Hodge structures on analytic moduli of real pluriharmonic bundles

J.P.Pridham^{∗} January 21, 2009

Abstract

We define and construct the real analytic moduli stack of pluriharmonic bundles on a compact Kähler manifold X, and show how this is equipped with Hodge and quaternionic structures. This stack maps to the de Rham moduli stack, giving rise to preferred sections of the Deligne-Hitchin twistor stack. We then show how the non-abelian mixed Hodge structures on Malcev homotopy types X can be extended to objects over the pluriharmonic moduli stack.

Contents

Introduction 2 1 The de Rham moduli stack 3 2 The de Rham twistor stack 4 3 The pluriharmonic moduli stack 6 4 The pluriharmonic twistor stack 7 5 Local structure of the moduli stacks 9 6 Relative homotopy types over analytic moduli stacks 12

6.1 Formality and splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7 Unitary actions 15 Bibliography 17

∗This work was supported by Trinity College, Cambridge; and the Engineering and Physical Sciences Research Council [grant number EP/F043570/1].

Introduction

This paper is a sequel to [Pri2], in which we showed how to put mixed Hodge and mixed twistor structures on schematic (or, more generally, relative Malcev) homotopy types of compact Kähler manifolds.

In [Sim4] and [Sim5], Simpson defined the coarse Betti, de Rham and Dolbeault moduli spaces of a smooth projective complex variety. These are all algebraic spaces, and he constructed a complex analytic isomorphism between the Betti and de Rham moduli spaces, then a homeomorphism between the de Rham and Dolbeault moduli spaces. The key to the latter was the correspondence between semisimple local systems, pluriharmonic bundles and Higgs bundles.

The de Rham moduli stack 𝔐_{dR,X,n} parametrises pairs (𝒱 , ∇) on X, where 𝒱 is a complex rank n C^{∞} vector bundle on X, and ∇ a flat connection. This is a complex analytic stack, equipped with an antiholomorphic involution τ, corresponding to complex conjugation of 𝒱 . It extends to a stack 𝔗_{dR,X,n} over ℂ⊗_{ℝ}ℂ−{0}, which we will call the de Rham twistor stack. The fibre over α⊗1+β ⊗i is given by pairs (𝒱 , ∇), where we now take ∇ to be a flat (αd + βd^{c})-connection. Thus the fibre of 𝔗_{dR,X,n} over 1 ⊗ 1 is just 𝔐_{dR,X,n}.

Moreover, the stack 𝔗_{dR,X,n} has an analytic action of (ℂ ⊗_{ℝ} ℂ)^{∗} over ℂ ⊗_{ℝ} ℂ − {0}, with (α ⊗ 1 + β ⊗ i) ⋄ (𝒱 , ∇) := (𝒱 , α∇ + βad_{J}∇). As explained in Remark 2.7, this means that 𝔗_{dR,X,n} is a non-abelian Hodge filtration on 𝔐_{dR,X,n}. Moreover, taking the quotient by ℂ^{∗} gives us a stack [𝔗_{dR,X,n}/ℂ^{∗}] over ℙ^{1}(ℂ) ≅ (ℂ ⊗_{ℝ} ℂ − {0})/ℂ^{∗}. This is precisely the Deligne-Hitchin twistor stack, which is usually defined in terms of gluing rather than as a quotient (Remark 2.6).

The pluriharmonic moduli stack 𝔐_{H,X,n} is defined over 𝔐_{dR,X,n}, with the fibre over (𝒱 , ∇) parametrising hermitian metrics on 𝒱 with respect to which the connection ∇ is pluriharmonic. This stack is real (not complex) analytic, and has an analytic action of the circle group U_{1} (Definition 3.5), which we can regard as giving it a pure Hodge structure of weight 0. The pluriharmonic twistor stack 𝔗_{H,X,n} is an extension of 𝔐_{H,X,n} over the non-zero quaternions ℍ^{∗}. This stack admits a compatible multiplication by ℍ^{∗}, making it isomorphic to the trivial bundle 𝔐_{H,X,n} × ℍ^{∗}.

There is a morphism Υ : 𝔗_{H,X,n} → 𝔗_{dR,X,n} of stacks given by forgetting the pluriharmonic structure, and this lies over a standard isomorphism ℂ ⊗_{ℝ} ℂ ≅ ℍ. Thus 𝔐_{H,X,n} gives sections of the map 𝔗_{dR,X,n} → ℂ ⊗_{ℝ} ℂ − {0}. Many of the known structures on the Deligne-Hitchin twistor space come from combining the actions of (ℂ ⊗_{ℝ} ℂ)^{∗} and ℍ^{∗} with the injectivity of Υ on objects (see Remark 4.12 for one example).

Rather than just working with vector bundles, we generalise to work with R-bundles, with R any real reductive pro-algebraic group (or groupoid) equipped with a Cartan involution. It then makes sense to consider relative Malcev homotopy types as in [Pri1]. There is a natural object over the stack 𝔐_{dR,X,R}, whose fibre over the point [ρ] (associated to a representation ρ : π_{f}X → R(ℂ) of the fundamental groupoid) is the relative Malcev homotopy type X^{ρ,Mal}. We denote the pullback of this universal object to 𝔐_{H,X,R} by X^{R,univ}, and show that this admits a natural algebraic mixed twistor structure (Proposition 6.4). We then show that this can be enhanced to an analytic mixed Hodge structure (Proposition 7.4), extending the weight 0 Hodge structure on 𝔐_{H,X,R}. Taking the fibre at a point [ρ] yields the mixed twistor and mixed Hodge structures on X^{ρ,Mal} of [Pri2] (Remarks 7.5).

The structure of the paper is as follows.

In Sections 1 and 2, we define the de Rham moduli and twistor stacks 𝔐_{dR,X,R} and 𝔗_{dR,X,R} of R-bundles on X, and construct the (ℂ ⊗_{ℝ} ℂ)^{∗}-action on the twistor stack. Sections 3 and 4 contain the pluriharmonic analogues with these, including the ℍ^{∗}-action on 𝔗_{dR,X,n}, together with properties of the forgetful functor Υ : 𝔗_{H,X,R} → 𝔗_{dR,X,R}. In Section 5, the local structures of the twistor stacks are described, including that singularities of 𝔗_{dR,X,R} in the image of Υ are all quadratic (Proposition 5.4), which is a generalisation of Goldman and Millson’s corresponding result ([GM]) for 𝔐_{dR,X,R}.

The universal Malcev homotopy type X^{R,univ} over 𝔐_{H,X,R} is defined in Section 6, and endowed with an algebraic mixed twistor structure. In Section 7, this is enhanced to an analytic mixed Hodge structure. These both split on tensoring with the weight 0 algebra S := ℝ[x] when the Hodge filtration on S ⊗ ℂ is given by powers of (x − i) (Propositions 6.8 and 7.6)

We will adopt the terminology and notation of [Pri2] without further comment. Throughout this paper, a real analytic stack will be understood to mean an Artin stack on the real analytic site, in the sense of [Noo].

1 The de Rham moduli stack

Fix a compact Kähler manifold X, and a real pro-algebraic groupoid R.

We define the de Rham moduli stack 𝔐_{dR,X,R} to be the real analytic stack parametrising flat connections on principal complex C^{∞} R-bundles on X. Explicitly:

Definition 1.1. Given a sheaf 𝒢 of groupoids on a topological space X, define a principal 𝒢 -bundle to consist of a pair (f, 𝒫), for f ∈ Γ(X, Ob 𝒢) a global section of the object set, and 𝒫 a principal 𝒢 (f, f)-bundle. A morphism from (f, 𝒫) to (f′, 𝒫′) consists of a section α ∈ Γ(X, 𝒢 (f, f′)) together with an isomorphism β : 𝒫α → 𝒫′ of 𝒢 (f′, f′)-bundles. Note that if 𝒢 is a group, this recovers the usual definition, and that if 𝒢 is a constant sheaf, then the groupoid of bundles is equivalent, but not isomorphic, to the category of functors from π_{f}X to 𝒢.

Definition 1.2. Given a real analytic space U and a real manifold Y, define the sheaf 𝒜_{Y ×U/U}^{0} to consist of C^{∞} functions on Y × U, analytic along U. It is clear that this is well-defined when U is smooth; if not, take a closed immersion i : U ↪ S, for S smooth, and set 𝒜_{Y ×U/U}^{0} := i^{−1}𝒜_{Y ×S/S}^{0} ⊂ 𝒜_{Y ×U}^{0}. Now set 𝒜_{Y ×U/U}^{n} := 𝒜_{Y ×U/U}^{0} ⊗_{𝒜_{Y}^{0}} 𝒜_{Y}^{n}.

Definition 1.3. Given a real analytic space U, define 𝔐_{dR,X,R}(U) to be the groupoid with objects (𝒫, ∇), for 𝒫 a principal R(𝒜_{X×U/U}^{0} ⊗ ℂ)-bundle on X, and

∇ : 𝒫 → ad𝒫 ⊗_{𝒜_{X×U/U}^{0}} 𝒜_{X×U/U}^{1},

which is a connection in the sense that ∇(p·g) = ad_{g}∇(p)+g^{−1} ·dg, for z, z′ ∈ Ob R, p ∈ 𝒫(z), g ∈ R(𝒜_{X×U/U}^{0})(z, z′). We also require that ∇ be flat, i.e. that

d ∘ ∇ = 0 : 𝒫 → ad𝒫 ⊗_{𝒜_{X×U/U}^{0}} 𝒜_{X×U/U}^{2}.

An isomorphism from (𝒫, ∇) to (𝒫′, ∇′) is an isomorphism f : 𝒫 → 𝒫′, such that ∇ ∘ f = adf ∘ ∇′.

Define the involution τ on 𝔐_{dR,X,R} to be given by complex conjugation of R(ℂ).

Lemma 1.4. Suppose that U, W are open balls in ℝ^{n} centred at the origin, and S a real analytic subspace of an open disc V in ℝ^{N}, with all embedded components passing through a point s ∈ S. If 𝒫 is an R(𝒜_{U×W ×S/S}^{0})-torsor on U × W × S, equipped with a flat connection

∇ : 𝒫 → (ad𝒫) ⊗_{𝒜_{U}^{0}} 𝒜_{U}^{1},

then

ν_{S} : H^{0}(U × W × S, 𝒫)^{∇} → H^{0}({0} × W × S, 𝒫|_{{0}×W ×S})

is an isomorphism.

Proof. If S is smooth, then the result is well known. In particular it is true when S is a point, and hence for an Artinian scheme (replacing 𝒫 by pr_{1∗}𝒫). The argument of [Sim5] Lemma 7.4 adapts to give injectivity of ν.

For surjectivity, we may assume that U = U′ × U′′, with dim U′ = 1. Replacing U with U′ and W with W′ := U′′ × W, and arguing inductively, we see that it suffices to consider the case dim U = 1. The flatness condition is then vacuous, allowing us to extend (𝒫, ∇) to (𝒫̃, ˜∇) on U × W × V (similarly to [Sim5] Lemma 7.4). Since ν_{V} is known to be an isomorphism, this gives the required surjectivity of ν_{S}.

Proposition 1.5. For any contractible real analytic space U, there is an equivalence of categories between the groupoid of R_{ℂ}(U)-representations of π_{1}(X), and 𝔐_{dR,X,R}(U). Moreover, R(U)-representations correspond to the τ-invariant locus of 𝔐_{dR,X,R}(U).

Proof. This is really just the observation that R(U)-representations of π_{1}(X) correspond to R(𝒪_{U})-torsors on X × U. The proof now proceeds as for [Sim5] Theorem 7.1, substituting Lemma 1.4 for [Sim5] Lemma 7.4. Given a connection (𝒫, ∇), we have an R(𝒪_{U})-torsor ker ∇. To an R(𝒪_{U})-torsor P, we associate the pair (P ×^{R(𝒪_{U})} R(𝒜_{X×U/U}^{0}), ∇), where ∇ is given by ∇(p, g) = g^{−1} · dg, for z, z′ ∈ Ob R, p ∈ P (z), g ∈ R(𝒜_{X×U/U}^{0})(z, z′).

Definition 1.6. Given a point ρ ∈ 𝔐_{dR,X,R}(ℝ), say that ρ is stable if the corresponding representation is pro-reductive, and let the stable de Rham stack 𝔐_{dR,X,R}^{s} ⊂ 𝔐_{dR,X,R} be the (open) substack on stable points.

2 The de Rham twistor stack

Definition 2.1. Define C to be the real affine scheme ∏_{ℂ/ℝ} 𝔸^{1} obtained from 𝔸_{ℂ}^{1} by restriction of scalars, so for any real algebra A, we have C(A) = 𝔸_{ℂ}^{1}(A ⊗_{ℝ} ℂ) ≅ A ⊗_{ℝ} ℂ. 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 [Del] 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.

S × C → C
(λ, w) ↦ (λ^{−1}w).

Choosing i ∈ ℂ gives an isomorphism C ≅ 𝔸_{ℝ}^{2}, giving co-ordinates u, v on C so that the isomorphism A × A ≅ A ⊗_{ℝ} ℂ is written (u, v) ↦ u ⊗ 1 + v ⊗ i. Thus the algebra O(C) associated to C is the polynomial ring C = ℝ[u, v]. S is isomorphic to the scheme 𝔸_{ℝ}^{2} − {(u, v) : u^{2} + v^{2} = 0}.

Definition 2.2. Let C^{∗}(ℂ) be the real analytic space of complex-valued points of C^{∗} (isomorphic to ℂ ⊗_{ℝ} ℂ − {0}). On C^{∗}(ℂ), we denote the complex co-ordinates (u, v) by (α, β), corresponding to α ⊗ 1 + β ⊗ i ∈ ℂ ⊗_{ℝ} ℂ.

C^{∗} has an involution τ given by complex conjugation of the co-ordinates ℂ, so τ(α, β) = (ᾱ, β̄).

There is an isomorphism C(ℂ) ≅ ℂ^{2} −{0} given by (α, β) ↦ (α +iβ, α −iβ). Under this isomorphism, S(ℂ) ⊂ C(ℂ) maps to ℂ^{∗} × ℂ^{∗}.

We will now define the twistor stack 𝔗_{dR,X,R} to be the analytic stack over C^{∗}(ℂ) whose fibre over (α, β) classifies flat αd + βd^{c}-connections on principal C^{∞} R-bundles on X. The explicit construction follows.

Definition 2.3. Given a real analytic space U over C^{∗}(ℂ), define 𝔗_{dR,X,R}(U) to be the groupoid with objects (𝒫, ∇), for 𝒫 a principal R(𝒜_{X×U/U}^{0})-bundle on X, and

∇ : 𝒫 → ad𝒫 ⊗_{𝒜_{X×U/U}^{0}} 𝒜_{X×U/U}^{1},

which is an αd+βd^{c}-connection in the sense that ∇(p·g) = ad_{g}∇(p)+g^{−1} ·(αd+βd^{c})g, for z, z′ ∈ Ob R, p ∈ 𝒫(z), g ∈ R(𝒜_{X×U/U}^{0})(z, z′). We also require that ∇ be flat, i.e. that

(αd + βd^{c}) ∘ ∇ = 0 : 𝒫 → ad𝒫 ⊗_{𝒜_{X×U/U}^{0}} 𝒜_{X×U/U}^{2}.

An isomorphism from (𝒫, ∇) to (𝒫′, ∇′) is an isomorphism

f : 𝒫 ×^{R(𝒜_{X×U/U}^{0} )} R(𝒜_{X×U/U}^{0} ⊗ ℂ) → 𝒫′ ×^{R(𝒜_{X×U/U}^{0} )} R(𝒜_{X×U/U}^{0} ⊗ ℂ)

of R(𝒜_{X×U/U}^{0} ⊗ ℂ)-bundles, such that ∇ ∘ f = adf ∘ ∇′.

Define the involution τ on 𝔗_{dR,X,R} to be given by complex conjugation, lifting τ on C^{∗}(ℂ).

Remarks 2.4. Note that we may extend this definition to give a stack over C(ℂ), but that this would not be analytic (i.e. not have a presentation), since the fibre over 0 is too large.

Note that 𝔐_{dR,X,R} = 𝔗_{dR,X,R} ×_{C^{∗}(ℂ)} (1, 0).

Definition 2.5. We may define a real analytic ℂ^{∗} × ℂ^{∗} ≅ S(ℂ)-action on 𝔗_{dR,X,R}(U) over C^{∗}(ℂ) by (α + iβ, α − iβ) ⋄ (𝒫, ∇) := (𝒫, α∇ + βJ ¯∇). Here (α, β) are the standard co-ordinates on C, as in Definition 2.2. This is τ-equivariant in the sense that τ((λ, μ) ⋄ y) = (μ̄, λ̄) ⋄ τ(y), since τ(α + iβ, α − iβ) = (ᾱ + iβ̄, ᾱ + iβ̄).

Remark 2.6. In fact, 𝔗_{dR,X,R} is the real analytic stack underlying a complex analytic stack, since the constructions above can all be defined for complex analytic spaces, with τ becoming an antiholomorphic involution. We have concentrated on the real structure to facilitate comparisons in the rest of the paper.

There is a canonical embedding 𝔾_{m} ↪ S, corresponding on ℂ-valued points to the diagonal inclusion ℂ^{∗} ↪ ℂ^{∗} ×ℂ^{∗} when making use of the isomorphism in Definition 2.5. Since [C^{∗}/𝔾_{m}] ≅ ℙ^{1}, this gives us a morphism [𝔗_{dR,X,R}/ℂ^{∗}] → ℙ^{1}(ℂ) of real analytic stacks. If R = GL_{n}, then the coarse moduli space associated to this stack is precisely the Deligne-Hitchin twistor space, as constructed in [Hit] and described in [Sim2] §3.

Remark 2.7. In [Pri2], an algebraic Hodge filtration on an object Z was defined to be an S-equivariant extension of X over C^{∗}. We may therefore regard 𝔗_{dR,X,R} as a kind of Hodge filtration on 𝔐_{dR,X,R}.

3 The pluriharmonic moduli stack

Definition 3.1. Define an involution C on a real pro-algebraic groupoid R to be a Cartan involution if for all x, y ∈ Ob R, Cτ is is complex conjugation with respect to a compact real form of the complex scheme R_{ℂ}(x, y), where τ denotes complex conjugation. Note that this extends the standard definition (from e.g. [Sim3] §4) for pro-algebraic groups.

Examples 3.2. 1. GL_{n} has a Cartan involution, given by C(A) = (A^{⊤})^{−1}.

2. By [Sim3] Theorem 7, ϖ_{1}(X, x)_{ℝ}^{red} has a Cartan involution, corresponding to −1 ∈ U_{1} for the unitary action on Higgs bundles. More generally, if ρ : π_{f}X → R(ℝ) is any Zariski-dense morphism, then R is a quotient of ϖ_{f}(X)^{red}, and C descends to R (by Tannakian duality, since C(V) ≅ V^{∨}), so the pair (R, C) satisfies the conditions above.

From now on, assume that the real pro-algebraic groupoid R is equipped with a Cartan involution C.

For an R(𝒜^{0} ⊗ ℂ)-torsor 𝒫, observe that the set of principal R_{ℂ}^{Cτ}(𝒜^{0})-subbundles 𝒬 of 𝒫 with 𝒫 = 𝒬 ×^{R_{ℂ}^{Cτ}(𝒜^{0})} R_{ℂ}(𝒜^{0}) is isomorphic to the set of global sections s of 𝒫/R_{ℂ}^{Cτ}(𝒜^{0}). Under this correspondence, 𝒬 is the inverse image of s, while s is the image of 𝒬. There is then an action of Cτ on 𝒫, given by Cτ(q, r) = (q, Cτ(r)), for r ∈ R(𝒜^{0}). This action is (R(𝒜^{0} ⊗ ℂ), Cτ)-equivariant, in the sense that Cτ(p · r) = Cτ(p) · Cτ(r).

Definition 3.3. Given an R(𝒜^{0} ⊗ ℂ)-torsor 𝒫 on a Kähler manifold X, equipped with a global section s of 𝒫/R_{ℂ}^{Cτ}(𝒜^{0}), and a flat connection D : 𝒫 → (ad𝒫) ⊗_{𝒜^{0}} 𝒜^{1}, define the d^{c}-connection D^{c} to be Jad_{Cτ} D : 𝒫 → (ad𝒫) ⊗_{𝒜^{0}} 𝒜^{1}, for J the complex structure.

Explicitly, if we decompose D into antihermitian and hermitian parts (i.e. ±1eigenspaces for Cτ) as D = d^{+} + ϑ, so

d^{+} = \frac{1}{2}(D + Cτ ∘ D ∘ Cτ), ϑ = \frac{1}{2}(D − Cτ ∘ D ∘ Cτ)

(noting that d^{+} is a connection and ϑ is R(𝒜^{0} ⊗ ℂ)-equivariant), then into (1, 0) and (0, 1) types as d^{+} = ∂ + ∂^{†} and ϑ = θ + θ^{†}, then we have

D^{c} = i∂ − i∂^{†} − iθ + iθ^{†}.

Definition 3.4. Define the pseudocurvature of the section s by G_{K} := [D, D^{c}] = DD^{c} + D^{c}D. The bundle is called harmonic (as in [Sim3] §1) if ΛG_{K} = 0, for Λ the formal adjoint to the Kähler form ω. The bundle is said to be pluriharmonic (as in [Moc] Remark 2.2) if G_{K} = 0.

Definition 3.5. Given t ∈ U_{1} (i.e. t ∈ ℂ and |t| = 1), and a flat pluriharmonic connection D, define t♣D := d^{+} + t ⋄ ϑ.

Lemma 3.6. The U_{1}-action given by ♣ preserves the set of flat pluriharmonic connections, and τ(t♣D) = t♣(τD), where τ denotes complex conjugation.

Definition 3.7. Given a real analytic space U, define 𝔐_{H,X,R}(U) to be the groupoid with objects (𝒫, s, ∇), for 𝒫 a principal R(𝒜_{X×U/U}^{0} ⊗ ℂ)-bundle on X, equipped with a global section s of 𝒫/R_{ℂ}^{Cτ}(𝒜^{0}), and

∇ : 𝒫 → ad𝒫 ⊗_{𝒜_{X×U/U}^{0}} 𝒜_{X×U/U}^{1}

a flat pluriharmonic connection in the sense of Definition 3.4.

An isomorphism from (𝒫, s, ∇) to (𝒫′, s′, ∇′) is an isomorphism f : 𝒫 → 𝒫′, such that f(s) = s′ and ∇ ∘ f = adf ∘ ∇′.

Lemma 3.8. There is a canonical map Υ : 𝔐_{H,X,R} → 𝔐_{dR,X,R}, and this maps isomorphism classes of ℝ- valued points isomorphically to the reductive representations.

Proof. This comes from forgetting the section s. Note that the image of this map is characterised in [Sim3] Theorem 1 as the semisimple local systems when (R, C) = (GL_{n}, transpose). This is because a harmonic metric on a local system 𝕍 defines a C^{∞} U_{n}-torsor. By Tannakian duality, it follows that the image in general comprises the reductive representations.

Moreover, it follows from [Moc] Theorem 1.1 that the harmonic metric on 𝕍 is unique up to conjugation by Aut_{X}(𝕍) (by uniqueness up to a scalar for irreducible local systems). This shows that Υ is injective on isomorphism classes when R = GL_{n}, and hence in general.

Remark 3.9. Note that the functor Υ is not full on ℝ-valued points: if X is a point, then 𝔐_{H,X,R} ≃ R^{Cτ}, while 𝔐_{dR,X,R} ≃ R.

4 The pluriharmonic twistor stack

Definition 4.1. Define Q to be the real scheme Q(A) = ℍ⊗A representing the quaternions, so Q ≅ 𝔸_{ℝ}^{4}. We have a real analytic isomorphism ȷ : C(ℂ) → Q(ℝ), given by mapping (α, β) to α + βj. Let Q^{∗} = Q − {0}.

Definition 4.2. Given a real analytic space U over Q^{∗}, define 𝔗_{H,X,R}(U) to be the groupoid with objects (𝒫, s, ∇), for 𝒫 a principal R(𝒜_{X×U/U}^{0})-bundle on X, equipped with a global section s of 𝒫/R^{C}(𝒜^{0}), and

∇ : 𝒫 → ad𝒫 ⊗_{𝒜_{X×U/U}^{0}} 𝒜_{X×U/U}^{1} ⊗ ℂ

an αd + βd^{c}-connection. We also require that ∇ be flat and pluriharmonic, i.e. that

(αd + βd^{c}) ∘ ∇ = 0 : 𝒫 → ad𝒫 ⊗_{𝒜_{X×U/U}^{0}} 𝒜_{X×U/U}^{2},

and that the commutator [∇, ∇^{c}] := ∇∇^{c} + ∇^{c}∇ vanishes, for ∇^{c} = Jad_{Cτ} ∇. An isomorphism from (𝒫, s, ∇) to (𝒫′, s′, ∇′) is an isomorphism

f : 𝒫 ×^{R(𝒜_{X×U/U}^{0} )} R(𝒜_{X×U/U}^{0} ⊗ ℂ) → 𝒫′ ×^{R(𝒜_{X×U/U}^{0} )} R(𝒜_{X×U/U}^{0} ⊗ ℂ)

of R(𝒜_{X×U/U}^{0} ⊗ ℂ)-bundles, such that ∇ ∘ f = adf ∘ ∇′ and ∇ ∘ f = adf ∘ ∇′.

Remarks 4.3. Note that we may extend this definition to give a stack over Q, but that this would not be analytic (i.e. not have a presentation), since the fibre over 0 is too large.

Observe that 𝔐_{H,X,R} = 𝔗_{H,X,R} ×_{Q^{∗}} (1, 0).

Definition 4.4. Define the involution τ on 𝔗_{H,X,R} to be given by complex conjugation, lifting the map τ = ad_{j} on Q^{∗} given by α + βj ↦ ᾱ + β̄j.

Definition 4.5. We may define a real analytic Q^{∗}-action on 𝔗_{H,X,R}(U) over Q^{∗} by (α + βj) ⋆ (𝒫, ∇) := (𝒫, α∇ + βJad_{Cτ} ∇). This is τ-equivariant in the sense that τ((α + βj) ⋆ y) = (ᾱ + β̄j) ⋆ τ(y).

Remark 4.6. Observe that, although multiplication on the scheme Q^{∗} does not have an inverse, its ℝ-valued points have a group structure, since they coincide with ℝ-valued points of the open subscheme |α|^{2} + |β|^{2} ≠ 0.

Proposition 4.7. 𝔗_{H,X,R} ≅ Q^{∗} × 𝔐_{H,X,R} as a real analytic stack over Q^{∗}.

Proof. The isomorphism is given by z ↦ (π(z), π(z)^{−1} ⋆ z), for π : 𝔗_{H,X,R} → Q^{∗}. The inverse is (q, y) ↦ q ⋆ y.

Definition 4.8. Define Υ : 𝔗_{H,X,R} → 𝔗_{dR,X,R} by (𝒫, s, ∇) ↦ (𝒫, ∇); this is a morphism over ȷ^{−1} : Q^{∗}(ℝ) → C^{∗}(ℂ).

Definition 4.9. Let j := 1 ⊗ i ∈ ℂ ⊗_{ℝ} ℂ. For γ ∈ C(ℂ), this gives γ = α + βj in terms of the standard co-ordinates. Note that j commutes with ℂ ⊗ 1, and that the isomorphism ȷ is then given by α + βj ↦ α + βj.

Observe that S(ℂ) then consists of α + βj for which α^{2} + β^{2} is invertible, and that on a (p, q)-form, the ⋄-action of α + βj ∈ S(ℂ) is (α + iβ)^{p}(α − iβ)^{q}.

Proposition 4.10. For (α+βj) ∈ S(ℂ), such that |α+iβ| = |α−iβ|, and y in 𝔐_{H,X,R}

(α + βj) ⋄ Υ(y) = Υ((α + βj) ⋆ (\frac{α + iβ}{α − iβ} ♣y)),

for ♣ as in Definition 3.5. In particular, for β = 0 (corresponding to 𝔾_{m}(ℂ) ⊂ S(ℂ)), the ⋆ and ⋄-actions agree.

Proof. This follows from the observation that for D a d-connection (associated to an object of 𝔐_{H,X,R}),

(α + βj) ⋆ D = (α + βJ)d^{+} + (α − βJ)ϑ
= (α + iβ)(∂ + θ^{†}) + (α − iβ)(∂^{†} + θ)
= (α + βj) ⋄ (d^{+} + (\frac{α^{2} + β^{2} + 2αβj}{α^{2} + β^{2}} ) ⋄ ϑ).

Thus

(α + βj) ⋄ Υ(\frac{α − iβ}{α + iβ} ♣z) = Υ((α + βj) ⋆ z),

and the result follows by setting z = \frac{α+iβ}{α−iβ} ♣y.

Remark 4.11. Making use of the isomorphism ℙ_{ℂ}^{1} ≅ [C_{ℂ}^{∗}/𝔾_{m,ℂ}], and the compatibility of the ℂ^{∗} actions for ⋆ and ⋄, we obtain an analytic morphism

Υ : [𝔐_{H,X,R}/𝔾_{m}(ℂ)] → [𝔐_{dR,X,R}/𝔾_{m}(ℂ)]

over ℙ^{1}(ℂ) = [C^{∗}(ℂ)/𝔾(ℂ)].

Υ is τ-equivariant, where τ is the involution τ(α : β) = (ᾱ : β̄). Making use of the co-ordinate change (λ : μ) = (α + iβ : α − iβ) as Definition 2.2, τ corresponds to the circular involution (λ : μ) ↦ (μ̄ : λ̄) of [Sim1] p.12 . The antipodal involution σ(λ : μ) = (iλ̄, −iμ̄) of [Sim1] corresponds in our co-ordinates to σ(α : β) = (−β̄ : ᾱ) — this lifts to [𝔐_{H,X,R}/ℂ^{∗}] as j⋆. Remark 4.12. We may also use these actions to describe the discrete ℂ^{∗}-action given in [Sim3] on the points of 𝔐_{H,X,R}. λ ∈ ℂ^{∗} maps y to

(1 − ij) ⋆ (Υ^{−1}(\frac{λ + 1}{2} + \frac{λ − 1}{2i}j) ⋄ Υ(\frac{1 + ij}{2} ⋆ y)),

making use of the injectivity of Υ on isomorphism classes of objects, and the fact that Im Υ is preserved by the ⋄-action.

This maps D = ∂ + ∂^{†} + θ + θ^{†} to a connection D′ with (∂^{†})′ = ∂^{†} and θ′ = λθ. If λ ∈ U_{1} ⊂ ℂ^{∗}, this expression reduces to λ♣D.

5 Local structure of the moduli stacks

Fix an ℝ-valued object y of 𝔗_{H,X,R}, i.e. a point α + βj ∈ ℍ^{∗}, together with a triple (𝒫, s, ∇) as in Definition 4.2, for ∇ a flat pluriharmonic αd + βd^{c}-connection. We now describe the local structure of 𝔗_{H,X,R} over Q^{∗} about y, and of 𝔗_{dR,X,R} over C^{∗}(ℂ) about Υ(y).

As in [GM], §3.1, the analytic germs are determined by evaluating the stack at Artinian local ℝ-algebras. Explicitly, let C_{ℝ} be the category of Artinian local ℝ-algebras with residue field ℝ. Then the germ of 𝔗_{H,X,R} at y is determined by the groupoid-valued functor

(𝔗_{H,X,R})_{y} : C_{ℝ} → Gpd
A ↦ 𝔗_{H,X,R}(Spec A) ×_{𝔗_{H,X,R}(Spec ℝ)} y.

We may also describe Q^{∗} in this way, with

Q_{α+βj}^{∗} : C_{ℝ} → Set
A ↦ Q^{∗}(Spec A) ×_{Q^{(}Spec ℝ)} (α + βj);

this is pro-represented by { \atop O(Q)_{α+βj}}, the complete local ring of Q at α + βj, which is isomorphic to the real power series ring in 4 variables.

There is a similar description for 𝔗_{dR,X,R} (noting that C^{∗}(ℂ) is a smooth 4dimensional real analytic space).

In [GM] §2, a deformation groupoid is associated to every real differential graded Lie algebra (DGLA) L.

Definition 5.1. Fix a real DGLA L. The Maurer-Cartan functor MC_{L} : C_{ℝ} → Set is defined by

MC_{L}(A) = \{x ∈ L^{1} ⊗ 𝔪(A)|dx + \frac{1}{2}[x, x] = 0\}.

Define the gauge functor G_{L} : C_{ℝ} → Grp by G_{L}(A) = exp(L^{0} ⊗ 𝔪(A)). This acts on MC_{L} by the gauge action g(x) := gxg^{−1} − (dg)g^{−1}. Define the deformation functor 𝔇𝔢𝔣_{L} : C_{ℝ} → Gpd by 𝔇𝔢𝔣_{L} := [MC_{L}/G_{L}], and say that L governs a functor F if F is equivalent to 𝔇𝔢𝔣_{L}.

Definition 5.2. Given y = (α + βj, 𝒫, s, ∇) ∈ (𝔗_{H,X,R})(ℝ), define

A^{•}(X, ady) := H^{0}(X, (ad𝒫 ⊗_{𝒜_{X×U/U}^{0}} 𝒜_{X×U/U}^{∗}, ∇))
H^{∗}(X, ady) := H^{∗}(A^{•}(X, ady)).

Lemma 5.3. Given y = (α + βj, 𝒫, s, ∇) ∈ (𝔗_{H,X,R})(ℝ), the functor (𝔗_{dR,X,R})_{Υ(y)} : C_{ℝ} → Gpd is governed by the DGLA

L^{i} := \begin{cases}A^{i}(X, ady) & i ≠ 1 \\ ℂ ⊗_{ℝ} ℂ ⊕ A^{1}(X, ady) & i = 1, \end{cases}

with the standard differential and Lie bracket on A^{•}(X, ady), while d(ℂ ⊗_{ℝ} ℂ) = 0, and [u + vj, w] = \frac{(uᾱ−vβ̄)∇w+(vα−uβ)∇^{c}w}{|α|^{2}+|β|^{2}}, for u + vj ∈ ℂ ⊗_{ℝ} ℂ and w ∈ A^{i}(X, ady).

Proof. Fix A ∈ C_{ℝ}, and denote the maximal ideal by 𝔪(A). Since C^{∞}-bundles do not deform, any A-valued deformation of (𝒫, ∇) is isomorphic to a deformation of the form (α̃ + β̃j, 𝒫 ×^{R} R(A), ˜∇). Here, α̃ + β̃j ∈ A ⊗_{ℝ} ℂ ⊗_{ℝ} ℂ, congruent to α + βj mod 𝔪(A). Similarly,

˜∇ : 𝒫 → ad𝒫 ⊗_{𝒜_{X×U/U}^{0}} 𝒜_{X×U/U}^{1} ⊗ A,

is a flat α̃d + β̃d^{c}-connection, congruent to ∇ mod 𝔪(A).

Now, given an element (u + vj, ω) ∈ MC_{L}(A) ⊂ L^{1} ⊗ 𝔪(A) of the Maurer-Cartan space, define

(α̃ + β̃j) := (α + βj) + (u + vj)
˜∇ := ȷ(α̃ + β̃j) ⋆ ȷ(α + βj)^{−1} ⋆ ∇ + ω.

This gives the required correspondence, noting that flatness of ˜∇ is equivalent to the Maurer-Cartan equation

d(u + vj, ω) + \frac{1}{2}[(u + vj, ω), (u + vj, ω)] = 0.

The treatment of isomorphisms is similar (as in [GM] §6).

The following result generalises the formality results of [GM], and shows that the moduli stack has only quadratic singularities in the image of Υ.

Proposition 5.4. Given y = (α + βj, 𝒫, s, ∇) ∈ (𝔗_{dR,X,R})(ℝ), then the germ (𝔗_{dR,X,R}, Υ(y)) is isomorphic to (C^{∗}(ℂ), α + βj) × 𝔇𝔢𝔣_{(H^{∗}(X,ady))} over C^{∗}(ℂ).

Proof. By Lemma 5.3, the germ is given by 𝔇𝔢𝔣_{L}. Now, we have quasi-isomorphisms

H^{∗}(L) ← (ker ∇^{c}, ∇) → L

of DGLAs by the principle of two types (similarly to [Pri2] Lemma 4.11), so 𝔇𝔢𝔣_{L} ≅ 𝔇𝔢𝔣_{H∗(L)}. The quasi-isomorphisms also respect the augmentation maps from these DGLAs to ℂ⊗_{ℝ}ℂ[−1], the DGLA governing (C^{∗}(ℂ), α+βj). Now, H^{∗}(L) ≅ \frac{ker ∇∩ker ∇^{c}}{Im ∇∇^{c}}, so H^{∗}(L) = H^{∗}(X, ady) × ℂ ⊗_{ℝ} ℂ[−1].

Definition 5.5. Given a ∈ A^{∗}(X, ady), set a^{c} := JCτa.

Observe that the description H^{∗}(X, ady) ≅ \frac{ker ∇∩ker ∇^{c}}{Im ∇∇^{c}} ensures that JCτ is a welldefined automorphism of H^{∗}(X, ady).

Proposition 5.6. The isomorphism classes of the germ (𝔐_{H,X,R}, y) have tangent space H^{1}(X, ady) and obstruction space H^{2}(X, ady) ⊕ H^{2}(X, ady)^{−JCτ}, while the automorphism group of y has tangent space H^{0}(X, ady)^{Cτ}. The action of the automorphism group on the tangent and obstruction spaces comes from exponentiating the Lie bracket, and the primary obstruction map is

H^{1}(X, ady) → H^{2}(X, ady) ⊕ H^{2}(X, ady)^{−JCτ}
ω ↦ ([ω, ω], [ω, ω^{c}]).

Proof. Observe that a flat d-connection ∇ is pluriharmonic if and only if ∇ + ∇^{c} is a flat d + d^{c}-connection. Thus (𝔐_{H,X,R}, y) is governed by the DGLA M ⊂ A^{•}(X, ady) × (A^{∗}(X, ady), ∇ + ∇^{c}), given by

M^{i} := \begin{cases}\{(a, a) : a ∈ A^{0}(X, ady)^{Cτ} \} & i = 0, \\ \{(a, a + a^{c}) : a ∈ A^{1}(X, ady)\} & i = 1, \\ \{(a, b) : b − a − a^{c} ∈ A^{2}(X, ady)^{−JCτ} \} & i = 2, \\ A^{i}(X, ady) × A^{i}(X, ady) & i > 2. \end{cases}

This follows because A^{•}(X, ady) × (A^{∗}(X, ady), ∇ + ∇^{c}) governs deformations (∇′, E′), of the pair (∇, ∇ + ∇^{c}), and M imposes the additional constraint that E′ = ∇′ + (∇′)^{c}.

We now calculate the cohomology groups of this DGLA. Observe that H^{0}(M) = H^{0}(X, ady)^{Cτ}. For H^{1}, take (ω, ω +ω^{c}) ∈ Z^{1}(M), so (∇ω, ∇ω +∇^{c}ω +∇ω^{c} +∇^{c}ω^{c}) = 0, or equivalently ∇ω = ∇^{c}ω + ∇ω^{c} = 0. By the principle of two types, we may write ∇^{c}ω = ∇^{c}∇α, for some α ∈ A^{0}(X, ady). Applying JCτ, we also have ∇ω^{c} = ∇∇^{c}α^{c}, so (∇^{c}α, ∇^{c}α + (∇^{c}α)^{c}) ∈ Z^{1}(M). Decompose α as α = α^{+} + α^{−} into ±1-eigenvectors for Cτ, and observe that (∇α)^{c} = ∇^{c}α^{+} − ∇^{c}α^{−}. Thus 0 = ∇(∇α)^{c} + ∇^{c}(∇α) = 2∇^{c}∇α^{−}, which means that ∇α^{−} ∈ Im (∇∇^{c}) = 0, by the principle of two types, so we may assume that α^{−} = 0. Since ω − ∇α ∈ ker(∇) ∩ ker(∇^{c}), we have

Z^{1}(M) = {(a, a + a^{c}) : a ∈ (ker(∇) ∩ ker(∇^{c})) ⊕ ∇(M^{0})},

so H^{1}(M) ≅ ker(∇) ∩ ker(∇^{c}) ≅ H^{1}(X, ady).

To calculate H^{2}, take z ∈ H^{2}(M), represented by (a, b) ∈ Z^{2}(M). We may choose e ∈ A^{1}(X, ady) such that a + ∇e ∈ ker ∇^{c}, so without loss of generality, we may assume that a ∈ ker ∇ ∩ ker ∇^{c}. Then

∇^{c}(b − a − a^{c}) = (−∇(b − a − a^{c})^{c})^{c} = (∇(b − a − a^{c}))^{c} = 0,

so b ∈ ker ∇ ∩ ker ∇^{c}. By the principle of two types, any other such representative is of the form (a + ∇e, b + (∇ + ∇^{c})(e + e^{c})), for e ∈ ker ∇^{c}∇. Thus:

H^{2}(M) ≅ \frac{\{(a, b) ∈ (ker ∇ ∩ ker ∇^{c}) × ker(∇ + ∇^{c}) : b − a − a^{c} ∈ A^{2}(X, ady)^{−JCτ} \}}{\{∇e, (∇ + ∇^{c})(e + e^{c})) : e ∈ ker ∇^{c}∇}.

If we now change co-ordinates on A^{∗}(X, ady)^{2}, by the transformation (a, b) ↦ (a, b − a − a^{c}), then in these new co-ordinates we have

H^{2}(M) ≅ \frac{(ker ∇ ∩ ker ∇^{c}) × ker(∇ + ∇^{c})^{−JCτ}}{\{∇e, ∇e^{c} + ∇^{c}e) : e ∈ ker ∇^{c}∇\}}

Since ker ∇^{c}∇ = ker ∇^{c} + ker ∇, we may write e = f + g with respect to this decomposition, giving (a + ∇f, β + ∇g^{c} + ∇^{c}g). Now ∇(ker ∇^{c}) = ∇^{c}(ker ∇) = ∇^{c}∇A^{0}(X, ady), so the denominator is

Im (∇^{c}∇) × {v − v^{c} : v ∈ Im (∇^{c}∇)} = Im (∇^{c}∇) × Im (∇^{c}∇)^{−JCτ} .

Hence

H^{2}(M) ≅ \frac{(ker ∇ ∩ ker ∇^{c}) × ker(∇ + ∇^{c})^{−JCτ}}{Im (∇^{c}∇) × Im (∇^{c}∇)^{−JCτ}} ≅ H^{2}(X, ady) ⊕ H^{2}(X, ady)^{−JCτ} .

The remaining statements follow by computing the Lie bracket on H^{∗}(M).

6 Relative homotopy types over analytic moduli stacks

The following is a partial generalisation of [Pri2] Definition 2.42 to arbitrary ringed topoi:

Definition 6.1. Given a ringed topos Y, define DG_{ℤ}Alg_{Y} (R) to be the category of R-representations in quasi-coherent ℤ-graded 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} (R)) to be the homotopy category obtained by localising at weak equivalences. Define the categories dg_{ℤ}Aff_{Y} (R), Ho(dg_{ℤ}Aff_{Y} (R)) to be the opposite categories.

Remark 6.2. If, for the R-representation Y in schemes of [Pri2] Definition 2.42, we let Z be the algebraic stack

[ \xymatrix{∐_{x,x′∈Ob R} R(x, x′) × Y (x′) \ar[r] \ar[r] & ∐_{x∈Ob R} Y (x) \ar[l]}]

in the notation of [LMB] 2.4.3, then the categories DG_{ℤ}Alg_{Z} and DG_{ℤ}Alg_{Y} (R) are equivalent, and similarly for the other constructions in Definition 6.1

Similarly, we have DG_{ℤ}Alg_{Y} (R) ≃ DG_{ℤ}Alg_{Y ×R} for the site Y of Definition 6.1, regarding R as an algebraic stack (with the lisse-étale site).

Definition 6.3. Define X^{R,univ} ∈ Ho(dg_{ℤ}Aff_{[𝔐_{H,X,R}/⟨τ⟩]}(R)) to be given by the flat object Spec A^{•}(X, O), defined as follows. For f : V → 𝔐_{H,X,R} given by (𝒫, s, ∇) as in Definition 3.7, we set 𝒜^{n}(𝒫) := 𝒫 ×^{R(𝒜_{X×V/V}^{0} ⊗ℂ)} (𝒜_{X×V/V}^{n} ⊗ ℂ ⊗ O(R)). Now, for V contractible define

Γ(V, A^{∗}(X, O)) := Γ(X, 𝒜^{∗}(𝒫)),

with differential ∇. Note that this definition does not involve s, so X^{R,univ} is the pullback of an object on 𝔐_{dR,X,R}.

Theorem 6.4. Given a compact Kähler manifold X, and a real pro-algebraic groupoid R with a Cartan involution C, there is a canonical object

X_{MTS}^{R,univ} ∈ Ho(dg_{ℤ}Aff_{[𝔐_{H,X,R}/⟨τ⟩]×[𝔸^{1}/𝔾_{m}]×[C^{∗}/𝔾_{m}]}(R)),

which on pulling back along the point [ρ] : Spec ℝ → [𝔐_{H,X,R}/⟨τ⟩] (corresponding to a real Zariski-dense representation ρ : π_{f} → R(ℝ)) gives the mixed twistor structure X_{MTS}^{ρ,Mal} ∈ Ho(dg_{ℤ}Aff_{[𝔸^{1}/𝔾_{m}]×[C^{∗}/𝔾_{m}]}(R)) of [Pri2] Theorem 5.1.

Moreover, X_{MTS}^{R,univ} ×_{[𝔸^{1}/𝔾_{m}]×[C^{∗}/𝔾_{m}],(1,1)}^{R} Spec ℝ ≅ X^{R,univ}.

Proof. We begin by defining the object

X_{𝕋}^{R,univ} = X_{MTS}^{R,univ} ×_{[𝔸^{1}/𝔾_{m}],1}^{R} Spec ℝ ∈ Ho(dg_{ℤ}Aff_{[𝔐_{H,X,R}/⟨τ⟩]×[C^{∗}/𝔾_{m}]}(R)).

Since 𝔖 := [𝔐_{H,X,R}/⟨τ⟩] × [C^{∗}/𝔾_{m}] is a product of an analytic stack with an algebraic stack, a little care has to be taken with this site. A base for the site is given by objects of the form [V/⟨τ⟩] × [W/𝔾_{m}], for f : V → 𝔐_{H,X,R} a τ-equivariant smooth map of analytic stacks (i.e. a smooth map [V/⟨τ⟩] → 𝔐_{H,X,R}/⟨τ⟩ over B⟨τ⟩), and g : W → C^{∗} a smooth 𝔾_{m}-equivariant map of algebraic spaces. The structure sheaf 𝒪_{𝔖} on this site is given by 𝒪_{𝔖}|_{V ×W} = (pr_{V}^{−1}𝒪_{V}) ⊗_{ℝ} (pr_{W}^{−1}𝒪_{W}).

Let f : V → 𝔐_{H,X,R} be given by (𝒫, s, ∇) as in Definition 3.7, and take g : W → C^{∗}, 𝔾_{m}-equivariant. Now, for V contractible and W affine, define the sheaf Ã^{n}(X, O) locally by

Γ(V × W , Ã^{n}(X, O)) := Γ(X, 𝒜^{n}(𝒫)) ⊗_{ℝ} Γ(W, 𝒪_{W}).

This has a τ-action given by combining the τ-action on 𝒫 (or equivalently, on V) with complex conjugation on ℂ. It also has a 𝔾_{m}-action, given by combining the 𝔾_{m}-action on W with the action setting 𝒜^{n}(𝒫) to be pure of weight n. We therefore define Ã^{n}(X, O) on [V/⟨τ⟩] × [W/𝔾_{m}] to be given by ⟨τ⟩ × 𝔾_{m}-invariants:

Γ([V/⟨τ⟩] × [W/𝔾_{m}], Ã^{n}(X, O)) := Γ(V × W, Ã^{n}(X, O))^{⟨τ⟩,𝔾_{m}}.

To see that this is a quasi-coherent sheaf, first consider the moduli stack 𝔐_{C^{∞},X,R} of complex principal C^{∞} R-bundles on X. This is a discrete stack, so provided V is connected, we may assume that there is a principal R(𝒜_{X}^{0} ⊗ ℂ)-bundle ℬ, with 𝒫 = ℬ ×^{R(𝒜_{X}^{0}⊗ℂ)} R(𝒜_{X×V/V}^{0} ⊗ ℂ). Then

𝒜^{n}(𝒫) = ℬ ×^{R(𝒜_{X}^{0}⊗ℂ)} (𝒜_{X×V/V}^{n} ⊗ ℂ ⊗ O(R)),

and compactness of X then gives

Γ(X, 𝒜^{n}(𝒫)) = Γ(X, ℬ ×^{R(𝒜_{X}^{0}⊗ℂ)} (𝒜_{X}^{n} ⊗ ℂ ⊗ O(R))) ⊗ Γ(V, 𝒪_{V}),

since we may regard 𝒜_{X×V/V}^{n} as consisting of C^{∞} functions from X to 𝒪_{V}. This gives

Ã^{n}(X, O) = Γ(X, ℬ ×^{R(𝒜_{X}^{0}⊗ℂ)} (𝒜_{X}^{n} ⊗ ℂ ⊗ O(R))) ⊗ 𝒪_{𝔖}.

We now define the differential on Ã^{∗}(X, O). The co-ordinates u, v on C^{∗} give elements of 𝒪_{W}, and we now set the differential on Γ(V × W, Ã^{∗}(X, O)) to be u∇ + v∇^{c}. Since this is ⟨τ⟩ × 𝔾_{m}-equivariant, it descends to Γ([V/⟨τ⟩] × [W/𝔾_{m}], Ã^{n}(X, O)), so we have defined

Ã^{•}(X, O) ∈ DG_{ℤ}Alg_{[𝔐_{H,X,R}/⟨τ⟩]×[C^{∗}/𝔾_{m}]}(R),

and may set X_{𝕋}^{R,univ} := Spec Ã^{•}(X, O).

We now define the weight filtration W on Ã^{•}(X, O) to be given by good truncation W_{r} = τ_{≤r}, and we define O(X_{MTS}^{R,univ}) to be the Rees algebra Rees(Ã^{•}(X, O), W) of this filtration. Since this complex is bounded and flat, derived pullbacks agree with ordinary pullbacks, so

Spec Rees(Ã^{•}(X, O), W) ×_{C^{∗},1}^{R} Spec ℝ = Spec Rees(A^{•}(X, O), W),

as required.

6.1 Formality and splitting

Lemma 6.5. On the [𝔐_{H,X,R}/⟨τ⟩]-sheaf A^{∗}(X, O) of Definition 6.3, the operators ∇, ∇^{c} satisfy the principle of two types

ker ∇ ∩ ker ∇^{c} ∩ (Im ∇ + Im ∇^{c}) = Im ∇∇^{c}.

Proof. It suffices to show that for any real analytic space U, and any U-valued family (𝒱 , ⟨⟩, ∇) of pluriharmonic vector bundles on X, that the operators ∇, ∇^{c} satisfy the principle of two types on A^{∗}(X, 𝒱 ). The proof of the Hodge decomposition on forms adapts to this generality, observing that the Green’s operator preserves real analytic families of forms.

Definition 6.6. Define \underline{gr}X_{MTS}^{R,univ} ∈ Ho(dg_{ℤ}Aff_{[𝔐_{H,X,R}/⟨τ⟩]×B𝔾_{m}}(R)) by setting

O(\underline{gr}X_{MTS}^{R,univ}) := H^{∗}(A^{•}(X, O)).

Definition 6.7. Define an S-action on SL_{2} to be given on the top row as right multi-

plication by λ ↦ ( \begin{matrix}ℜλ & ℑλ \\ −ℑλ & ℜλ \end{matrix} )^{−1} , and on the bottom row by λ ↦ ( \begin{matrix}ℜλ & −ℑλ \\ ℑλ & ℜλ \end{matrix} ) .

Let row_{1} : SL_{2} → C^{∗} be the S-equivariant map given by projection onto the first row. Under the equivalence between S-equivariant C^{∗}-bundles and Hodge filtrations of [Pri2] Corollary 1.8, row_{1} corresponds to the algebra

S := ℝ[x],

with filtration F^{p}(S ⊗ ℂ) = (x − i)^{p}ℂ[x], by [Pri2] Lemma 1.17.

Proposition 6.8. There is a canonical isomorphism

(X_{MTS}^{R,univ}) ×_{[C^{∗}/𝔾_{m}],row_{1}}^{R} [SL_{2}/𝔾_{m}] ≃ \underline{gr}X_{MTS}^{R,univ} ×_{B𝔾_{m}}^{R} ([𝔸^{1}/𝔾_{m}] × [SL_{2}/𝔾_{m}]),

in Ho(dg_{ℤ}Aff_{[𝔐_{H,X,R}/⟨τ⟩]×[𝔸^{1}/𝔾_{m}]×[SL_{2}/𝔾_{m}]}(R)), which gives the splitting of [Pri2] Theorem 5.1 on pulling back along [ρ].

Proof. The proof of [Pri2] Corollary 4.11 carries over, by using the principle of two types from Lemma 6.5.

7 Unitary actions

We now seek to describe additional structure on X_{MTS}^{R,univ} generalising the U_{1}-action of [Pri2] Proposition 5.14. The ⋄-action of λ ∈ S(ℝ) on A^{n}(X) gives a discrete action on Ã^{•}(X, O), when combined with the standard action on C^{∗} and the ♣-action of \frac{λ̄}{λ} on 𝔐_{H,X,R} (this is essentially the ♠-action of [Pri2] Proposition 4.25). However, we wish to capture the analytic properties of this action. Informally, we would like to work on the site

([𝔐_{H,X,R}/⟨τ⟩] × C^{∗})/ιS,

with ι : S → U_{1} × S (given by λ ↦ (\frac{λ̄}{λ}, λ)) acting on both terms, as in the description above. The problem with this is that [𝔐_{H,X,R}/⟨τ⟩] is analytic, with an analytic U_{1}- action, while C^{∗} is algebraic, with an algebraic S-action.

We circumvent this by considering that for an immersion G ↪ H of group schemes, with H acting on an algebraic stack 𝔑, there are canonical isomorphisms

[𝔑/G] ≅ [𝔑/H] ×_{BH} BG ≅ [𝔑/H] ×_{BH} [(G/H)/H],

so quasi-coherent sheaves on [𝔑/G] correspond to quasi-coherent q^{∗}𝒪_{[(G/H)/H]}-modules on [𝔑/H], for q : [𝔑/H] → BH. [Pri2] Lemma 1.24 is a special case of this, with 𝔑 = C^{∗}, G = 𝔾_{m} and H = S.

Now, the quotient of the map ι given above is isomorphic to S/𝔾_{m} ≅ U_{1}, motivating the following definition.

Definition 7.1. Observe that on the product B(U_{1}^{an}) × BS of an analytic stack with an algebraic stack, quasi-coherent sheaves correspond to vector spaces equipped with an analytic U_{1}-action commuting with an algebraic S-action.

Define 𝒰 to be the quasi-coherent sheaf on this site corresponding to the real algebra O(U_{1}) (of polynomial functions on U_{1}), equipped with its usual structure U_{1}-action, together with an S-action in which λ acts as \frac{λ}{λ̄}. Note that the image of ι corresponds to the kernel of these actions combined.

Definition 7.2. For the quotient map q : [𝔐_{H,X,R}/(⟨τ⟩×U_{1}^{an})] → B(U_{1}^{an})×BS, define

DG_{ℤ}Alg_{([𝔐_{H,X,R}/⟨τ⟩]×C^{∗})/ιS}(R) := (q^{∗}𝒰 )↓DG_{ℤ}Alg_{[𝔐_{H,X,R}/(⟨τ⟩×U_{1}^{an})]×[C^{∗}/S]}(R).

Lemma 7.3. To give an object of A ∈ DG_{ℤ}Alg_{([𝔐_{H,X,R}/⟨τ⟩]×C^{∗})/ιS}(R) is equivalent to giving, for all τ, U_{1}-equivariant smooth maps f : V → 𝔐_{H,X,R} a of analytic stacks, and all smooth S-equivariant maps g : W → C^{∗} of algebraic spaces, the compatible objects

A|_{V ×W} ∈ DG_{ℤ}Alg_{U×V} (R),

equipped with their natural S^{δ} = ℂ^{∗}-representations, subject to the conditions that

1. the action of 𝔾_{m} ⊂ S is algebraic, and compatible with the 𝔾_{m}-action on ℂ^{∗};

2. the action of U_{1} ⊂ S is analytic, and compatible with the U_{1}^{δ}-action on 𝔐_{H,X,R}×ℂ^{∗} given by t♠(y, c) = (t^{−2}♣y, tc).

Proof. We give the construction of A|_{V ×W}. The object A corresponds to some

B ∈ (q^{∗}𝒰 )↓DG_{ℤ}Alg_{[𝔐_{H,X,R}/(⟨τ⟩×U_{1}^{an})]×[C^{∗}/S]}(R),

by definition. The site [𝔐_{H,X,R}/(⟨τ⟩×U_{1}^{an})]×[C^{∗}/S] is generated by spaces of the form [V/⟨τ⟩ × U_{1}^{an}] × [W/S], so we may consider (f, g)^{∗}B ∈ (q^{∗}𝒰 ) ↓ DG_{ℤ}Alg_{V ×W} (R). This has an analytic U_{1}-action over W, and an algebraic S-action over W.

Now, the unit 1 ∈ U_{1} gives a map O(U_{1}) → ℝ, and hence an ιS-equivariant map q^{∗}𝒰 → 𝒪_{V ×W}, so we set

A|_{V ×Q} := (f, g)^{∗}B ⊗_{𝒰} 𝒪_{V ×W} ,

noting that this has an S-action given by ι, and satisfying the required properties.

Proposition 7.4. Given a compact Kähler manifold X, and a real pro-algebraic groupoid R with a Cartan involution C, there is a canonical object

X_{MHS}^{R,univ} ∈ Ho(dg_{ℤ}Aff_{[𝔸^{1}/𝔾_{m}]×[([𝔐_{H,X,R}/⟨τ⟩]×C^{∗})/ιS]}(R)).

Pulling back along

[𝔐_{H,X,R}/⟨τ⟩] × [C^{∗}/𝔾_{m}] = [([𝔐_{H,X,R}/⟨τ⟩] × C^{∗})/ι𝔾_{m}] ⟶^{ε} [([𝔐_{H,X,R}/⟨τ⟩] × C^{∗})/ιS]

gives ε^{∗}X_{MHS}^{R,univ} = X_{MTS}^{R,univ}.

Proof. We use the criteria of Lemma 7.3. We will enhance the structure of Ã^{•}(X, O) from Theorem 6.4 to give an object of DGAlg_{([𝔐_{H,X,R}/⟨τ⟩]×C^{∗})/ιS}(R). For V, W as in Lemma 7.3, consider

Γ(V × W , Ã^{n}(X, O)),

and observe that the analytic U_{1}-action on V allows us to extend the algebraic 𝔾_{m}-action of Theorem 6.4 to an analytic S-action, given by the ♠ formula. This satisfies the conditions of Lemma 7.3, allowing us to define X_{𝔽}^{R,univ} ∈ Ho(dg_{ℤ}Aff_{[([𝔐_{H,X,R}/⟨τ⟩]×C^{∗})/ιS]}(R)). Taking the Rees algebra construction gives X_{MHS}^{R,univ}.

Remarks 7.5. 1. Given a U_{1}^{an}-equivariant map [ρ] : Spec ℝ → [𝔐_{H,X,R}/⟨τ⟩], we may pull back X_{MHS}^{R,univ} to obtain an object

[ρ]^{∗}X_{MHS}^{R,univ} ∈ Ho(dg_{ℤ}Aff_{BU_{1}^{an}×[𝔸^{1}/𝔾_{m}]×[C^{∗}/S]}(R)↓(Spec 𝒰 ) × [C^{∗}/S]),

which is well-defined since O(X_{MHS}^{R,univ}) is flat and bounded. If [ρ] comes from a Zariski-dense representation ρ, then observe that U_{1}-equivariance amounts to giving a homomorphism α : U_{1}^{an} → R^{C} such that t♣ρ = ad_{α(t)}ρ. This gives a U_{1}-action (and hence an S-action) on R, as in [Pri2] Theorem 4.25. There is then an isomorphism R ⋊ S ≅ R × S, given by (r, s) ↦ (rα(\frac{s̄}{s}), s). We may thus regard X_{MHS}^{ρ,Mal} as an object of

Ho(dg_{ℤ}Aff_{𝔸^{1}×C^{∗}}(𝔾_{m} × R × S)) ≃ Ho(dg_{ℤ}Aff_{[𝔸^{1}/𝔾_{m}]×[C^{∗}/S]}(R))
≃ Ho(dg_{ℤ}Aff_{BU_{1}×[C^{∗}/S]}(R)↓Spec 𝒰 ) × [C^{∗}/S]).

Pulling back along BU_{1}^{an} → BU_{1} (i.e. taking the forgetful functor from algebraic to analytic U_{1}-representations), this recovers [ρ]^{∗}X_{MHS}^{R,univ}.

2. For any map (not necessarily U_{1}-equivariant) [ρ] : Spec ℝ → [𝔐_{H,X,R}/⟨τ⟩] coming from a Zariski-dense representation ρ, the pullback of X_{MHS}^{R,univ} along [ρ] and 1 ∈ C^{∗} gives X^{ρ,Mal}, by Theorem 6.4. The action of U_{1} ⊂ S then gives us an analytic map

X^{ρ,Mal} × U_{1}^{an} → X_{MHS}^{R,univ} ×_{[([𝔐_{H,X,R}/⟨τ⟩]×C^{∗})/ιS]} [𝔐_{H,X,R}/⟨τ⟩] × C^{∗};

this is essentially a generalisation of the analytic U_{1}-action of [Pri2] Proposition 5.14.

We also have the following extension of [Pri2] Corollary 4.11, showing that there is an analytic S-action on \underline{gr}X_{MTS}^{R,univ} (whose induced 𝔾_{m}-action is algebraic), and a splitting of X_{MHS}^{R,univ} over SL_{2}.

Proposition 7.6. There is a canonical object \underline{gr}X_{MHS}^{R,univ} of

Ho(dg_{ℤ}Aff_{[𝔐_{H,X,R}/⟨τ⟩×ιS]}(R)) := Ho(dg_{ℤ}Aff_{[𝔐_{H,X,R}/(⟨τ⟩×U_{1}^{an})]×BS}(R)↓Spec (q^{∗}𝒰 )),

with ε^{∗}\underline{gr}X_{MHS}^{R,univ} = \underline{gr}X_{MTS}^{R,univ}, for ε as in Proposition 7.4.

There is also an isomorphism

row_{1}^{∗}X_{MHS}^{R,univ} ≅ \underline{gr}X_{MHS}^{R,univ} ×_{[𝔐_{H,X,R}/⟨τ⟩×ιS]} [([𝔐_{H,X,R}/⟨τ⟩] × SL_{2})/ιS].

extending the isomorphism of Proposition 6.8.

Proof. To define \underline{gr}X_{MTS}^{R,univ}, adapt the proof of Proposition 7.4, replacing Ã^{•}(X, O) with A^{•}(X, O). The proof of Proposition 6.8 then adapts to give the splitting isomorphism.

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