Non-abelian real Hodge theory for proper varieties

J.P.Pridham^{∗}

January 21, 2009

Abstract

We show that if X is any proper complex variety, there is a weight decomposition on the real schematic homotopy type, in the form of an algebraic 𝔾_{m}-action. This extends to a real Hodge structure, in the form of a discrete ℂ^{∗}-action, such that ℂ^{∗} × X → X^{sch} is real analytic. If the fundamental group is algebraically good, and the higher homotopy groups have finite rank, this gives bigraded decompositions on the complexified homotopy groups. For smooth proper varieties, the Hodge structure can be recovered from the cohomology ring with coefficients in the universal semisimple local system.

Contents

Introduction 2

1 Review of pro-algebraic homotopy types 3 1.1 Pro-algebraic groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The pro-algebraic homotopy type of a topological space . . . . . . . . . 5 1.3 Relative Malcev homotopy types . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Simplicial Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.2 Chain Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.3 Cosimplicial algebras . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.4 Cochain algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Real Hodge structures 13 2.1 Review of classical real Hodge structures . . . . . . . . . . . . . . . . . . 13 2.2 Non-abelian real Hodge structures . . . . . . . . . . . . . . . . . . . . . 14 2.3 Variations of Hodge structure . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Classical homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Compact Kähler manifolds 19

∗The author is supported by Trinity College, Cambridge

3.1 Formality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 S-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Holomorphy of the S-action . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Action on the reductive fundamental groupoid . . . . . . . . . . 22 3.3.2 The full action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Proper complex varieties 27 4.1 Semisimple local systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 The S-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Bibliography 31

Introduction

In [Sim1], Simpson established the existence of a pure Hodge structure on the real reductive pro-algebraic fundamental group ϖ_{1}(X, x)^{red} of a compact Kähler manifold X. This took the form of a discrete action of the circle group U(1) on ϖ_{1}(X, x)^{red}, in such a way that the resulting map

U(1) × π_{1}(X, x) → ϖ_{1}(X, x)^{red}(ℝ)

was continuous. In fact, the results of [Sim2] imply that this map is real holomorphic (i.e. corresponds to a Gal(ℂ/ℝ)-equivariant holomorphic map on the ℂ-valued points of these schemes).

The purpose of this paper is to establish the existence of a real Hodge structure on the pro-algebraic homotopy types as defined in [Pri1] (or equivalently, on the schematic homotopy types of [Toë]) of complex varieties. The definition and main properties of pro-algebraic homotopy types are summarised in Section 1.

In [Del1] Definition 2.1.4, Deligne defined a real Hodge structure on a real vector space to be an algebraic action of the real algebraic group ℂ^{∗}. A pure Hodge structure of weight 0 is then a ℂ^{∗}-action for which ℝ^{∗} < ℂ^{∗} acts trivially, giving an action of ℂ^{∗}/ℝ^{∗} ≅ U(1). Accordingly, we look to extend Simpson’s U(1)-action on ϖ_{1}(X, x)^{red}(ℝ) to a ℂ^{∗}- action on the whole real pro-algebraic homotopy type G(X)^{alg} (the algebraisation of the path groupoid of X). Our notion of real real Hodge structure will be a real holomorphic map (or, rather, a homotopy class of maps)

ℂ^{∗} × G(X) → G(X)^{alg}(ℝ),

such that the induced map

ℂ^{∗} → Hom(G(X), G(X)^{alg}(ℝ)) ≅ End(G(X)^{alg})

is multiplicative. We also require that the reductive pro-algebraic fundamental groupoid ϖ_{f}(X)^{red} is of weight zero under this action, and that the induced map

ℝ^{∗} × G(X)^{alg} → G(X)^{alg}

is algebraic (i.e. a morphism of schemes). In Section 2, this definition is made precise, and several properties of non-abelian Hodge structures are given. In particular, if the group π_{1}(X, x) is algebraically good, and the higher homotopy groups have finite rank, this gives algebraic ℂ^{∗}-actions on the real vector spaces π_{n}(X, x) ⊗_{ℤ} ℝ (and hence bigraded decompositions on the π_{n}(X, x) ⊗_{ℤ} ℂ).

In Section 3, we establish the existence of a real Hodge structure on the homotopy type of a compact Kähler manifold. This is done by combining the U(1)-action of [Sim1] with Hodge theory for semisimple local systems. Formality of the pro-algebraic homotopy type (from the dd^{c}-lemma) means that it is determined by the cohomology ring of the universal semisimple local system. Characterisation of cohomology groups by harmonic forms then enables to construct a ℂ^{∗}-action on cohomology, by studying behaviour of the Laplacian under Simpson’s U(1)-action. However, we have not been able to compare this Hodge structure fully with the Hodge structures of [KPT2] and [Mor].

Section 4 is concerned with extending the non-abelian real Hodge structures from smooth proper varieties to singular proper varieties, in the spirit of Hodge III ([Del2]). The key idea is that we may replace any proper variety by a smooth proper simplicial variety. The machinery of [Pri1] is well-suited to dealing with simplicial spaces and cohomological descent, so functoriality of the non-abelian Hodge structure on smooth proper varieties allows it to extend naturally to all proper varieties.

1 Review of pro-algebraic homotopy types

Here we give a summary of the results from [Pri1] which will be needed in this paper. Fix a field k of characteristic zero.

1.1 Pro-algebraic groupoids

We first recall some definitions from [Pri1] §§2.1–2.3.

Definition 1.1. Define a pro-algebraic groupoid G over k to consist of the following data:

1. A discrete set Ob (G).

2. For all x, y ∈ Ob (G), an affine scheme G(x, y) (possibly empty) over k.

3. A groupoid structure on G, consisting of an associative multiplication morphism m : G(x, y) × G(y, z) → G(x, z), identities Spec k → G(x, x) and inverses G(x, y) → G(y, x)

Note that a pro-algebraic group is just a pro-algebraic groupoid on one object. We say that a pro-algebraic groupoid is reductive (resp. pro-unipotent) if the pro-algebraic groups G(x, x) are so for all x ∈ Ob (G). An algebraic groupoid is a pro-algebraic groupoid for which the G(x, y) are all of finite type.

If G is a pro-algebraic groupoid, let O(G(x, y)) denote the global sections of the structure sheaf of G(x, y).

Definition 1.2. Given morphisms f, g : G → H of pro-algebraic groupoids, define a natural isomorphism η between f and g to consist of morphisms

η_{x} : Spec k → H(f(x), g(x))

for all x ∈ Ob (G), such that the following diagram commutes, for all x, y ∈ Ob (G):

G(x, y) ⟶^{f(x,y)} H(f(x), f(y))
g(x,y)↓  ↓^{·η_{y}}
H(g(x), g(y)) ⟶^{η_{x}·} H(f(x), g(y)).

A morphism f : G → H of pro-algebraic groupoids is said to be an equivalence if there exists a morphism g : H → G such that fg and gf are both naturally isomorphic to identity morphisms. This is the same as saying that for all y ∈ Ob (H), there exists x ∈ Ob (G) such that H(f(x), y)(k) is non-empty (essential surjectivity), and that for all x_{1}, x_{2} ∈ Ob (G), G(x, y) → G(f(x_{1}), f(x_{2})) is an isomorphism.

Definition 1.3. Given a pro-algebraic groupoid G, define a finite-dimensional linear G-representation to be a functor ρ : G → FDVect_{k} respecting the algebraic structure. Explicitly, this consists of a set {V_{x}}_{x∈Ob (G)} of finite-dimensional k-vector spaces, together with morphisms ρ_{xy} : G(x, y) → Hom(V_{y}, V_{x}) of affine schemes, respecting the multiplication and identities.

A morphism f : (V, ρ) → (W, ϱ) of G-representations consists of f_{x} ∈ Hom(V_{x}, W_{x}) such that

f_{x} ∘ ϱ_{xy} = ρ_{xy} ∘ f_{y} : G(x, y) → Hom(V_{x}, W_{y}).

Definition 1.4. Given a pro-algebraic groupoid G, define the reductive quotient G^{red} of G by setting Ob (G^{red}) = Ob (G), and

G^{red}(x, y) = G(x, y)/R_{u}(G(y, y)) = R_{u}(G(x, x))\G(x, y),

where R_{u}(G(x, x)) is the pro-unipotent radical of the pro-algebraic group G(x, x). The equality arises since if f ∈ G(x, y), g ∈ R_{u}(G(y, y)), then fgf^{−1} ∈ R_{u}(G(x, x)), so both equivalence relations are the same. Multiplication and inversion descend similarly. Observe that G^{red} is then a reductive pro-algebraic groupoid. Representations of G^{red} correspond to semisimple representations of G.

Definition 1.5. Let AGpd denote the category of pro-algebraic groupoids over k, and observe that this category is contains all (inverse) limits. There is functor from AGpd to Gpd, the category of abstract groupoids, given by G ↦ G(k). This functor preserves all limits, so has a left adjoint, the algebraisation functor, denoted Γ ↦ Γ^{alg}. This can be given explicitly by Ob (Γ)^{alg} = Ob (Γ), and

Γ^{alg}(x, y) = Γ(x, x)^{alg} ×^{Γ(x,x)} Γ(x, y),

where Γ(x, x)^{alg} is the pro-algebraic completion of the group Γ(x, x).

The finite-dimensional linear representations of Γ (as in Definition 1.3) correspond to those of Γ^{alg}, and these can be used to recover Γ^{alg}, by Tannakian duality.

Definition 1.6. Given a pro-algebraic groupoid G, and U = {U_{x}}_{x∈Ob (G)} a collection of pro-algebraic groups parametrised by Ob (G), we say that G acts on U if there are morphisms U_{x} × G(x, y) ⟶^{∗} U_{y} of affine schemes, satisfying the following conditions:

1. (uv) ∗ g = (u ∗ g)(v ∗ g), 1 ∗ g = 1 and (u^{−1}) ∗ g = (u ∗ g)^{−1}, for g ∈ G(x, y) and u, v ∈ U_{x}.

2. u ∗ (gh) = (u ∗ g) ∗ h and u ∗ 1 = u, for g ∈ G(x, y), h ∈ G(y, z) and u ∈ U_{x}.

If G acts on U, we write G ⋉ U for the groupoid given by

1. Ob (G ⋉ U) := Ob (G).

2. (G ⋉ U)(x, y) := G(x, y) × U_{y}.

3. (g, u)(h, v) := (gh, (u ∗ h)v) for g ∈ G(x, y), h ∈ G(y, z) and u ∈ U_{y}, v ∈ U_{z}.

Definition 1.7. Given a pro-algebraic groupoid G, define R_{u}(G) to be the collection R_{u}(G)_{x} = R_{u}(G(x, x)) of pro-unipotent pro-algebraic groups, for x ∈ Ob (G). G then acts on R_{u}(G) by conjugation, i.e.

u ∗ g := g^{−1}ug,

for u ∈ R_{u}(G)_{x}, g ∈ G(x, y).

Proposition 1.8. For any pro-algebraic groupoid G, there is a Levi decomposition G = G^{red} ⋉ R_{u}(G), unique up to conjugation by R_{u}(G).

Proof. [Pri1] Proposition 2.17.

1.2 The pro-algebraic homotopy type of a topological space

We now recall the results from [Pri1] §2.4.

Definition 1.9. Let 𝕊 be the category of simplicial sets, and sGpd the category of simplicial groupoids on a constant set of objects (as in [GJ]). Let Top denote the category of compactly generated Hausdorff topological spaces.

A map f : X → Y in Top is said to be a weak equivalence if it gives an isomorphism π_{0}X → π_{0}Y on path components, and for all x ∈ X, the maps π_{n}(f) : π_{n}(X, x) → π_{n}(Y, fx) are all isomorphisms. A map f : X → Y in 𝕊 is said to be a weak equivalence if the map |f| : |X| → |Y | is so. A map f : G → H in sGpd is a weak equivalence if the map on components π_{0}G_{0} → π_{0}H_{0} is an isomorphism, and for all objects x ∈ Ob G, the maps π_{n}(G(x, x)) → π_{n}(H(x, x)) are all isomorphisms.

For each of these categories, we define the corresponding homotopy categories Ho(𝕊), Ho(sGpd), Ho(Top) by localising at weak equivalences.

Note that there is a functor from Top to 𝕊 which sends X to the simplicial set

Sing(X)_{n} = Hom_{Top}(|Δ^{n}|, X).

this gives an equivalence of the corresponding homotopy categories, whose quasi-inverse is geometric realisation. From now on, we will thus restrict our attention to simplicial sets.

As in [GJ] Ch.V.7, there is a classifying space functor W̄ : sGpd → 𝕊, with left adjoint G : 𝕊 → sGpd, Dwyer and Kan’s path groupoid functor ([DK]), and these give equivalences Ho(𝕊) ∼ Ho(sGpd). The geometric realisation of |G(X)| is weakly equivalent to the path space of |X|. These functors have the additional properties that Ob G(X) = X_{0}, (W̄ G)_{0} = Ob (G), π_{0}G(X) ≅ π_{0}|X|, π_{0}(|W̄ G|) ≅ π_{0}G_{0}, π_{n}(G(X)(x, x)) ≅ π_{n+1}(|X|, x) and π_{n+1}(|W̄ G|, x) = π_{n}(G(x, x)). This allows us to study simplicial groupoids instead of topological spaces.

Definition 1.10. Given a simplicial object G_{•} in the category of pro-algebraic groupoids, with Ob (G_{•}) constant, define the fundamental groupoid π_{f}(G_{•}) of G_{•} to have objects Ob (G), and for x, y ∈ Ob (G), set

π_{f}(G)(x, y) := G_{0}(x, y)/ ∼,

where ∼ is the equivalence relation generated by ∂_{0}h ∼ ∂_{1}h for h ∈ G(x, y). This is also pro-algebraic.

Definition 1.11. Define a pro-algebraic simplicial groupoid to consist of a simplicial complex G_{•} of pro-algebraic groupoids, such that Ob (G_{•}) is constant and for all x ∈ Ob (G), G(x, x)_{•} ∈ sAGp, i.e. the maps G_{n}(x, x) → π_{0}(G)(x, x) are pro-unipotent extensions of pro-algebraic groups. We denote the category of pro-algebraic simplicial groupoids by sAGpd.

Define a morphism f : G_{•} → H_{•} in sAGpd to be a weak equivalence if the map π_{f}(f) : π_{f}(G_{•}) → π_{f}(H_{•}) is an equivalence of pro-algebraic groupoids, and the maps π_{n}(f, x) : π_{n}(G_{•}(x, x)) → π_{n}(H_{•}(fx, fx)) are isomorphisms for all n and for all x ∈ Ob (G). We define Ho(sAGpd) to be the localisation of sAGpd at weak equivalences.

There is a forgetful functor (k) : sAGpd → sGpd, given by sending G_{•} to G_{•}(k). This functor has a left adjoint G_{•} ↦ (G_{•})^{alg}. We can describe (G_{•})^{alg} explicitly. First let (π_{f}(G))^{alg} be the pro-algebraic completion of the abstract groupoid π_{f}(G), then let (G^{alg})_{n} be the relative Malcev completion (defined in [Hai] for pro-algebraic groups) of the morphism

G_{n} → (π_{f}(G))^{alg}.

In other words, G_{n} → (G^{alg})_{n} ⟶^{f} (π_{f}(G))^{alg} is the universal diagram with f a prounipotent extension.

Proposition 1.12. The functors (k) and ^{alg} give rise to a pair of adjoint functors

\xymatrix{Ho(sGpd) \ar[r]^{𝕃^{alg}} & Ho(sAGpd) \ar[l]^{(k)}|{⊥}},

with 𝕃^{alg}G(X) = G(X)^{alg}, for any X ∈ 𝕊.

Proof. [Pri1] Proposition 2.26.

Definition 1.13. Given a simplicial set (or equivalently a topological space), define the pro-algebraic homotopy type of X over k to be the object

G(X)^{alg}

in Ho(sAGpd). Define the pro-algebraic fundamental groupoid by ϖ_{f}(X) := π_{f}(G(X)^{alg}). Note that π_{f}(G^{alg}) is the pro-algebraic completion of the fundamental groupoid π_{f}(G).

We then define the higher homotopy groups ϖ_{n}(X) (as ϖ_{f}X-representations) by

ϖ_{n}(X) := π_{n−1}(G(X)^{alg}),

where π_{n}(G) is the representation x ↦ π_{n}(G(x, x)), for x ∈ Ob (G).

1.3 Relative Malcev homotopy types

Definition 1.14. Assume we have an abstract groupoid G, a reductive pro-algebraic groupoid R, and a representation ρ : G → R(k) which is an isomorphism on objects and Zariski-dense on morphisms (i.e. ρ: G(x, y) → R(k)(ρx, ρy) is Zariski-dense for all x, y ∈ Ob G). Define the Malcev completion (G, ρ)^{Mal} (or G^{ρ,Mal}) of G relative to ρ to be the universal diagram

G → (G, ρ)^{Mal} ⟶^{p} R,

with p a pro-unipotent extension, and the composition equal to ρ. Explicitly, Ob (G, ρ)^{Mal} = Ob G and

(G, ρ)^{Mal}(x, y) = (G(x, x), ρ)^{Mal} ×^{G(x,x)} G(x, y).

If G and R are groups, observe that this agrees with the usual definition.

If ϱ : G → R(k) is any any Zariski-dense representation (i.e. essentially surjective on objects and Zariski-dense on morphisms) to a reductive pro-algebraic groupoid (in most examples, we take R to be a group), we can define another reductive groupoid R̃ by setting Ob R̃ = Ob G, and R̃(x, y) = R(ϱx, ϱy). This gives a representation ρ : π_{f}X ⟶^{ρ} R̃ satisfying the above hypotheses, and we define the Malcev completion of G relative to ϱ to be the Malcev completion of G relative to ρ. Note that R̃ → R is an equivalence of pro-algebraic groupoids.

Definition 1.15. Given a Zariski-dense morphism ρ : π_{f}X → R(k), let the Malcev completion G(X, ρ)^{Mal} of X relative to ρ be the pro-algebraic simplicial group (G(X), ρ)^{Mal}. Observe that the Malcev completion of X relative to (π_{f}X)^{red} is just G(X)^{alg}. Let ϖ_{f}(X, ρ)^{Mal} = π_{f}G(X, ρ)^{Mal} and ϖ_{n}(X, ρ)^{Mal} = π_{n−1}G(X, ρ)^{Mal}. Note that π_{f}((X, ρ)^{Mal}) is the relative Malcev completion of π_{f}ρ : π_{f}X → R(k).

Definition 1.16. Define a groupoid Γ to be good with respect to a Zariski-dense representation ρ : Γ → R(k) to a reductive pro-algebraic groupoid if the map

H^{n}(Γ^{ρ,Mal}, V) → H^{n}(Γ, V)

is an isomorphism for all n and all finite-dimensional Γ^{ρ,Mal}-representations Γ.

Lemma 1.17. Assume that for all x ∈∈ Ob Γ, Γ(x, x) is finitely presented, with H^{n}(Γ, −) commuting with filtered direct limits of Γ^{ρ,Mal}-representations, and H^{n}(Γ, V) finite-dimensional for all finite-dimensional Γ^{ρ,Mal}-representations V.

Then Γ is good with respect to ρ if and only if for any finite-dimensional Γ^{ρ,Mal}- representation V, and α ∈ H^{n}(Γ, V), there exists an injection f : V → W_{α} of finitedimensional Γ^{ρ,Mal}-representations, with f(α) = 0 ∈ H^{n}(Γ, W_{α}).

Proof. As for [KPT1] Lemma 4.15.

Theorem 1.18. If X is a topological space with fundamental groupoid Γ, equipped with a Zariski-dense representation ρ : Γ → R(k) to a reductive pro-algebraic groupoid for which:

1. Γ is good with respect to ρ,

2. π_{n}(X, −) is of finite rank for all n > 1,

3. and the Γ-representation π_{n}(X, −) ⊗_{ℤ} k is an extension of R-representations (i.e. a Γ^{ρ,Mal}-representation),

then the canonical map

π_{n}(X, −) ⊗_{ℤ} k → ϖ_{n}(X^{ρ,Mal}, −)

is an isomorphism for all n > 1.

Proof. [Pri1] Theorem 3.21.

1.4 Equivalent formulations

Fix a reductive pro-algebraic groupoid R.

1.4.1 Simplicial Lie algebras

Definition 1.19. Given a k-algebra A, let N̂_{A} be opposite to the category of indconilpotent Lie coalgebras, and let N̂_{A}(R) be the category of R-representations in N̂_{A}.Write sN̂_{A}(R) for the category of simplicial objects in N̂_{A}(R). A weak equivalence in sN̂_{A}(R) is a map which gives isomorphisms on cohomology groups of the duals (which are just A-modules). We denote by Ho(sN̂_{A}(R)) the localisation of sN̂_{A}(R) at weak equivalences. For k = A, we will usually drop the subscript, so N̂ (R) := N̂_{k}(R), and so on.

Definition 1.20. Define E(R) to be the full subcategory of AGpd↓R consisting of those morphisms ρ : G → R of proalgebraic groupoids which are pro-unipotent extensions. Similarly, define sE(R) to consist of the pro-unipotent extensions in sAGpd ↓ R, and Ho(sE(R)) to be the localisation of sE(R) at weak equivalences.

Definition 1.21. Given a pro-algebraic groupoid R, define the category sM_{A}(R) to have the same objects as sN̂_{A}(R), with morphisms given by

Hom_{sM_{A}(R)}(𝔤, 𝔥) = Hom_{Ho(sN̂_{A}(R))}(𝔤, 𝔥)/ exp(𝔥_{0}^{R}),

where 𝔥_{0}^{R} is the sub-Lie algebra of 𝔥_{0}(A) fixed by R, acting by conjugation on the set of homomorphisms.

Proposition 1.22. For any reductive pro-algebraic groupoid R, the categories Ho(sE(R)) and sM(R) are equivalent, sending 𝔤 ∈ sN̂ (R) to R ⋉ exp(𝔤).

Proof. [Pri1] Proposition 1.22.

Definition 1.23. We can now define the Malcev homotopy type of X relative to ρ to be the image of G(X, ρ)^{Mal} in Ho(sE(R̃)), or equivalently R_{u}G(X, ρ)^{Mal} in sM(R̃). Since R̃ → R is an equivalence of groupoids, there is an equivalence sN̂ (R) → sN̂ (R̃), so may regard the Malcev homotopy type as an object of sM(R) (or of Ho(sE(R))). It pro-represents the functor

𝔤 ↦ Hom_{Ho(𝕊↓BR(k))}(X, W̄ (R(k) ⋉ exp(𝔤))),

for 𝔤 ∈ sM(R).

1.4.2 Chain Lie algebras

Definition 1.24. Let dgN̂_{A} be opposite to the category of non-negatively graded indconilpotent cochain Lie coalgebras over A. Define dgN̂_{A}(R) to be the category of Rrepresentations in dgN̂_{A}. A weak equivalence in dgN̂_{A}(R) is a map which induces isomorphisms on cohomology groups of the duals. We denote by Ho(dgN̂_{A}(R)) the localisation of dgN̂_{A}(R) at weak equivalences. For k = A, we will usually drop the subscript, so dgN̂ (R) := dgN̂_{k}(R), and so on.

Definition 1.25. We say that a morphism f : 𝔤 → 𝔥 in dgN̂ (R) is free if there exists a (pro-finite-dimensional) sub-R-representation V ⊂ 𝔥 such that 𝔥 is the free pro-nilpotent graded Lie algebra over 𝔤 on generators V.

Proposition 1.26 (Minimal models). For every object 𝔤 of dgN̂ (R), there exists a free chain Lie algebra 𝔪 with d = 0 on the abelianisation 𝔪/[𝔪, 𝔪], unique up to non-unique isomorphism, together with a weak equivalence 𝔪 → 𝔤.

Proof. [Pri1] Proposition 4.7.

Definition 1.27. Let dgM_{A}(R) be the category with the same objects as dgN̂_{A}(R), and morphisms given by

Hom_{dgM_{A}(R)}(𝔤, 𝔥) = Hom_{Ho(dgN̂_{A}(R))}(𝔤, 𝔥)/ exp(𝔥_{0}^{R}).

Proposition 1.28. There is a normalisation functor N : sN̂_{A}(R) → dgN̂_{A}(R) such that

H_{i}(N𝔤) ≅ π_{i}(𝔤),

giving equivalences Ho(sN̂_{A}(R)) ≃ Ho(dgN̂_{A}(R)), and sM_{A}(R) ≃ dgM_{A}(R).

Proof. [Pri1] Propositions 4.12 and 5.11.

1.4.3 Cosimplicial algebras

Definition 1.29. Let cAlg(R) be the category of of R-representations in cosimplicial k-algebras. A weak equivalence in cAlg(R) is a map which induces isomorphisms on cohomology groups. We denote by Ho(cAlg(R)) the localisation of cAlg(R) at weak equivalences.

Definition 1.30. Let sAff(R) denote the category of simplicial affine schemes over k, i.e. the category opposite to cAlg(R). Similarly, let Ho(sAff(R)) be the category opposite to Ho(cAlg(R)).

Definition 1.31. Given V, W ∈ Rep(R), define V ⊗^{R} W := Hom_{Rep(R)}(k, V ⊗ W).

Definition 1.32. Given A ∈ cAlg(R) and 𝔤 ∈ sN̂ (R), define the Maurer-Cartan space MC(A, G) to consist of sets {ω_{n}}_{n≥0}, with ω_{n} ∈ exp(A^{n+1} ˆ⊗^{R}𝔤_{n}), such that

∂_{i}ω_{n} = \begin{cases}∂^{i+1}ω_{n−1} & i > 0 \\ (∂^{1}ω_{n−1}) · (∂^{0}ω_{n−1})^{−1} & i = 0, \end{cases}
σ_{i}ω_{n} = σ^{i+1}ω_{n+1},
σ^{0}ω_{n} = 1,

where exp(A^{n+1} ˆ⊗^{R}𝔤_{n}) is the group whose underlying set is the Lie algebra A^{n+1} ˆ⊗𝔤_{n−1}, with multiplication given by the Campbell-Baker-Hausdorff formula.

Definition 1.33. Given A ∈ cAlg(R) and 𝔤 ∈ sN̂ (R), define the gauge group Gg(A, 𝔤) ≤ ∏_{n} exp(A^{n} ˆ⊗^{R}𝔤_{n}) to consist of those g satisfying

∂_{i}g_{n} = ∂^{i}g_{n−1} ∀i > 0,
σ_{i}g_{n} = σ^{i}g_{n+1} ∀i.

This has an action on MC(A, 𝔤) given by

(g ∗ ω)_{n} = (∂_{0}g_{n+1}) · ω_{n} · (∂^{0}g_{n}^{−1}),

and we define the torsor space by

π(A, 𝔤) := MC(A, 𝔤)/Gg(A, 𝔤).

Definition 1.34. Let cAlg(R)_{0} be the full subcategory of cAlg(R) whose objects satisfy H^{0}(A) ≅ k. Let Ho(cAlg(R)_{0}) be the full subcategory of Ho(cAlg(R)_{0}) with objects in cAlg(R)_{0}. Let sAff(R)_{0} be the category opposite to cAlg(R)_{0}, and Ho(sAff(R)_{0}) opposite to Ho(cAlg(R)_{0}).

Proposition 1.35. There is a pair of equivalences

\xymatrix{Ho(sAff(R))_{0} \ar[r]^{Ḡ} & sM(R) \ar[l]^{W̄}},

given by

Hom_{Ho(sAff(R))}(Spec A, W̄ 𝔤) = Hom_{sM(R)}(Ḡ(A), 𝔤) = π(A, 𝔤).

Proof. [Pri1] Proposition 3.48.

Definition 1.36. Given a topological space X, and a sheaf ℱ on X, define

C^{n}(X, ℱ) := ∏_{f:Δ^{n}→X} Γ(Δ^{n}, f^{−1}ℱ).

Together, these form a cosimplicial complex C^{•}(X, ℱ).

Definition 1.37. Recall that O(R) has the natural structure of an R×R-representation. Since every R-representation has an associated semisimple local system on |BR(k)|, we will also write O(R) for the R-representation in semisimple local systems on |BR(k)| corresponding to the R × R-representation O(R). We then define the R-representation 𝕆(R) in semisimple local systems on X by 𝕆(R) := ρ^{−1}O(R).

Proposition 1.38. Under the equivalences of Propositions 1.22 and 1.35, the relative Malcev homotopy type G(X)^{ρ,Mal} of a topological space X corresponds to

C^{•}(X, 𝕆(R)) ∈ cAlg(R).

Proof. [Pri1] Theorem 3.55

Corollary 1.39. Pro-algebraic homotopy types are equivalent to the schematic homotopy types of [Toë], in the sense that the full subcategory of the homotopy category Ho(sPr) on objects X^{sch} is equivalent to the full subcategory of Ho(sAGpd) on objects G(X)^{alg}. Under this equivalence, pro-algebraic homotopy groups are isomorphic to schematic homotopy groups.

Proof. [Pri1] Corollary 3.57.

1.4.4 Cochain algebras

Definition 1.40. Define DGAlg(R) to be the category of R-representations in nonnegatively graded cochain k-algebras. A weak equivalence in DGAlg(R) is a map which induces isomorphisms on cohomology groups. We denote by Ho(DGAlg(R)) the localisation of DGAlg(R) at weak equivalences. Define dgAff(R) to be the category opposite to DGAlg(R), and Ho(dgAff(R)) opposite to Ho(DGAlg(R)).

Let DGAlg(R)_{0} be the full subcategory of DGAlg(R) whose objects A satisfy H^{0}(A) = k. Let Ho(DGAlg(R))_{0} be the full subcategory of Ho(DGAlg(R)) on the objects of DGAlg(R)_{0}. Let dgAff(R)_{0} and Ho(dgAff(R))_{0} be the opposite categories to DGAlg(R)_{0} and Ho(DGAlg(R))_{0}, respectively.

Proposition 1.41. There is a denormalisation functor D : DGAlg(R) → cAlg(R) such that

H^{i}(DA) ≅ H^{i}(A),

giving an equivalence Ho(cAlg(R)) ≃ Ho(DGAlg(R)).

Proof. [Pri1] 4.27.

Definition 1.42. Given a cochain algebra A ∈ DGAlg(R), and a chain Lie algebra 𝔤 ∈ dgN̂ (R), define

MC(A, 𝔤) := \{ω ∈ ⨁_{n} A^{n+1} ˆ⊗^{R}𝔤_{n} | dω + \frac{1}{2}[ω, ω] = 0\}.

Definition 1.43. Given A ∈ DGAlg(R) and 𝔤 ∈ dgN̂ (R), we define the gauge group by

Gg(A, 𝔤) := exp(∏_{n} A^{n} ˆ⊗^{R}𝔤_{n}).

Define a gauge action of Gg(A, 𝔤) on MC(A, 𝔤) by

g(ω) := g · ω · g^{−1} − (dg) · g^{−1}.

We define the torsor space by

π(A, 𝔤) = MC(A, 𝔤)/Gg(A, 𝔤).

Theorem 1.44. We have the following commutative diagram of equivalences of categories:

\xymatrix{Ho(dgAff(R))_{0} \ar[r]^{Spec D} \ar[d]^{Ḡ} & Ho(sAff(R))_{0} \ar[d]^{Ḡ} \\ dgM(R) \ar[u]^{W̄} & sM(R), , \ar[u]^{W̄} \ar[l]^{N}}

with the pair

\xymatrix{Ho(dgAff(R))_{0} \ar[r]^{Ḡ} & dgM(R) \ar[l]^{W̄}},

given by

Hom_{Ho(dgAff(R))}(Spec A, W̄ 𝔤) = Hom_{dgM(R)}(Ḡ(A), 𝔤) = π(A, 𝔤).

Proof. [Pri1] Theorem 4.39, Corollary 4.41 and Theorem 4.44.

Definition 1.45. Given a manifold X, denote the sheaf of real n-forms on X by 𝒜^{n}. Given a real sheaf ℱ on X, write

A^{n}(X, ℱ) := Γ(X, ℱ ⊗_{ℝ} 𝒜^{n}).

Proposition 1.46. The real Malcev homotopy type of a manifold X relative to ρ : π_{f}X → R(ℝ) is given in DGAlg(R) by A^{•}(X, 𝕆(R)).

Proof. [Pri1] Proposition 4.50.

1.5 Weights

Definition 1.47. Given 𝔤 ∈ sN̂_{R} and a k-algebra A, define

Out_{R}(𝔤)(A) := Aut_{sM_{A}(R)}(𝔤 ˆ⊗A).

Given G ∈ sE(R), define ROut(G) := Out_{R}(R_{u}(G)), noting that ROut(G)(k) ≅ Aut_{Ho(sE(R))}(G). For G ∈ sAGpd, set ROut(G) := Out_{G^{red}}(R_{u}(G)).

Definition 1.48. By [Pri1] Theorem 5.13, if H^{i}(X, V) is finite-dimensional for all finitedimensional irreducible R-representations V and ρ : G(X) → R, then

ROut(G(X)^{ρ,Mal})

is represented by a pro-algebraic group over k, and we define a weight decomposition on G(X)^{ρ,Mal} to be a morphism

𝔾_{m} → ROut(G(X)^{ρ,Mal})

of pro-algebraic groups.

Definition 1.49. Given a reductive pro-algebraic groupoid R, we say that A ∈ DGAlg(R) is formal if it is weakly equivalent to its cohomology algebra H^{∗}(A). We say that the Malcev homotopy type (X, ρ)^{Mal} of a topological space X relative to a Zariski-dense homomorphism ρ : π_{f}X → R is formal if the corresponding DG algebra (given by Theorem 1.44) is formal.

Proposition 1.50. If G(X, ρ)^{Mal} is formal, then it has a canonical weight decomposition, for which H^{i}(X, 𝕍) is pure of weight i.

Proof. [Pri1] Corollary 5.17.

2 Real Hodge structures

From now on, we will take k = ℝ. We now wish to study the Hodge structure on cohomology.

2.1 Review of classical real Hodge structures

In this section, we recall the standard definitions of real Hodge theory, and fix some notation.

Definition 2.1. Recall from [Del1] Definition 2.1.4 that a real Hodge structure on a real vector space V is an action of the real algebraic group S := ℂ^{∗}, obtained from 𝔾_{m} by Weil restriction of scalars from ℂ to ℝ. Explicitly, for any real algebra A,

S(A) = {(x, y) ∈ A^{2} | x^{2} + y^{2} is invertible}.

Therefore

S(ℂ) ≅ ℂ^{∗} × ℂ^{∗}
(a, b) ↦ (a + ib, a − ib).

For a vector space V, with complexification V_{ℂ} this corresponds to a Hodge decomposition

V_{ℂ} = ⨁_{p+q} V_{ℂ}^{pq}

by setting

λ′μ′′(∑ v_{pq}) = ∑ λ^{p}μ^{q}v_{pq},

for (λ, μ) ∈ (ℂ^{∗})^{2} ≅ S(ℂ).

Since \overline{V_{ℂ}^{pq}} = \overline{V_{ℂ}}^{qp} is the condition for this Hodge structure to be real, note that

\overline{λ′μ′′v} = μ̄′λ̄′′v̄,

so that this does indeed descend to a real S-action.

Definition 2.2. For λ ∈ ℂ^{∗} ≅ S(ℝ), we will denote the action of λ on V by

λ ⋄ v := λ′λ̄′′v.

Given a ℂ-linear map F : V → V of type (a, b), observe that

λ ⋄ F v = λ^{a}λ̄^{b}F (λ ⋄ v).

2.2 Non-abelian real Hodge structures

Definition 2.3. For G ∈ sAGpd, define the group Out(G) of outer automorphisms of G to consist of pairs θ = (θ^{red}, R_{u}(θ)), with θ^{red} : G^{red} → G^{red} an automorphism, and

R_{u}(θ) ∈ Hom_{Ho(sAGpd↓G^{red})}(θ_{∗}^{red}G, G)

an isomorphism, where θ_{∗}^{red}G is the composition G → G^{red} ⟶^{θ^{red}} G^{red}.

We define the group structure on Out(G) by

θ ∘ ϕ = (θ^{red} ∘ ϕ^{red}, R_{u}(θ) ∘ θ_{∗}^{red}R_{u}(ϕ)).

Definition 2.4. Given a real affine scheme X, define the ring O(X)^{hol} of real holomorphic functions on X by

O(X)^{hol} := H^{0}(X(ℂ), 𝒪_{X(ℂ)}^{hol})/Gal(ℂ/ℝ),

where 𝒪_{X(ℂ)}^{hol} is the sheaf of complex holomorphic functions on X(ℂ).

Given real affine schemes X, Y, define a real holomorphic map f : X → Y to be a Gal(ℂ/ℝ)-equivariant holomorphic map f : X(ℂ) → Y (ℂ). Observe that this is equivalent to an element of Y (O(X)^{hol}).

Definition 2.5. Given X ∈ 𝕊, H ∈ sAGpd_{ℝ}, and a k-algebra A, we define the set OHom(X, W̄ H)(A) of A-valued outer homomorphisms to be the coequaliser

\xymatrix{Hom_{sGpd}(X, W̄ P H(A)) \ar[r] \ar[r] & Hom_{𝕊}(X, W̄ H(A)) \ar[r] & OHom(X, W̄ H)(A)}.

Here, P H := H^{I} ×_{(H^{red})^{I}} H^{red}, for H^{I} a path object for H in sAGpd; equivalently P H is a path object for H in sAGpd↓H^{red}.

Note that if ρ : π_{f}X → R is as in Definition 1.14, then

Out(G(X)^{ρ,Mal}) ⊂ OHom(X, W̄ G(X)^{ρ,Mal})(ℝ).

For a real holomorphic space Z, we define the set OHom(X, W̄ H)(Z) of Z-valued outer homomorphisms by

OHom(X, W̄ H)(Z) := OHom(X, W̄ H)(O(Z)^{hol}).

Lemma 2.6.

OHom(X, W̄ H)(A) = MC(X, H(A))/Gg(X, R_{u}(H)(A)).

Definition 2.7. Given X ∈ 𝕊 and a Zariski-dense homomorphism, isomorphic on objects, ρ : π_{f}X → H(ℝ) to a reductive pro-algebraic groupoid over ℝ, such that H^{i}(X, ρ^{∗}V) is finite-dimensional for all i and all finite-dimensional irreducible Hrepresentations V, define a real Hodge structure on ρ to consist of an S-valued outer homomorphism

h ∈ OHom(X, W̄ G(X)^{ρ,Mal})(S_{hol}),

such that

1. for all s ∈ S(ℝ), the morphism h(s)^{red} : ϖ_{f}(X)^{red} → H descends to an automorphism of H, and the induced map

h^{δ} : S(ℝ) → OHom(G(X)^{ρ,val}, G(X)^{ρ,Mal})

is a monoid homomorphism, or equivalently

h^{δ} : S(ℝ)^{δ} → Out(G(X)^{ρ,Mal})

is a group homomorphism;

2. the action of S(ℝ) on (G(X)^{ρ,Mal})^{red} = H is pure of weight 0, i.e. the composition

𝔾_{m}(ℝ) ↪ S(ℝ) → Out(G(X)^{ρ,Mal}) ⟶^{p} Aut(H)

is trivial;

3. the map

h : 𝔾_{m} → ROut(G(X)^{ρ,Mal}) = ker p

is a homomorphism of pro-algebraic groups. In other words, the restriction of h to 𝔾_{m} is a weight decomposition on G(X)^{ρ,Mal} in the sense of §1.5.

Define a real Hodge structure on X to be a real Hodge structure on the canonical map ρ : π_{f}G → ϖ_{f}(X)^{red}(ℝ), in which case G(X)^{ρ,Mal} = G(X)^{alg}.

Remark 2.8. Note that, unlike the definition of pure Hodge structures in [Sim1], we do not have any hypothesis on Hodge type. This is because this would not be satisfied by singular varieties.

On passing to homotopy groups, we obtain the following.

Proposition 2.9. For any point x in a space X with a real Hodge structure, there are real holomorphic maps

S × π_{n}(X, x) → ϖ_{n}(X, x),

unique up to conjugation by R_{u}(ϖ_{1}(X, x)), such that the induced map

S(ℝ) → End(ϖ_{n}(X, x)(ℝ))/R_{u}(ϖ_{1}(X, x))

is a monoid homomorphism, and such that the composition

𝔾_{m} ↪ S → Out(ϖ_{n}(X, x)(ℝ))

is a homomorphism of pro-algebraic groups. In particular, this gives weight decomposition on the pro-algebraic homotopy groups, unique up to conjugation by R_{u}(ϖ_{1}(X, x)).

2.3 Variations of Hodge structure

The following definition is adapted from [Pri2]:

Definition 2.10. Given a discrete group Γ acting on a pro-algebraic groupoid G, for which the action on Ob G is trivial, define ^{Γ}G to be the maximal quotient of G on which Γ acts algebraically. This is the inverse limit \underleftarrow{lim}_{α} G_{α} over those surjective maps

G → G_{α},

with G_{α} algebraic, for which the Γ-action descends to G_{α}. Equivalently, O(^{Γ}G) is the sum of those finite-dimensional Γ-representations of O(G) which are closed under the coproduct.

Definition 2.11. Given a real Hodge structure on ρ : π_{f}X → H(ℝ), define

VHSH := ^{S^{δ}}H.

Remarks 2.12. This notion is analogous to the definition given in [Pri2] of the maximal quotient on which Frobenius acts algebraically. In the same way that representations of that group corresponded to semisimple subsystems of local systems underlying Weil sheaves, representations of ^{VHS}ϖ_{f}(X)^{red} will correspond to local systems underlying variations of Hodge structure. Also note that since 𝔾_{m}(ℝ) < S(ℝ) acts trivially on ϖ_{f}(X)^{red}, the S(ℝ)-action factors through the circle group U(1) ≅ S/𝔾_{m}.

Proposition 2.13. The action of S/𝔾_{m} on ^{VHS}H is algebraic, in the sense that

S/𝔾_{m} ×^{VHS}H → ^{VHS}H

is a morphism of schemes.

It is also an inner action, coming from a morphism

S/𝔾_{m} → ( ∏_{x∈X_{0}} ^{VHS}H(x, x))/Z(^{VHS}H)

of pro-algebraic groupoids, where Z denotes the centre of the groupoid,

Z(^{VHS}H) = {z ∈ ∏_{x∈X_{0}} ^{VHS}H(x, x) : z_{x}f = fz_{y} ∀f ∈ ^{VHS}H(x, y)}.

Proof. As in [Sim1] Lemma 5.1, the map

Aut(G_{α}) → Hom(π_{f}X, G_{α})

is a closed immersion of schemes, so the map

(S/𝔾_{m})^{δ} → Aut(G_{α})

is analytic, hence continuous. Since S/𝔾_{m} is isomorphic to U(1), this means that it defines a one-parameter subgroup, so is algebraic. Therefore the map

S/𝔾_{m} ×^{VHS}H → ^{VHS}H

is algebraic, as ^{VHS}H = \underleftarrow{lim} G_{α}.

Since ϖ_{f}(X)^{red} is equivalent to a disjoint union of reductive proalgebraic groups, G_{α} is equivalent to a disjoint union of reductive algebraic groups. This implies that the connected component Aut(G_{α})^{0} of the identity in Aut(G_{α}) is given by

Aut(G_{α})^{0} = ∏_{x∈X_{0}} G_{α}(x, x)/Z(G_{α}).

Since

∏_{x∈X_{0}} ^{VHS}H(x, x)/Z(^{VHS}H) = \underleftarrow{lim} ∏_{x∈X_{0}} G_{α}(x, x)/Z(G_{α}),

we have an algebraic map

S/𝔾_{m} → ∏_{x∈X_{0}} ^{VHS}H(x, x)/Z(^{VHS}H),

as required.

Proposition 2.14. The following conditions are equivalent:

1. V is a representation of ^{VHS}H;

2. V is a representation of H such that λ ⋄ V ≅ V for all λ ∈ (S/𝔾_{m})(ℝ);

3. V is a representation of H such that λ ⋄ V ≅ V for some non-torsion λ ∈ (S/𝔾_{m})(ℝ).

Proof.

1. ⟹ 2. If V is a representation of^{VHS}H, then it is a representation of H, so is a semisimple representation of ϖ_{f}X. By Lemma 2.13, λ ∈ S/𝔾_{m} is an inner automorphism of VHSH, coming from g ∈ ∏_{x∈X_{0}} ^{VHS}H(x, x), say. Then multiplication by g gives the isomorphism λ ⋄ V ≅ V.

2. ⟹ 3. Trivial.

3. ⟹ 1. Let M be the monodromy groupoid of V; this is a quotient of H. The isomorphism λ ⋄ V ≅ V gives an element g ∈ Aut(M), such that g is the image of λ in Hom(π_{f}X, M), using the standard embedding of Aut(M) as a closed subscheme of Hom(π_{f}X, M). The same is true of g^{n}, λ^{n}, so the image of S/𝔾_{m} in Hom(π_{f}X, M) is just the closure of {g^{n}}_{n∈ℤ}, which is contained in Aut(M), as Aut(M) is closed. For any μ ∈ S/𝔾_{m}, this gives us an isomorphism μ ⋄ V ≅ V, as required.

Lemma 2.15. If ρ : π_{f}X → H(ℝ) has a real Hodge structure, then the obstruction to a surjective map α : H → G factoring through ^{VHS}H lies in H^{1}(X, ρ^{∗}adα).

Proof. We have a real holomorphic map

S/𝔾_{m} × π_{f}X → G,

and α will factor through ^{VHS}H if and only if the induced map

S/𝔾_{m} ⟶^{ϕ} Hom(π_{f}X, G)/Aut(G)

is constant. Since G is reductive and S/𝔾_{m} connected, it suffices to replace Aut(G) by the group of inner automorphisms. On tangent spaces, we then have a map

iℝ ⟶^{D_{1}ϕ} H^{1}(X, ρ^{∗}adα);

let φ ∈ H^{1}(X, ρ^{∗}adα) be the image of i.

If ϕ is constant, then φ = 0. Conversely, observe that for t ∈ (S/𝔾_{m})(R), D_{t}ϕ = tD_{1}ϕt^{−1}, making use of the discrete action of (S/𝔾_{m})(R) on Hom(π_{f}X, G). If φ = 0, this implies that D_{t}ϕ = 0 for all t ∈ (S/𝔾_{m})(ℝ), so ϕ is constant, as required.

Remark 2.16. For compact Kähler manifolds, we will have φ = [iθ − iθ̄], for θ the Higgs form.

Definition 2.17. Define^{VHS}G(X)^{ρ,Mal} to be the relative Malcev completion of G(X) → VHSH. If ρ : π_{f} → ϖ_{f}(X)^{red}(ℝ) is the canonical map, we write ^{VHS}G(X)^{alg} for VHSG(X)^{ρ,Mal}.

Lemma 2.18. Given λ ∈ S(ℝ) such that λ is not torsion in S/𝔾_{m}, lift the image of λ in Out(G(X)^{ρ,Mal}) to some α ∈ Aut(𝔪 ⋊ H), where (𝔪 ⋊ H) is some minimal model for G(X)^{ρ,Mal}. Then

VHSG(X)^{ρ,Mal} = ^{α}(𝔪 ⋊ H).

Proof. Combine Proposition 2.14 with [Pri2] Corollary ?? and Lemma ??.

2.4 Classical homotopy groups

Proposition 2.19. If a topological space X has a real Hodge structure and satisfies the conditions of Theorem 1.18 for R = (π_{f}X)^{red} (or any quotient to which the U(1)- action descends), then the homotopy groups π_{n}(X, x) for n ≥ 2 carry natural real Hodge structures in the sense of [Del1], i.e. algebraic ℂ^{∗}-actions on the real vector spaces

π_{n}(X, x) ⊗_{ℤ} ℝ.

These are unique up to conjugation by R_{u}(ϖ_{1}(X, x)) (in other words, by the unipotent radical of the Zariski closure of π_{1}(X, x) → GL(π_{n}(X, x) ⊗_{ℤ} ℝ)).

Proof. The real Hodge structure on X defines a discrete action of ℂ^{∗} on ϖ_{n}(X, x) ≅ π_{n}(X, x) ⊗_{ℤ} ℝ, for which the action of ℝ^{∗} is algebraic, and the map

ℂ^{∗} × π_{n}(X, x) → π_{n}(X, x) ⊗_{ℤ} ℝ

is real holomorphic. Since π_{n}(X, x) ⊗_{ℤ} ℝ is finite-dimensional, the argument of Proposition 2.13 adapts to show that the action of U(1) < ℂ^{∗} must be algebraic. Therefore the action of ℂ^{∗} = (U(1) × ℝ^{∗})/(−1, −1) is algebraic.

3 Compact Kähler manifolds

3.1 Formality

As in [Pri1] Theorem 5.5, we know that the real pro-algebraic homotopy type of a compact Kähler manifold is formal. We recall the proof here.

Given a semisimple complex local system 𝕎 on a compact Kähler manifold X, there exists a harmonic metric K on 𝕎 ([Sim1] Theorem 1). In fact, this metric is unique up to a scalar on complex irreducible local systems. In [Sim1] §1, the connection D and metric K are used to define connections D′_{K}, D′′_{K} on 𝒜_{X}^{0}(𝕎). Uniqueness implies that the restrictions of D′_{K}, D′′_{K} to any irreducible subsystem are independent of the choice of harmonic metric on 𝕍, so D′_{K}, D′′_{K} are independent of the choice of harmonic metric on 𝕎, and we will simply denote them by D′, D′′.

Now define an operator D^{c} := iD′ − iD′′ on the bundle 𝒜_{X}^{0}(𝕎). This is a real operator; in other words, if 𝕍 is a real semisimple local system, then D^{c} : 𝕍 ⊗ ℂ → 𝒜_{X}^{1}(𝕍 ⊗ ℂ) descends to an ℝ-linear map D^{c} : 𝕍 → 𝒜_{X}^{1}(𝕍).

Now, the principle of two types ([Sim1] Lemmas 2.1 and 2.2) gives quasiisomorphisms

A^{•}(X, 𝕍) ← (Z_{D^{c}}(A^{∗}(X, 𝕍)), d) → (H_{D^{c}}^{∗}(X, 𝕍), 0)

for any semisimple local system 𝕍. As in Proposition 1.50, this gives a natural weight decomposition on G(X)^{alg}, defined by setting H^{i}(X, 𝕍) to be of weight i.

3.2 S-action

In this section, we will show how to construct the map

S(ℝ) → Out(G(X)^{alg}),

for X a compact Kähler manifold.

To every complex semisimple local system 𝕍, there corresponds a Higgs bundle (ℰ , θ) with a harmonic metric K ([Sim1] Theorem 1). Here, ℰ is a locally free 𝒪_{X}-module on X, and θ : ℰ → ℰ ⊗_{𝒪_{X}} Ω_{X}^{1} is a linear map such that θ^{2} = 0. The conditions for a metric to be harmonic are given in [Sim1] §1 or [Moc] §2.5.

We will proceed by studying the harmonic forms of [Sim1] §1. On the space of forms

A^{pq}(X, ℰ) := Γ(X, 𝒜_{ℂ}^{pq} ⊗_{𝒪_{X}} ℰ),

we have an operator ¯∂ of type (0, 1) coming from the holomorphic structure of ℰ. The Higgs form θ then defines a linear operator of type (1, 0) on A^{∗}(X, ℰ).

If we set D′′ = ¯∂ + θ, the Laplacian is defined by

Δ_{θ} = D′′(D′′)^{∗} + (D′′)^{∗}D′′,

where (D′′)^{∗} is the formal adjoint of D′′ with respect to the metric K. The cohomology groups H^{i}(X, 𝕍) are isomorphic to the space

H^{i}(ℰ , θ) := ker Δ_{θ} < A^{i}(X, ℰ)

of harmonic forms.

Definition 3.1. Define an action of the circle group U(1) on the space of semisimple complex local systems by defining t𝕍 to be the local system corresponding to the Higgs bundle (ℰ , tθ), where (ℰ , θ) is the Higgs bundle corresponding to 𝕍.

Since Δ_{θ} is not of type (0, 0), the space H^{i}(ℰ , θ) will not be preserved by the Saction. However, we have the following lemma:

Lemma 3.2. Given λ ∈ ℂ^{∗}, and a semisimple complex local system 𝕍 on X,

λ ⋄ H^{i}(𝕍) = H^{i}(\frac{λ}{λ̄}𝕍).

Proof. First take t ∈ U(1); we can then decompose Δ_{θ} = Δ_{θ}^{0,0} + Δ_{θ}^{1,−1} + Δ_{θ}^{−1,1} by type, so that

Δ_{θ}^{0,0} = ¯∂¯∂^{∗} + ¯∂^{∗} ¯∂ + θθ^{∗} + θ^{∗}θ
Δ_{θ}^{1,−1} = θ ¯∂^{∗} + ¯∂^{∗}θ
Δ_{θ}^{−1,1} = ¯∂θ^{∗} + θ^{∗} ¯∂.

As remarked in [Sim1] Lemma 4.4, for t ∈ U(1), (ℰ , tθ) is another Higgs bundle for which K is a harmonic metric (note that for general t ∈ ℂ^{∗}, this is not the case, although there will exist another harmonic metric). We then have

Δ_{tθ}^{0,0} = Δ_{θ}^{0,0}
Δ_{tθ}^{1,−1} = tΔ_{θ}^{1,−1}
Δ_{tθ}^{−1,1} = t^{−1}Δ_{θ}^{−1,1}

However, if λ ∈ ℂ^{∗}, then

λ ⋄ Δ_{θ}(λ^{−1} ⋄ v) = Δ_{θ}^{0,0} + tΔ_{θ}^{1,−1} + t^{−1}Δ_{θ}^{−1,1},

for t = \frac{λ}{λ̄}.

Therefore

λ ⋄ H^{i}(ℰ , θ) = H^{i}(ℰ , \frac{λ}{λ̄}θ),

as required.

Proposition 3.3. There is a family of isomorphisms

H^{i}(X, 𝕍) ⟶^{λ⋄} H^{i}(X, \frac{λ}{λ̄}(𝕍)),

pure of weight i, parametrised by λ ∈ S(ℝ) ≅ ℂ^{∗}. This respects cup-products, in the sense that

(λ ⋄ a) ∪ (λ ⋄ b) = λ ⋄ (a ∪ b) ∈ H^{∗}(X, \frac{λ}{λ̄}(𝕍 ⊗ 𝕎)).

Proof. This is defined by using the isomorphism between harmonic forms and cohomology. Observe that if λ is real, then this action is just multiplication by λ^{i}, so is pure.

For the last part, given a ∈ H^{i}(X, 𝕍), b ∈ H^{j}(X, 𝕎), let c ∈ H^{i+j}(X, 𝕍 ⊗ 𝕎) be uniquely determined by the property that [c] = [a ∧ b] ∈ H^{i+j}(X, 𝕍 ⊗ 𝕎). Then the principle of two types ([Sim1] Lemma 2.1) implies that a ∧ b − c ∈ Im D′_{θ}D′′_{θ}, where D′_{θ} = ∂ + θ̄. Since

λ ⋄ (D′_{θ}D′′_{θ}x) = λλ̄D′_{\frac{λ}{λ̄} θ}D′′_{\frac{λ}{λ̄} θ}(λ ⋄ x),

we have [λ ⋄ D′_{θ}D′′_{θ}x] = 0, so

λ ⋄ ([a] ∪ [b]) = [λ ⋄ c] = [λ ⋄ (a ∧ b)] = [λ ⋄ a] ∪ [λ ⋄ b].

By [Sim1] Lemma 2.11, the complex conjugate bundle to (ℰ , θ) is isomorphic to (ℰ^{∨}, θ^{t}), with the isomorphism given by the metric, which means that

\overline{(t𝕍)} ≅ t\overline{𝕍},

for all t ∈ U(1).

Proposition 3.4. There is a commutative diagram

H^{i}(X, 𝕍) ⟶^{λ⋄} H^{i}(X, \frac{λ}{λ̄}(𝕍))
↓  ↓
H^{i}(X, \overline{𝕍}) ⟶^{λ⋄} H^{i}(X, \frac{λ}{λ̄}(\overline{𝕍})),

where the vertical arrows combine complex conjugation with the isomorphisms \overline{(t𝕍)} ⟶^{C_{t}} t(\overline{𝕍}). In particular, the action of S(ℝ) defines a real Hodge structure.

Proof. Note that

\overline{λ′λ̄′′v} = λ′λ̄′′v̄,

for all forms.

Definition 3.5. We now define the S-action on π_{f}(X)^{red} by Tannakian duality, setting λ ⋄ 𝕍 := \frac{λ}{λ̄}𝕍, for a real semisimple π_{f}(X)^{red}-representation 𝕍, noting that this is welldefined by the above discussion.

This action gives us a canonical isomorphism λ ⋄ O(π_{f}(X)^{red}) ≅ O(π_{f}(X)^{red}), allowing us to define an S action on H^{∗}(X, 𝕆(π_{f}(X)^{red})), as the composition

H^{∗}(X, 𝕆(π_{f}(X)^{red})) ⟶^{λ⋄} H^{∗}(X, λ ⋄ 𝕆(π_{f}(X)^{red})) ≅ H^{∗}(X, 𝕆(π_{f}(X)^{red})).

We have therefore defined an action S(ℝ) → Out(G(X)^{alg}), by universality. We already know (from [Pri1] §5) that 𝔾_{m} < S acts algebraically. It only remains to show holomorphy.

Remark 3.6. In [KPT2], a discrete ℂ^{∗}-action was defined on the complex schematic homotopy type G(X)^{alg} ⊗_{ℝ} ℂ of X, which leads to the question of whether that action can be compared to the S(ℝ) ≅ ℂ^{∗}-action given here on the real schematic homotopy type. On cohomology groups, the answer is that, for λ ∈ U(1) < ℂ^{∗}, the action defined in [KPT2] corresponds to λ′′ in our notation.

Just by considering ϖ_{f}(X)^{red}, we can see that rest of the action is incomparable with ours in general. However, on the quotient ^{VHS}G(X)^{alg} ⊗_{ℝ} ℂ of G(X)^{alg} ⊗_{ℝ} ℂ, our S-action extends (by algebraicity) to an action of S(ℂ) = ℂ^{∗} × ℂ^{∗}. For λ ∈ ℂ^{∗}, the action defined in [KPT2] there corresponds on cohomology groups to (1, λ) ∈ ℂ^{∗} × ℂ^{∗}.

3.3 Holomorphy of the S-action

We now construct the whole Hodge structure, in the form of an element of OHom(X, W̄ G(X)^{alg})(S_{hol}). Initially, this entails defining an element of Hom(X, Bϖ_{f}^{red}(X))((S/𝔾_{m})_{hol}), for which we adapt the approach of [Sim2] §7.

3.3.1 Action on the reductive fundamental groupoid

Let R := ϖ_{f}(X)^{red}, and let 𝔹 be the universal R-torsor on X, coming from the canonical element of Hom(π_{f}X, R(ℝ)). By Tannakian duality, the adjoint bundle ad𝔹 is given by

ad𝔹 = {α ∈ ∏_{𝕍}End(𝕍) : α_{𝕍⊗𝕎} = α_{𝕍} ⊗ id ⊕ id ⊗ α_{𝕎}, α_{𝕍⊕𝕎} = α_{𝕍} ⊕ α_{𝕎}, fα_{𝕍} = α_{𝕎}f},

where 𝕍 ranges over all semisimple local systems on X, and f over all elements of Hom(𝕍, 𝕎). The Higgs forms thus combine to give

θ + θ̄ ∈ ad𝔹 ⊗ 𝒜_{X}^{1},

and we also have a connection

∂ + ¯∂: 𝔹(𝒜_{X}^{0}) → ad𝔹 ⊗ 𝒜_{X}^{1}.

Let

𝒜_{X}^{0}𝒪_{S/𝔾_{m}} ⊗ ℂ :=

ker(¯∂_{(S/𝔾_{m})(ℂ)} : 𝒜_{X×(S/𝔾_{m})(ℂ)}^{0} ⊗ ℂ → 𝒜_{X×(S/𝔾_{m})(ℂ)}^{0} ⊗_{𝒜_{(S/𝔾m)(ℂ)}^{0}} 𝒜_{(S/𝔾_{m})(ℂ)}^{1} ⊗ ℂ),

𝒜_{X}^{n}𝒪_{S/𝔾_{m}} ⊗ ℂ := (𝒜_{X}^{0}𝒪_{S/𝔾_{m}} ⊗ ℂ) ⊗_{𝒜_{X}^{0}} 𝒜_{X}^{n},

the sheaf of those smooth functions on X × (S/𝔾_{m})(ℂ) which are holomorphic along S/𝔾_{m}. Write t ∈ O(S/𝔾_{m})^{hol} ⊗ ℂ for the canonical isomorphism t : (S/𝔾_{m})(ℂ) → ℂ^{∗}, given by (a, b) ↦ \frac{a+ib}{a−ib} in the notation of §2.

We now define a connection

D_{h} : 𝔹(𝒜_{X}^{0}𝒪_{S/𝔾_{m}} ⊗ ℂ) → ad𝔹 ⊗ 𝒜_{X}^{1}𝒪_{S/𝔾_{m}} ⊗ ℂ

on the R(𝒜_{X}^{0}𝒪_{S/𝔾_{m}} ⊗ ℂ)-torsor 𝔹(𝒜_{X}^{0}𝒪_{S/𝔾_{m}} ⊗ ℂ) by

D_{h} = ∂ + ¯∂ + tθ + t^{−1}θ̄.

ker D_{h} is then an R(𝒪_{(S/𝔾_{m})}^{hol} ⊗ ℂ)-torsor on X × (S/𝔾_{m})(ℂ), with the isomorphisms 𝔹_{x} ≅ R(x, x) for each x ∈ X giving isomorphisms (ker D_{h})_{x} ≅ R(𝒪_{S/𝔾_{m}}^{hol} ⊗ ℂ)(x, x) on (S/𝔾_{m})(ℂ).

If σ denotes the complex conjugation map on (S/𝔾_{m})(ℂ), corresponding to t ↦ t̄^{−1}, then there is a canonical isomorphism \overline{ker D_{h}} ≅ σ_{∗} ker D_{h}. Using this isomorphism, let

𝔹_{h} := {b ∈ π_{X∗} ker D_{h} : b = b̄}.

This is then an R((S/𝔾_{m})^{hol})-torsor on X, giving

ρ_{h} ∈ Hom(π_{f}X, R(O(S/𝔾_{m})^{hol})).

Remark 3.7. Observe that the ℂ^{∗}-action of [Sim1] does not extend this to a Gal(ℂ/ℝ)- equivariant S/𝔾_{m}(ℂ) ≅ ℂ^{∗}-action on ϖ_{f}^{ℂ,red}X, since, for λ ∈ ℂ^{∗}, λ̄(𝕍) = λ𝕍̄, whereas if the action were a S/𝔾_{m}(ℂ)-action, it would satisfy \overline{(λ𝕍)} = λ̄^{−1}𝕍̄. However, there is a holomorphic map S(ℂ) → End(ϖ_{f}^{ℂ}X), but this is generally not multiplicative.

3.3.2 The full action

Writing G(X)^{alg} = R ⋉ U, it only remains to define an element of

OHom(X, W̄ G(X)^{alg})(S_{hol}) = MC(X, (R ⋉ U)(O(S)^{hol}))/Gg(X, U(O(S)^{hol}).

We already have a canonical element ω of MC(X, R ⋉ U), and the argument above defines an element ω_{h}^{red} of

MC(X, R(O(S/𝔾_{m})^{hol})).

The fibre of

OHom(X, W̄ G(X)^{alg})(S_{hol}) → Hom(X, W̄ R)(S_{hol})

over ω_{h}^{red} is

π(C^{•}(X, 𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})^{hol}} O(S)^{hol}, U),

in the notation of Definition 1.43, where we write

𝕆(𝔹_{h}) := (O(R) ⊗ O(S/𝔾_{m})^{hol}) ×^{R((S/𝔾_{m})_{hol})} 𝔹_{h}.

Definition 3.8. Define the complex 𝒜_{X}^{•}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) of sheaves on X by

𝒜_{X}^{n}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) := {v ∈ π_{X∗}(𝒜_{X}^{n}𝒪_{S/𝔾_{m}} ⊗ ℂ ⊗ 𝕆(𝔹)) : v̄ = v},

with differential given by D = ∂ + ¯∂ + θ + θ̄. Define the complex 𝒜_{X}^{•}(𝕆(𝔹_{h})) by

𝒜_{X}^{n}(𝕆(𝔹_{h})) := 𝒜_{X}^{n}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}),

with differential given by D_{h} := ∂ + ¯∂ + tθ + t^{−1}θ̄. Note that 𝒜_{X}^{•}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) is a flabby resolution of 𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}, while 𝒜_{X}^{•}(𝕆(𝔹_{h})) is a flabby resolution of 𝕆(𝔹_{h}).

Let

A_{X}^{•}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) := Γ(X, 𝒜_{X}^{•}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}))
A_{X}^{•}(𝕆(𝔹_{h})) := Γ(X, 𝒜_{X}^{•}(𝕆(𝔹_{h}))).

Lemma 3.9. A_{X}^{0}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) ⊗ ℂ is the space of those functions f : X × (S/𝔾_{m})(ℂ) → 𝔹 ⊗_{ℝ} ℂ over X for which

f(−, s) : X → 𝔹 ⊗_{ℝ} ℂ

is infinitely differentiable for all s ∈ (S/𝔾_{m})(ℂ), and for which all the partial derivatives

\frac{∂^{n}}{∂x_{α_{1}}∂x_{α_{2}} ⋯ ∂x_{α_{n}}} f(x, −) : (S/𝔾_{m})(ℂ) → 𝔹 ⊗_{ℝ} ℂ

are complex analytic, for all x ∈ X.

Proof. It is immediate that all the functions in A_{X}^{0}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) ⊗ ℂ satisfy these properties. For the converse, it suffices to show that a function satisfying these properties is infinitely differentiable. This follows because complex analyticity implies infinite differentiability, and continuity of partial derivatives implies differentiability.

Lemma 3.10. Given f ∈ A_{X}^{0}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) ⊗ ℂ and s_{0} ∈ (S/𝔾_{m})(ℂ), there exists h ∈ A_{X}^{0}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) ⊗ ℂ such that

f(x, s) = f(x, s_{0}) + (t(s) − t(s_{0}))\frac{∂(f ∘ t^{−1})}{∂t(s)}(x, t(s)) + (t(s) − t(s_{0}))^{2}h(x, s).

Proof. Set

h(x, s) = \begin{cases}\frac{f(x,s)−f(x,s_{0})−(t(s)−t(s_{0})) \frac{∂(f∘t^{−1})}{∂t(s)} (x,s)}{(t(s)−t(s_{0}))^{2}} & s ≠ s_{0} \\ \frac{1}{2} \frac{∂^{2}(f∘t^{−1})}{∂t(s)^{2}}(x, t(s_{0})) & s = s_{0} \end{cases}

It follows from the preceding lemma that this has the required properties.

Proposition 3.11. Given an ℝ-linear operator F : A_{X}^{∗}(𝕆(𝔹)) → A_{X}^{∗}(𝕆(𝔹)), we may extend F to an O(S/𝔾_{m})^{hol}-linear operator on A_{X}^{∗}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) by the formula

F (f)(−, s) := F (f(−, s)),

for s ∈ (S/𝔾_{m})(ℂ).

Similarly, given an ℝ-bilinear operator B : A_{X}^{∗}(𝕆(𝔹)) ⊗ A_{X}^{∗}(𝕆(𝔹)) → ℝ, we may extend B to an O(S/𝔾_{m})^{hol}-bilinear operator

A_{X}^{∗}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) ⊗ A_{X}^{∗}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) → O(S)^{hol},

by the formula

B(f, g)(s) := B(f(−, s), g(−, s)).

Proof. Observe that the previous lemmas imply that F (f) ∈ A_{X}^{∗}(𝕆(𝔹)) and B(f, g) ∈ O(S)^{hol}, as required.

Definition 3.12. Let

H_{X}^{n}(𝕆(𝔹)⊗O(S/𝔾_{m})^{hol}) := ker(Δ_{θ} : A_{X}^{n}(𝕆(𝔹)⊗O(S/𝔾_{m})^{hol}) → A_{X}^{n}(𝕆(𝔹)⊗O(S/𝔾_{m})^{hol}),

and let

H_{X}^{n}(𝕆(𝔹_{h})) := ker(Δ_{tθ} : A_{X}^{n}(𝕆(𝔹_{h})) → A_{X}^{n}(𝕆(𝔹_{h}))).

Proposition 3.13. There is a direct sum decomposition

A_{X}^{n}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) = H_{X}^{n}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}) ⊕ Δ_{θ}A_{X}^{n}(𝕆(𝔹) ⊗ O(S/𝔾_{m})^{hol}).

Proof. We extend the standard inner product on A_{X}^{∗}(𝕆(𝔹)) to A_{X}^{n}(𝕆(𝔹)⊗O(S/𝔾_{m})^{hol}) by Proposition 3.11, noting that Δ_{θ} is self-adjoint with respect to this product. Thus, if (Δ_{θ})^{2}f = 0 then

(Δ_{θ}f, Δ_{θ}f) = (v, (Δ_{θ})^{2}f) = 0,

so Δ_{θ}f(−, s) = 0 for all s, hence Δ_{θ}f = 0, which shows that the two summands have zero intersection. The Green’s operator G also extends by Proposition 3.11, and f − Δ_{θ}Gf is harmonic, as required.

Definition 3.14. Define λ, λ̄ ∈ O(S) ⊗ ℂ by a + ib, a − ib respectively, in the notation of §2. Note that t = \frac{λ}{λ̄}.

Definition 3.15. Define an O(S/𝔾_{m})^{hol} ⊗_{O(S/𝔾_{m})} O(S)-linear automorphism

λ⋄ : A_{X}^{∗}(𝕆(𝔹)⊗O(S/𝔾_{m})^{hol})⊗_{O(S/𝔾_{m})}O(S) → A_{X}^{∗}(𝕆(𝔹)⊗O(S/𝔾_{m})^{hol})⊗_{O(S/𝔾_{m})}O(S)

by the formula of Definition 2.2.

Corollary 3.16. There is a direct sum decomposition

A_{X}^{n}(𝕆(𝔹_{h}))⊗_{O(S/𝔾_{m})}O(S) = H_{X}^{n}(𝕆(𝔹_{h}))⊗_{O(S/𝔾_{m})}O(S)⊕Δ_{tθ}A_{X}^{n}(𝕆(𝔹_{h}))⊗_{O(S/𝔾_{m})}O(S).

Proof. The key observation is that λ ⋄ Δ_{θ}f = Δ_{tθ}(λ ⋄ f), with the same calculation as in Lemma 3.2, so conjugating by λ⋄ sends the decomposition of Proposition 3.13 to this decomposition.

Proposition 3.17. The cosimplicial R-representation C^{•}(X, 𝕆(𝔹_{h}) ⊗_{O(S/𝔾_{m})} O(S)) in algebras is formal.

Proof. As 𝒜_{X}^{•}(𝕆(𝔹_{h}) ⊗_{O(S/𝔾_{m})} O(S) is a flabby resolution of 𝕆(𝔹_{h}) ⊗_{O(S/𝔾_{m})} O(S), the argument of Proposition 1.46 shows that C^{•}(X, 𝕆(𝔹_{h}) ⊗_{O(S/𝔾_{m})} O(S)) is weakly equivalent to DA_{X}^{•}(𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})} O(S).

Corollary 3.16 then implies that the DD^{c} lemma holds for this complex, giving quasi-isomorphisms

[figure: (H_{D_{h}^{c}}^{∗}(A_{X}^{•}(𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})} O(S)), 0); Z_{D_{h}^{c}}(A_{X}^{•}(𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})} O(S)), D_{h}),; A_{X}^{•}(𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})} O(S)]

where D_{h}^{c} is the conjugate of D_{h} under the action of i⋄, and Z_{D_{h}^{c}} := ker(D_{h}^{c}).

Corollary 3.18. The fibre of

OHom(X, W̄ G(X)^{alg})(S) → Hom(X, W̄ R)(S)

over ω_{h}^{red} is isomorphic to

π(H^{∗}(X, 𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})^{hol}} O(S)^{hol}, NU).

Definition 3.19. Using the canonical isomorphisms

H^{∗}(X, 𝕆(𝔹)) ≅ H_{X}^{∗}(𝕆(𝔹)), H^{∗}(X, 𝕆(𝔹)_{h})⊗_{O(S/𝔾_{m})}O(S) ≅ H_{X}^{∗}(𝕆(𝔹_{h}))⊗_{O(S/𝔾_{m})}O(S),

define

λ⋄ : H^{∗}(X, 𝕆(𝔹)) → H^{∗}(X, 𝕆(𝔹)_{h}) ⊗_{O(S/𝔾_{m})} O(S),

noting that the argument of Proposition 3.3 shows that this preserves cup products.

Definition 3.20. Recall that ω ∈ MC(X, R ⋉ U) corresponds to the canonical map X → W̄ G(X)^{alg}. By §3.1, this gives

ω ∈ π(H^{∗}(X, 𝕆(𝔹)), NU),

over ω^{red}.

Now define

ω_{h} := λ ⋄ ω

in

π(H^{∗}(X, 𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})^{hol}} O(S)^{hol}, NU) ≅ OHom(X, W̄ G(X)^{alg})(S_{hol})_{ω_{h}^{red}}.

Thus we have proved:

Theorem 3.21. There is a natural real Hodge structure on any compact Kähler manifold X.

Proof. Definition 3.20 provides the Hodge structure

h ∈ OHom(X, W̄ G(X)^{alg})(S_{hol}),

such that ϖ_{f}(X)^{red} is pure of weight zero. By §3.2, this gives a group homomorphism S(ℝ) → Out(G(X)^{alg}), and by §3.1, 𝔾_{m} < S gives a weight decomposition.

Corollary 3.22. If a compact Kähler manifold X satisfies the conditions of Theorem 1.18 for R = (π_{f}X)^{red} (or any quotient to which the U(1)-action descends), then the homotopy groups π_{n}(X, x) for n ≥ 2 carry natural real Hodge structures in the sense of [Del1], i.e. algebraic ℂ^{∗}-actions on the real vector spaces

π_{n}(X, x) ⊗_{ℤ} ℝ.

Proof. Apply Proposition 2.19.

4 Proper complex varieties

In this section, we will show how the techniques of cohomological descent allow us to extend real Hodge structures to all proper complex varieties. By [SD] Remark 4.1.10, the method of [Gro] §9 shows that a surjective proper morphism of topological spaces is universally of effective cohomological descent.

Lemma 4.1. If f : X → Y is a map of compactly generated Hausdorff topological spaces inducing an equivalence on fundamental groupoids, such that R^{i}f_{∗}𝕍 = 0 for all local systems 𝕍 on X and all i > 0, then f is a weak equivalence.

Proof. Without loss of generality, we may assume that X and Y are path-connected. If X̃ ⟶^{π} X, Ỹ ⟶^{π′} Y are the universal covering spaces of X, Y, then it will suffice to show that f̃: X̃ → Ỹ is a weak equivalence, since the fundamental groups are isomorphic.

As X̃, Ỹ are simply connected, it suffices to show that R^{i}f̃_{∗}ℤ = 0 for all i > 0. By the Leray-Serre spectral sequence, R^{i}π_{∗}ℤ = 0 for all i > 0, and similarly for Y. The result now follows from the observation that π_{∗}ℤ is a local system on X.

Proposition 4.2. If a : X_{•} → S is a morphism (of simplicial topological spaces) of effective cohomological descent, then |a| : |X_{•}| → S is a weak equivalence, where |X_{•}| is the geometric realisation of X_{•}.

Proof. We begin by showing that the fundamental groupoids are equivalent. Since H^{0}(|X_{•}|, ℤ) ≅ H^{0}(S, ℤ), we know that π_{0}|X_{•}| ≅ π_{0}S, so we may assume that |X_{•}| and S are both connected.

Now the fundamental group of |X_{•}| is isomorphic to the fundamental group of the simplicial set dSing(X_{•}) (the diagonal of the bisimplicial complex given by the singular sets of the X_{n}). For any group G, the groupoid of G-torsors on |X_{•}| is thus equivalent to the groupoid of pairs (T, ω), where T is a G-torsor on X_{0}, and the descent datum ω : ∂_{0}^{−1}T → ∂_{1}^{−1}T is a morphism of G-torsors satisfying

∂_{2}^{−1}ω ∘ ∂_{0}^{−1}ω = ∂_{1}^{−1}ω, σ_{0}^{−1}ω = 1.

Since a is effective, this groupoid is equivalent to the groupoid of G-torsors on S, so the fundamental groups are isomorphic.

Given a local system 𝕍 on |X_{•}|, there is a corresponding GL(V)-torsor T, which therefore descends to S. Since 𝕍 = T ×^{GL(V)} V and T = a^{−1}a_{∗}T, we can deduce that 𝕍 = a^{−1}a_{∗}𝕍, so R^{i}a_{∗}𝕍 = 0 for all i > 0, as required.

Corollary 4.3. Given a proper complex variety X, there exists a smooth proper simplicial variety X_{•}, unique up to homotopy, and a map a : X_{•} → X, such that |X_{•}| → X is a weak equivalence.

If Y is any complex variety, there exists a smooth proper simplicial variety X_{•}, a simplicial divisor D_{•} ⊂ X_{•} with normal crossings, and a map (X_{•} − D_{•}) → Y such that |X_{•} − D_{•}| → Y is a weak equivalence.

In fact, we may take each X_{n} to be projective, and these resolutions are unique up to homotopy.

Proof. Apply [Del2] 6.2.8, 6.4.4 and §8.2.

4.1 Semisimple local systems

In this section, we will define the real holomorphic U(1)-action on a suitable quotient of the real reductive pro-algebraic fundamental groupoid ϖ_{f}(X)^{red} of any proper complex variety (or, indeed, of any simplicial proper complex variety).

Recall that a local system on a simplicial complex X_{•} of topological spaces is equivalent to the category of pairs (𝕍, α), where 𝕍 is a local system on X_{0}, and α : ∂_{0}^{−1}𝕍 → ∂_{1}^{−1}𝕍 is an isomorphism of local systems satisfying

∂_{2}^{−1}α ∘ ∂_{0}^{−1}α = ∂_{1}^{−1}α, σ_{0}^{−1}α = 1.

Definition 4.4. Given a simplicial complex X_{•} of smooth proper varieties, define ϖ_{f}(X_{•})^{red,norm} to be the quotient of ϖ_{f}(X_{•})^{red} by the image of R_{u}(π_{f}(X_{0})). Its representations consist of normally semisimple local systems on X_{•}, i.e. semisimple local systems 𝕎 for which a_{0}^{−1}𝕎 is also semisimple, for a_{0} : X_{0} → X_{•}.

Lemma 4.5. If f: X_{•} → Y_{•} is a homotopy equivalence of simplicial smooth proper varieties, then ϖ_{f}(|X_{•}|)^{red,norm} ≅ ϖ_{f}(|Y_{•}|)^{red,norm}.

Proof. Without loss of generality, we may assume that the matching maps of f are faithfully flat and proper. Topological and algebraic effective descent then imply that f^{−1} induces an equivalence on the categories of local systems, and that f^{∗} induces an equivalence on the categories of quasi-coherent sheaves, and hence on the categories of Higgs bundles. Since representations of ϖ_{f}(|X_{•}|)^{red,norm} correspond to semisimple objects in the category of Higgs bundles on X_{•}, this completes the proof.

Definition 4.6. If X_{•} → X is a resolution as in Corollary 4.3, we therefore denote the corresponding reductive algebraic groupoid by ϖ_{f}(X)^{red,norm} := ϖ_{f}(|X_{•}|)^{red,norm}.

Proposition 4.7. If X is a proper complex variety with a smooth proper resolution a : X_{•} → X, then normally semisimple local systems on X_{•} correspond to semisimple local systems on X which remain semisimple on pulling back to the normalisation π : X^{norm} → X of X.

Proof. First observe that ϖ_{f}(|X_{•}|)^{red,norm} = ϖ_{f}(X)^{red}/⟨a_{0}R_{u}(ϖ_{f}(X_{0}))⟩. Lemma 4.5 ensures that ϖ_{f}(X_{•})^{red,norm} is independent of the choice of resolution X_{•} of X, so can be defined as ϖ_{f}(X)^{red}/⟨fR_{u}(ϖ_{f}(Y))⟩ for any smooth projective variety Y and proper faithfully flat f.

Now, since X^{norm} is normal, we may make use of an observation on [ABC^{+}] pp.9– 10 (due to M. Ramachandran). X^{norm} has a proper faithfully flat morphism g from a smooth variety Y with connected fibres over X^{norm}, so the map π_{f}g: π_{f}Y → π_{f}X^{norm} is full (from the long exact sequence of homotopy). Thus π_{f}g is surjective, so g(R_{u}(ϖ_{f}(Y)) = R_{u}ϖ_{f}(X^{norm}).

Taking f : Y → X to be the composition Y ⟶^{g} X^{norm} ⟶^{π} X, we see that fR_{u}(π_{f}(Y)) = πR_{u}(π_{f}(X^{norm})), so ϖ_{f}(X)^{red,norm} = ϖ_{f}(X)^{red}/⟨πR_{u}(π_{f}(X^{norm}))⟩, as required.

Proposition 4.8. If X is a proper complex variety, then there is an action of the circle group U(1) on ϖ_{f}(X)^{red,norm}, such that the composition U(1) × π_{f}X → ϖ_{f}(X)^{red,norm} is real holomorphic.

Proof. The key observation is that the U(1)-action defined in [Sim1] is functorial in X, and that semisimplicity is preserved by pullbacks between smooth proper varieties (since Higgs bundles pull back to Higgs bundles), so there is a canonical isomorphism t(∂_{i}^{−1}𝕍) ≅ ∂_{i}^{−1}(t𝕍); thus it makes sense for us to define

t(𝕍, α) := (t𝕍, t(α)).

By Tannakian duality, this defines a U(1)-action on ϖ_{f}(X)^{red,norm}.

Since X_{0}, X_{1} are smooth and proper, the actions of S/𝔾_{m} ≅ U(1) on their reductive pro-algebraic fundamental groupoids are real holomorphic, corresponding to maps

π_{f}(X_{i}) → ϖ_{f}(X_{i})^{red}(O(S/𝔾_{m})^{hol}).

The morphisms ϖ_{f}(X_{i}) → ϖ_{f}(X) then give us maps

π_{f}(X_{i}) → ϖ_{f}(X)^{red,norm}(O(S/𝔾_{m})^{hol}),

compatible with π_{f}(∂_{j}), π_{f}(σ_{j}). Since

\xymatrix{π_{f}(X_{1}) \ar[r]^{∂_{0}} \ar[r]_{∂_{1}} & π_{f}(X_{0}) → π_{f}X}

is a coequaliser diagram in the category of groupoids, this gives us a map

π_{f}(X) → ϖ_{f}(X)^{red,norm}(O(S/𝔾_{m})^{hol}),

as required.

Remark 4.9. Note that this holds for any space X which can be resolved by a simplicial smooth proper variety, so that we may take X to be the realisation of any simplicial proper variety, as in [Del2].

4.2 The S-action

Fix X as in Remark 4.9 We have now defined the S/𝔾_{m} ≅ U(1)-action on ϖ_{f}(X)^{red,norm}, and we must now extend it to the whole of G(X)^{norm} := G(X)^{ρ,Mal}, for ρ : π_{f}(X) → ϖ_{f}(X)^{red,norm}.

Definition 4.10. Write G(X)^{norm} = R ⋉ U, with R reductive and U unipotent. The representation

π_{f}X → R(O(S/𝔾_{m})^{hol})

from Corollary 4.8 corresponds to a R(O(S/𝔾_{m})^{hol})-torsor 𝔹_{h} on X

Similarly, write 𝔹 for the R(ℝ)-torsor on X corresponding to the canonical map π_{f} → ϖ_{f}(X)^{red,norm}(ℝ).

As in §3.3.2, we now need to define an element of

π(C^{•}(X, 𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})^{hol}} O(S)^{hol}, U),

Since X_{•} ⟶^{a} X is a cohomological descent morphism, the map

C^{•}(X, 𝕆(𝔹_{h})) ⟶^{a^{−1}} diag C^{•}(X_{•}, a^{−1}𝕆(𝔹_{h}))

is a quasi-isomorphism in cAlg(R).

Now, a_{n}^{−1}𝕍 is semisimple on X_{n} for any normally semisimple local system 𝕍 on X (being a pullback of a_{0}^{−1}𝕍). This means that the quasi-isomorphisms of Proposition 3.17 apply to give quasi-isomorphisms

C^{•}(X_{n}, a_{n}^{−1}𝕆(𝔹_{h})) ∼ DH^{∗}(X_{n}, a_{n}^{−1}𝕆(𝔹_{h}))

for all n, functorial in ∂_{i}, σ_{i}. These therefore combine to give a quasi-isomorphism

diag C^{•}(X_{•}, a^{−1}𝕆(𝔹_{h})) ∼ diag DH^{∗}(X_{•}, a^{−1}𝕆(𝔹_{h})).

Similarly, there are quasi-isomorphisms

C^{•}(X, 𝕆(𝔹)) ⟶^{a^{−1}} diag C^{•}(X_{•}, a^{−1}𝕆(𝔹)) ∼ diag DH^{∗}(X_{•}, a^{−1}𝕆(𝔹)).

Theorem 4.11. If X is a proper complex variety, or the realisation of a simplicial proper complex variety, then X admits a canonical real Hodge structure over ϖ_{f}(X)^{red,norm}.

Proof. For the weight decomposition, we just use the quasi-isomorphism

C^{•}(X, 𝕆(𝔹)) ∼ diag DH^{∗}(X_{•}, a^{−1}𝕆(𝔹)),

defining H^{n}(X_{m}) to be of weight n.

The adjunction map G(X) → G(X)^{alg}(ℝ) gives rise to an element

ω ∈ π(C^{•}(X, 𝕆(𝔹)), U).

This is equivalent to giving an element of

π(diag DH^{∗}(X_{•}, a^{−1}𝕆(𝔹)), U),

and we use the ⋄-action of §3.3.2 to set

ω_{h} := λ ⋄ ω ∈ π(diag DH^{∗}(X_{•}, a^{−1}𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})^{hol}} O(S)^{hol}, U)
≅ π(C^{•}(X, 𝕆(𝔹_{h})) ⊗_{O(S/𝔾_{m})^{hol}} O(S)^{hol}, U).

All of the necessary properties now follow from the corresponding results in §3.

Corollary 4.12. If a proper complex variety X satisfies the conditions of Theorem 1.18 for R = (π_{f}X)^{red,norm} (or any quotient to which the U(1)-action descends), then the homotopy groups π_{n}(X, x) for n ≥ 2 carry natural real Hodge structures in the sense of [Del1], i.e. algebraic ℂ^{∗}-actions on the real vector spaces

π_{n}(X, x) ⊗_{ℤ} ℝ,

unique up to automorphism by R_{u}ϖ_{1}(X, x).

Proof. Apply Proposition 2.19.

References

[ABC^{+}] J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo. Fundamental groups of compact Kähler manifolds, volume 44 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1996.

[Del1] Pierre Deligne. Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math., (40):5–57, 1971.

[Del2] Pierre Deligne. Théorie de Hodge. III. Inst. Hautes Études Sci. Publ. Math., (44):5–77, 1974.

[DK] W. G. Dwyer and D. M. Kan. Homotopy theory and simplicial groupoids. Nederl. Akad. Wetensch. Indag. Math., 46(4):379–385, 1984.

[GJ] Paul G. Goerss and John F. Jardine. Simplicial homotopy theory, volume 174 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1999.

[Gro] A. Grothendieck. Foncteurs fibres, supports, étude cohomologique des morphismes finis. In Théorie des topos et cohomologie étale des schémas. Tome 2, pages 366–412. Springer-Verlag, Berlin, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Mathematics, Vol. 270.

[Hai] Richard M. Hain. The Hodge de Rham theory of relative Malcev completion. Ann. Sci. École Norm. Sup. (4), 31(1):47–92, 1998.

[KPT1] L. Katzarkov, T. Pantev, and B. Toën. Algebraic and topological aspects of the schematization functor. arXiv math.AG/0503418 v1, 2005.

[KPT2] L. Katzarkov, T. Pantev, and B. Toën. Schematic homotopy types and non-abelian Hodge theory. Compos. Math., 144(3):582–632, 2008. arXiv math.AG/0107129 v5.

[Moc] Takuro Mochizuki. A characterization of semisimple local system by tame pure imaginary pluri-harmonic metric. arXiv math.DG/0402122, 2004.

[Mor] John W. Morgan. The algebraic topology of smooth algebraic varieties. Inst. Hautes Études Sci. Publ. Math., (48):137–204, 1978.

[Pri1] J. P. Pridham. Pro-algebraic homotopy types. Proc. London Math. Soc., 97(2):273–338, 2008. arXiv math.AT/0606107 v8.

[Pri2] J. P. Pridham. Weight decompositions on étale fundamental groups. Amer. J. Math., to appear. arXiv math.AG/0510245 v5.

[SD] Bernard Saint-Donat. Techniques de descente cohomologique. In Théorie des topos et cohomologie étale des schémas. Tome 2, pages 83–162. Springer-Verlag, Berlin, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Mathematics, Vol. 270.

[Sim1] Carlos T. Simpson. Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math., (75):5–95, 1992.

[Sim2] Carlos T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math., (80):5–79 (1995), 1994.

[Toë] Bertrand Toën. Champs affines. Selecta Math. (N.S.), 12(1):39–135, 2006. arXiv math.AG/0012219.