr W n F . Thus we may assume that E = O(m) and F = O(n). Since homomorphisms must respect the weight filtration, we have which is 0 for m > n and R for n = m, as required.
Definition 4.13. Define STS to be the category of pairs (V, β), where V is an G mrepresentation in real vector spaces and β : Note that the only difference between Definitions 4.5 and 4.13 is that the latter replaces S with G m throughout.
Observe that for (V, β), (V ′ , β ′ ) ∈ STS, Theorem 4.15. The categories MTS and STS are equivalent. This equivalence is additive, and compatible with tensor products and duals.
Proof. As in the proof of Theorem 4.8, every object (V, β) ∈ STS inherits a weight filtration W from V , and β gives rise to a G m -equivariant map respecting the weight filtration on V , with gr W N β = (id ⊗ N ).
For the projection row 1 : SL 2 → C * of Definition 4.1, we then get a G m -equivariant map row 1 * N β : row 1 * (V ⊗ O SL 2 ) → row 1 * (V ⊗ O SL 2 (−1)); Then ker(row 1 * N β ) is a G m -equivariant vector bundle on C * . Using the isomorphism C ∼ = A 2 of Remark 1.3 and the projection π : (A 2 − {0}) → P 1 , this corresponds to a vector bundle M (V, β) := (π * ker(row 1 * N β )) Gm on P 1 . Now, M (V, β) inherits a weight filtration W from V , and surjectivity of N β implies that is an exact sequence, so M is an exact functor. In particular, this gives gr W n M (V, β) = M (W n V, 0), which is just the vector bundle on P 1 corresponding to the G m -equivariant vector bundle (W n V )⊗O C * on C * . Since W n V has weight n for the G m -action, this means that gr W n M (V, β) has slope n, so we have defined an exact functor M : STS → MTS, which is clearly compatible with tensor products and duals. If we define Γ STS (V, β) := ker(β : V → V ⊗ row ♯ 2 O(A 2 )(−1)) Gm and R 1 Γ STS (V, β) := (coker β) Gm , then the proof of Theorem 4.8 gives us morphisms for i = 0, 1. These are automatically isomorphisms when β = 0, and the long exact sequences of cohomology then give that ρ i is an isomorphism for all (V, β). We therefore have isomorphisms β)), and arguing as in Theorem 4.8, this shows that M is an equivalence of categories, using Lemma 4.12 in the pure case.
Remark 4.16. Note that the Tannakian fundamental group (in the sense of [DMOS]) of the category STS is where Fr(V ) denotes the free pro-unipotent group generated by the pro-finite-dimensional vector space V .
The functor STS → MTS then gives a morphism Π(MTS) → Π(STS), but this is not unique, since it depends on a choice of natural isomorphism between the fibre functors (at 1 ∈ C * ) on MTS and on STS. This amounts to choosing a Levi decomposition for Π(MTS), or equivalently a functorial isomorphism E 1 ∼ = gr W E 1 of vector spaces for E ∈ MHS. A canonical choice of such an isomorphism is to take the fibre at I ∈ SL 2 .
We can think of Theorem 4.15 as an analogue of [Del4] for real mixed twistor structures, in that for any MTS E , it gives a canonical splitting of the weight filtration on E 1 , together with unique additional data required to recover E . 5. SL 2 splittings of non-abelian MTS/MHS and strictification 5.1. Simplicial structures.
Definition 5.1. Let sCat be the category of simplicially enriched small categories, which we will refer to as simplicial categories. Explicitly, an object C ∈ sCat consists of a set Ob C of objects, together with Hom C (x, y) ∈ S for all x, y ∈ Ob S, equipped with an associative composition law and identities.
Lemma 5.2. For a reductive pro-algebraic monoid M and an M -representation A in DG algebras, there is a cofibrantly generated model structure on DG Z Alg A (M ), in which fibrations are surjections, and weak equivalences are quasi-isomorphisms.
Proof. When M is a group, this is [Pri4,Lemma 3.38], but the same proof carries over to the monoid case.
Beware that DG Z Alg A (M ) does not then satisfy the axioms of a simplicial model category from [GJ] Ch. II, because Hom(−, B) : DG Z Alg A (M ) opp → S does not have a left adjoint. However, DG Z Alg A (M ) is a simplicial model category in the weaker sense of [Qui]. Now, as in [Hov,§5], for any pair X, Y of objects in a model category C, there is a derived function complex RMap C (X, Y ) ∈ S, defined up to weak equivalence. One construction is to take a cofibrant replacementX for X and a fibrant resolutionŶ • for Y in the Reedy category of simplicial diagrams in C, then to set RMap C (X, Y ) n := Hom C (X,Ŷ n ).
In fact, Dwyer and Kan showed in [DK] that RMap C is completely determined by the weak equivalences in C. In particular, π 0 RMap C (X, Y ) = Hom Ho(C) (X, Y ), where Ho(C) is the homotopy category of C, given by formally inverting weak equivalences.
To see that C ∆ • is a Reedy fibrant simplicial resolution of C in DG Z Alg A (M ), note that the matching object M n C ∆ • is given by so the matching map C ∆ n → M n C ∆ • is a fibration (i.e. surjective).
Therefore forB → B a cofibrant replacement, Definition 5.5. Given a simplicial category C, recall from [Ber] that the category π 0 C is defined to have the same objects as C, with morphisms Hom π 0 C (x, y) = π 0 Hom C (x, y).
A morphism in Hom C (x, y) 0 is said to be a homotopy equivalence if its image in π 0 C is an isomorphism.
If the objects of a simplicial category C are the fibrant cofibrant objects of a model category M, with Hom C = RMap M , then observe that homotopy equivalences in C are precisely weak equivalences in M.

Functors parametrising Hodge and twistor structures. The DG algebra
This is an acyclic resolution of the structure sheaf O C * , so regarded as an O(C)-algebra. This construction is moreover S-equivariant.
Definition 5.6. From now on, we will denote the DG algebra O(SL 2 ) N ǫ − − → O(SL 2 )(−1)ǫ by RO(C * ), thereby making a canonical choice of representative in the equivalence class RΓ(C * , O C * ). We also denote the sheaf j −1 RO(C * ) on C * by RO C * , giving a canonical acyclic resolution of O C * .
Proposition 5.7. For any R ′ acting on C * and any R ′ -equivariant algebra A, the functor j * : DG Z Alg A⊗RO(C * ) (R ′ ) → DG Z Alg Spec A×C * (R ′ ) induces an equivalence For any R ′ -representation B in A-algebras, this extends to an equivalence Proof. This is a special case of [Pri4,Proposition 3.45].
Remark 5.9. Since PT (A) * and PH(A) * are defined in terms of derived function complexes, it follows that a morphism in any of these categories is a homotopy equivalence (in the sense of Definition 5.5) if and only if it is weak equivalence in the associated model category, i.e. a quasi-isomorphism.
Remark 5.10. Let R[t] ∈ Alg(Mat 1 ) be given by setting t to be of weight 1. After applying Proposition 5.7 and taking fibrant cofibrant replacements, observe that a pointed algebraic non-abelian mixed twistor structure consists of together with an object O(X MTS ) ∈ PT * (R[t]) and a weak equivalence Likewise, a pointed algebraic non-abelian mixed Hodge structure consists of together with an object O(X MHS ) ∈ PH * (R[t]), and a weak equivalence in PH * (R).

Deformations.
5.3.1. Quasi-presmoothness. The following is [Pri3,Definition 2.22]: Definition 5.11. Say that a morphism F : A → B in sCat is is a 2-fibration if (F1) for any objects a 1 and a 2 in A, the map Hom A (a 1 , a 2 ) → Hom B (F a 1 , F a 2 ) is a fibration of simplicial sets; (F2) for any objects a 1 ∈ A, b ∈ B, and any homotopy equivalence e : F a 1 → b in B, there is an object a 2 ∈ C, a homotopy equivalence d : a 1 → a 2 in C and an isomorphism θ : The following are adapted from [Pri3]: Definition 5.12. Say that a functor D : Alg(Mat 1 ) → sCat is formally 2-quasi-presmooth if for all square-zero extensions A → B, the map is a 2-fibration. Say that D is formally 2-quasi-presmooth if D → • is so.
Proof. Apart from the augmentation maps, this is essentially the same as [Pri3,Proposition 3.14], which proves the corresponding statements for the functor on algebras given by sending A to the simplicial category of cofibrant DG (T ⊗ A)-algebras, for T cofibrant. The same proof carries over, the only change being to take Mat 1 × R × G m -invariants (resp. Mat 1 × R ⋊ S-invariants) of the André-Quillen cohomology groups. We now sketch the argument. Let P be PT * (resp. PH * ), and write S ′ for G m (resp. S). Fix a square-zero extension A → B in Alg(Mat 1 ). Thus an object C ∈ P(B) is a Mat , with the first map a cofibration and the second a fibration. Since C cofibrant, the underlying graded algebra is smooth over B ⊗ RO(C * ), so lifts essentially uniquely to give a smooth morphism reductive, these maps can all be chosen equivariantly. Now, choose some equivariant A-linear derivation δ onC lifting d C . The obstruction to lifting c ∈ P(B) to P(A) up to isomorphism is then the class This is because any other choice of (δ,p) amounts to adding the boundary of an element in HOM 1 C (Ω(C/(B ⊗ RO(C * ))), The key observation now is that the cotangent complex is an invariant of the quasiisomorphism class, so C lifts to P(A) up to isomorphism if and only if all quasi-isomorphic objects also lift. The treatment of morphisms is similar. Although augmentations are not addressed in [Pri3,Proposition 3.14], the same proof adapts. It is important to note that the André-Quillen characterisation of obstructions to lifting morphisms does not require the target to be cofibrant.

Strictification.
Proposition 5.14. Let P : Alg(Mat 1 ) → sCat be one of the functors PT * or PH * . Given an object E in P(R), an object P in P(R[t]), and a quasi-isomorphism f : P/tP → E in P(R), there is an object M ∈ P(R[t]), a quasi-isomorphism g : P → M , and an isomorphism θ : M/tM → E such that θ •ḡ = f .
Proof. If we replace R[t] with R[t]/t r , then the statement holds immediately from Proposition 5.13 and the definition of formal 2-quasi-presmoothness, since the extension R[t]/t r → R is nilpotent. Proceeding inductively, we get a system of objects M r ∈ P(R[t]/t r ), quasiisomorphisms g r : P/t r P → M r and isomorphisms φ r : We may therefore set M to be the inverse limit of the system in the category of Mat 1 -representations. Explicitly, this says that the maps are isomorphisms for all n. In particular, beware that the forgetful functor from Mat 1representations to vector spaces does not preserve inverse limits. Let M(A) be one of the model categories so P(A) is the full simplicial subcategory on fibrant cofibrant objects. The maps g r give a morphism g : P → M in M(R[t]) and the maps φ r give an isomorphism θ : M/tM → E in P(R). We need to show that M is fibrant and cofibrant (so M ∈ P(R[t])) and that g is a quasi-isomorphism. Fibrancy is immediate, since the deformation of a surjection is a surjection.
Given an object A ∈ M(R[t]), the Mat 1 -action gives a weight decomposition A = n≥0 W n A, and Therefore M is cofibrant. To show that g is a quasi-isomorphism, observe that for A ∈ M(R[t]), the map W n A → W n (A/t r A) is an isomorphism for n < r. Since g r is a quasi-isomorphism for all r, this means that g induces quasi-isomorphisms W n P → W n M for all n, so g is a quasiisomorphism.
for RO C * as in Definition 5.6.
Since M is flat over RO(C * ) ⊗ O(A 1 ), it follows that K is flat over C * × A 1 . Moreover, for 0 ∈ A 1 , we have 0 * K = K/tK, so Thus K satisfies the opposedness condition, so by Lemma 1.14 (resp. Lemma 1.8) it corresponds to an ind-MTS (resp. ind-MHS) on the R-equivariant O(R)-augmented DGA algebra (1, 1) * K given by pulling back along (1, 1) : Spec R → A 1 × C. Letting this ind-MTS (resp. ind-MHS) be M completes the proof. 5.3.3. Homotopy fibres. In Proposition 5.14, it is natural to ask how unique the model M is. We cannot expect it to be unique up to isomorphism, but only up to quasi-isomorphism. As we will see in Corollary 5.20, that quasi-isomorphism is unique up to homotopy, which in turn is unique up to 2-homotopy, and so on.
Definition 5.17. Recall from [Ber] Theorem 1.1 that a morphism F : C → D in sCat is said to be a weak equivalence (a.k.a. an ∞-equivalence) whenever (W1) for any objects a 1 and a 2 in C, the map Hom C (a 1 , a 2 ) → Hom D (F a 1 , F a 2 ) is a weak equivalence of simplicial sets; (W2) the induced functor π 0 F : π 0 C → π 0 D is an equivalence of categories. A morphism F : C → D in sCat is said to be a fibration whenever (F1) for any objects a 1 and a 2 in C, the map Hom C (a 1 , a 2 ) → Hom D (F a 1 , F a 2 ) is a fibration of simplicial sets; (F2) for any objects a 1 ∈ C, b ∈ D, and homotopy equivalence e : F a 1 → b in D, there is an object a 2 ∈ C and a homotopy equivalence d : a 1 → a 2 in C such that F d = e.
Definition 5.18. Given functors A F − → B G ← − C between categories, define the 2-fibre product A × (2) B C as follows. Objects of A × (2) B C are triples (a, θ, c), for a ∈ A, c ∈ C and θ : F a → Gc an isomorphism in B. A morphism in A × (2) B C from (a, θ, c) to (a ′ , θ ′ , c ′ ) is a pair (f, g), where f : a → a ′ is a morphism in A and g : c → c ′ a morphism in C, satisfying the condition that Remark 5.19. This definition has the property that A × B C is a model for the 2-fibre product in the 2-category of categories. However, we will always use the notation A × (2) B C to mean the specific model of Definition 5.18, and not merely any equivalent category.
Also note that A × Proof. By Proposition 5.13, P(R[t]/t r ) → P(R) is a 2-fibration in sCat. Moreover, the proof of Proposition 5.14 shows that the map to the inverse 2-limit is an equivalence, so P(R[t]) → P(R) is also a 2-fibration. Therefore P(R[t]) × (2) P(R) P(R) → P(R) is a fibration in the sense of Definition 5.17, so as required.

SL 2 -splittings.
Definition 5.21. An S-splitting (or SL 2 -splitting) of a mixed Hodge structure (X, x) ρ,Mal MHS on a relative Malcev homotopy type is an isomorphism , giving row * 1 of the opposedness isomorphism on pulling back along {0} → A 1 .
An S-splitting (or SL 2 -splitting) of a mixed twistor structure (X, x) ρ,Mal MTS on a relative Malcev homotopy type is an isomorphism , giving row * 1 of the opposedness isomorphism on pulling back along {0} → A 1 .
Proof. By Theorem 5.16, we have an R-equivariant O(R)-augmented DGA M in the category of ind-MTS (resp. ind-MHS) of non-negative weights, with (X, x) R,Mal MTS (resp. (X, x) R,Mal MHS ) quasi-isomorphic in the category of algebraic mixed twistor (resp. mixed Hodge) structures to Spec ζ(M ).
By Theorem 4.15 (resp. Theorem 4.8) and Lemma 4.7, there is a unique R × G mequivariant (resp. R ⋊ S-equivariant) derivation β : gr W M → (gr W M ) ⊗ row ♯ 2 O(A 2 )(−1), with the corresponding object isomorphic to the object M from the proof of Theorem 5.16 (with gr W M canonically isomorphic to E). In Since Spec A 1 ×C * M is by construction quasi-isomorphic to (X, x) R,Mal MTS (resp. (X, x) R,Mal MHS ), with Spec gr W M quasi-isomorphic to gr(X, x) R,Mal MTS (resp. gr(X, x) R,Mal MHS ), this gives us a quasi-isomorphism Corollary 5.23. If a pointed Malcev homotopy type (X, x) R,Mal admits a mixed twistor structure (X, x) R,Mal MTS , then there is a canonical family Proof. Take the fibre of the SL 2 -splitting row * 1 gr(X, x) R,Mal MTS ≃ A 1 × gr(X, x) R,Mal MTS × SL 2 over (1, 1) ∈ A 1 × C * . The fibre of SL 2 → C * over 1 is 1 0 A 1 0 , giving the family of quasi-isomorphisms.
These structures are compatible with the action of ̟ 1 on ̟ n , with the Whitehead bracket and with the Hurewicz maps ̟ n (X ρ,Mal ) → H n (X, O(R)) ∨ (n ≥ 2) and R u ̟ 1 (X ρ,Mal ) → H 1 (X, O(O(R))) ∨ , for O(R) as in Definition 1.18.
Note that Theorems 4.15 and 4.8 now show that the various homotopy groups have associated objects in STS or SHS, giving canonical SL 2 -splittings. These splittings will automatically be the same as those constructed in [Pri4,Theorem 4.21] from the splitting on the homotopy type. Explicitly, they give canonical isomorphisms compatible with weight filtrations and with twistor or Hodge filtrations, and similarly for O(̟ 1 (X, x) ρ,Mal ).
It is natural to ask whether the relative Malcev homotopy groups ̟ n (Y, y) Y,Mal are related to classical homotopy groups π n (Y, y). We now give conditions under which this is true.
Definition 5.25. Say that a group Γ is n-good with respect to a Zariski-dense representation ρ : Γ → R(k) to a reductive pro-algebraic group if for all finite-dimensional Γ ρ,Mal -representations V , the map is an isomorphism for all i ≤ n and an inclusion for i = n + 1.
To see how to compare homotopy groups when the goodness hypotheses are not satisfied, apply [Pri4,Theorem 3.10] to the universal cover of (Y, y).

5.4.
Quasi-projective varieties. Fix a smooth projective complex variety X, a divisor D locally of normal crossings, and set Y := X − D. Let j : Y → X be the inclusion morphism. Take a Zariski-dense representation ρ : π 1 (Y, y) → R(R), for R a reductive pro-algebraic group, with ρ having unitary monodromy around local components of D.
Definition 5.27. Define a functor G from DG algebras to pro-finite-dimensional chain Lie algebras as follows. First, write σA ∨ [1] for the brutal truncation (in non-negative degrees) of A ∨ [1], and set G(A) = Lie(σA ∨ [1]), the free pro-finite-dimensional pro-nilpotent graded Lie algebra, with differential defined on generators by d A + ∆, with ∆ : A ∨ → (A ⊗ A) ∨ here being the coproduct on A ∨ .
Given a DGA A with A 0 = R, define π n (A) := H n−1 G(A).
Moreover, there are canonical S-linear isomorphisms compatible with weight and twistor filtrations.
Proof. This just combines Theorem 3.19 (or Theorem 2.21 for a simpler proof whenever ρ has trivial monodromy around the divisor) with Corollary 5.24. The splitting comes from Corollary 5.22, making use of the isomorphism gr̟ n (Y, y) R,Mal MTS = gr W ̟ n (Y, y) R,Mal . induced by the exact functor gr W on MTS.
Corollary 5.29. If the local system on X associated to any R-representation underlies a polarisable variation of Hodge structure, then there are natural ind-MHS on the the duals (̟ n (Y, y) ρ,Mal ) ∨ of the relative Malcev homotopy groups for n ≥ 2, and on the Hopf algebra O(̟ 1 (Y, y) ρ,Mal ).
Moreover, there are canonical S-linear isomorphisms ̟ n (Y ρ,Mal,y ) ∨ ⊗ S ∼ = π n ( a,b compatible with weight and Hodge filtrations. Proof. This just combines Theorem 3.16 (or Theorem 2.22 for a simpler proof whenever ρ has trivial monodromy around the divisor) with Corollary 5.24, together with the splitting of Corollary 5.22.
Proof. The proof of [Pri4, Proposition 6.12] carries over to this generality.
Corollary 5.31. Assume that the (S 1 ) δ -action on ν ̟ 1 (Y, y) red descends to R, and that the group ̟ n (Y, y) ρ,Mal is finite-dimensional and spanned by the image of π n (Y, y). Then ̟ n (Y, y) ρ,Mal carries a natural S-split mixed Hodge structure, which extends the mixed twistor structure of Corollary 5.28.
Remark 5.32. If we are willing to discard the Hodge or twistor structures, then Corollary 5.23 gives a family of quasi-isomorphisms, and this copy of A 1 corresponds to Spec S. If we pull back along the morphism S → C given by x → i, the resulting complex quasi-isomorphism will preserve the Hodge filtration F (in the MHS case), but notF . This splitting is denoted by b i in Remark 4.11, and comparison with [Del4,Remark 1.3] shows that this is Deligne's functor a F . [Pri4, adapts to show that when R = 1, the mixed Hodge structure in Corollary 5.29 is the same as that of [Mor,Theorem 9.1]. Since a F was the splitting employed in [Mor], we deduce that when R = 1, the complex quasi-isomorphism at i ∈ A 1 (or equivalently at ( 1 0 i 0 ) ∈ SL 2 ) is precisely the quasi-isomorphism of [Mor,Corollary 9.7].
Whenever the discrete S 1 -action on ̟ n (Y, y) R,Mal MTS (from Proposition 3.20) is algebraic, it defines an algebraic mixed Hodge structure on ̟ n (Y, y) R,Mal . In the projective case (D = ∅), [KPT] constructed a discrete C × -action on ̟ n (X, x) C ; via [Pri4,Remark 6.4], the comments above show that whenever the C × -action is algebraic, it corresponds to the complex I pq decomposition of the mixed Hodge structure, with λ ∈ C × acting on I pq as multiplication by λ p . 5.4.1. Deformations of representations. For Y = X − D as above, and some real algebraic group G, take a reductive representation ρ : π 1 (Y, y) → G(R), with ρ having unitary monodromy around local components of D. Write g for the Lie algebra of G, and let adB ρ be the local system of Lie algebras on Y corresponding to the adjoint representation adρ : π 1 (Y, y) → Aut(g). and J 0 R ρ ⊂ R ρ by replacing Z with J 0 Z. These functors can be characterised as consisting of deformations for which the conjugacy classes of monodromy around the divisors remain unchanged -these are the functors studied in [Fot]. 5.4.2. Simplicial and singular varieties. As in §3.4, let X • be a simplicial smooth proper complex variety, and D • ⊂ X • a simplicial divisor with normal crossings. Set Y • = X • −D • , assume that |Y • | is connected, and pick a point y ∈ |Y • |. Let j : |Y • | → |X • | be the natural inclusion map.
Take ρ : π 1 (|Y • |, y) → R(R) Zariski-dense, and assume that for every local system V on |Y • | corresponding to an R-representation, the local system a −1 0 V on Y 0 is semisimple, with unitary monodromy around the local components of D 0 .
Moreover, there are canonical S-linear isomorphisms ̟ n (|Y • | ρ,Mal,y ) ∨ ⊗ S ∼ = π n (Th ( , d 1 )))) ⊗ S compatible with weight and twistor filtrations. If a −1 0 V underlies a polarisable variation of Hodge structure on Y 0 for all V as above, then the ind-MTS above all become ind-MHS, with the S-linear isomorphisms above compatible with Hodge filtrations.
Proof. The proofs of Corollaries 5.28 and 5.29 carry over, substituting Theorems 3.21 and 3.22 for Theorems 3.19 and 3.16.
Proof. This is essentially the same as Corollary 5.31, replacing Proposition 3.19 with Proposition 3.25.
Remark 5.37. When R = 1, [Pri4,Proposition 9.15] adapts to show that the mixed Hodge structure of Corollary 5.35 agrees with that of [Hai,Theorem 6.3.1]. 5.4.3. Projective varieties. In [Pri4, Theorems 5.14 and 6.1], explicit SL 2 splittings were given for the mixed Hodge and mixed twistor structures on a connected compact Kähler manifold X. Since any MHS or MTS has many possible SL 2 -splittings, it is natural to ask whether those of [Pri4] are the same as the canonical splittings of Corollary 5.22. Apparently miraculously, the answer is yes: Theorem 5.38. The quasi-isomorphisms row * 1 (X, x) R,Mal MTS ≃ A 1 × Spec (gr(X, x) R,Mal MTS ) × SL 2 and row * 1 (X, x) R,Mal MHS → A 1 × Spec (gr(X, x) R,Mal MHS ) × SL 2 of Corollary 5.22 are homotopic to the corresponding quasi-isomorphisms of [Pri4, Theorems 5.14 and 6.1].
Proof. Given a MTS or MHS V , an SL 2 -splitting row * 1 ξ(V ) ∼ = (gr W V ) ⊗ O(SL 2 ) gives rise to a derivation β : gr W V → gr W V ⊗ Ω(SL 2 /C * ), given by differentiation with respect to row * 1 ξ(V ). Since Ω(SL 2 /C * ) ∼ = O(SL 2 )(−1), this SL 2 -splitting corresponds to the canonical SL 2 -splitting of Theorem 4.15 or 4.8 if and only if β(gr W V ) ⊂ gr W V ⊗ row ♯ 2 O(C)(−1). Now, the formality quasi-isomorphisms of [Pri4, Theorems 5.14 and 6.1] allow us to transfer the derivation N : row If we write N β = id ⊗ N + β, for β : E → E ⊗ O(SL 2 )(−1), then the key observation to make is that the formality quasi-isomorphism coincides with the canonical quasiisomorphism of Corollary 5.22 if and only if for some choice of β in the homotopy class, we have [Pri4,Remark 4.22] characterises the homotopy class of derivations β in terms of minimal models, with [β] = [α + γ x ], where γ x characterises the basepoint, and α determines the unpointed structure. In [Pri4,Theorem 8.13], the operators α and γ x are computed explicitly in terms of standard operations on the de Rham complex.
For co-ordinates ( u v x y ) on SL 2 , it thus suffices to show that α and γ x are polynomials in x and y. The explicit computation expresses these operators as expressions inD = uD+vD c , D c = xD + yD c and h i = G 2 D * D c * Dc , where G is the Green's operator. However, each occurrence ofD is immediately preceded by eitherD c or by h i . Sincẽ D cD = (xD + yD c )(uD + vD c ) = (uy − vx)D c D = D c D, we deduce that α and γ x are indeed polynomials in x and y, so the formality quasiisomorphisms of [Pri4, Theorems 5.14 and 6.1] are just the canonical splittings of Corollary 5.22.