The structure of the pro-l-unipotent fundamental group of a smooth variety^{∗}

J.P.Pridham

May 5, 2005

Contents

Introduction 1

1 Functors on nilpotent Lie algebras 2 1.1 Differential graded algebras . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 SDCs over N_{k} 10 2.1 Computing SDCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The pro-unipotent fundamental group . . . . . . . . . . . . . . . . . . . 13 2.3 Equivalence between SDCs and DGAs in characteristic 0 . . . . . . . . . 18

3 Structure of the fundamental group 20 3.1 The topological fundamental group . . . . . . . . . . . . . . . . . . . . . 20 3.2 The algebraic fundamental group . . . . . . . . . . . . . . . . . . . . . . 21

Bibliography 26

Introduction

In [Pri04a], I used the theory of SDCs (developed in [Pri04b]) to prove results about the structure of the algebraic fundamental group of a smooth variety, similar to those proved in [GM88] for the structure of the fundamental group of a Kähler manifold. [GM88] was motivated by [DGMS75], which shows that the homotopy type of a compact Kähler manifold is a formal consequence of its cohomology, the idea being to replace statements about DGAs with those about DGLAs.

One consequence of [DGMS75] is that the de Rham fundamental group of a compact Kähler manifold is quadratically presented. In [Mor78], a weaker result is proved for smooth complex varieties. The primary aim of this paper is to prove an analogous result for smooth proper varieties in finite characteristic. The results in [DGMS75] and [Mor78] were inspired by the yoga of weights, motivated by the Weil Conjectures. With this in mind, it is reasonable to expect the characteristic zero results to have finite characteristic analogues.

∗The author is supported by Trinity College, Cambridge and the Isle of Man Department of Education.

In [Del75] is expressed the philosophy that, for algebraic varieties, there should be a weight decomposition not only on cohomology,but also on the minimal model (i.e. the homotopy theory), and that only non-zero weights are interesting. In Weil II ([Del80]), Deligne uses this idea to define the ℚ_{l}-homotopy type of a variety and prove that it is formal for smooth proper algebraic varieties defined over finite fields.

The results of [DGMS75] and [Mor78] give explicit information about the structure of the fundamental group. Both papers make use of the Sullivan minimal model, developed in [Sul77] and shown to determine the de Rham fundamental group π_{1}(X) ⊗ ℝ (and, indeed, the entire Whitehead graded Lie algebra). However, since the proof involves the topological methods of [Sul77], no such information has been obtained from the formality result in Weil II. By developing a theory of deformations over nilpotent Lie algebras in Section 1, I obtain similar explicit information on the structure of the algebraic fundamental group. The Weil Conjectures imply that, for X smooth and proper, the pro-nilpotent fundamental group π_{1}(X, x̄) ⊗_{ℤ_{l}} ℚ_{l} is quadratically presented. If X is merely smooth, then the pro-nilpotent fundamental group is defined by equations of bracket length at most four. The Hard Lefschetz Theorem implies that, for X smooth and projective, the pro-nilpotent fundamental group cannot be free.

1 Functors on nilpotent Lie algebras

This section extends the ideas of [Sch68] to a slightly different context.

Fix a field k, and define N_{k} to be the category of finite-dimensional nilpotent Lie algebras over k, and N̂_{k} to be the category of finitely generated (i.e. dim L/[L, L] < ∞) pro-nilpotent Lie algebras over k.

Given L ∈ N̂_{k}, let N_{L,k} be the category of pairs (N ∈ N_{k}, L ⟶^{ϕ} N), and { \atop N_{L,k}} the

category of pairs (N ∈ N̂_{k}, L ⟶^{ϕ} N). We will almost always consider the case L = 0 (note that N_{0,k} = N_{k}), although a few technical lemmas (Theorems 1.18 and 1.19 and Corollary 1.20) require the full generality of N_{L,k}. In { \atop N_{L,k}}, L is the initial object, and 0 the final object.

Definition 1.1. Given a finite dimensional vector space V/k, denote the free pronilpotent Lie algebra on generators V by L(V). Explicitly, let A(V) be the free complete associative algebra on generators V,

A(V) := lim_{\overleftarrow{n}}(⨁_{m} V^{⊗m})/(V)^{n},

(the universal enveloping algebra of L(V)), with coproduct Δ : A(V) → A(V) ˆ⊗A(V) defined on generators by Δ(v) = v ⊗ 1 + 1 ⊗ v. Then

L(V) := {l ∈ A(V) : Δ(l) = l ⊗ 1 + 1 ⊗ l}

Definition 1.2. Given Lie algebras N, M ∈ N̂_{k}, let N ∗ M be the completed free Lie algebra product, i.e. the completion with respect to the commutator filtration of the free product of the Lie algebras N and M. Note that ∗ is sum in N̂_{k} — the analogue in Ĉ_{Λ} is ˆ⊗.

Definition 1.3. Define the free Lie algebra L_{L}(V) := L ∗ L(V)

We require that all functors on N_{L,k} satisfy (H0) F (0) = •, the one-point set.

We adapt the following definitions and results from [Sch68] (with identical proofs): Definition 1.4. For p : N → M in N_{L,k} surjective, p is a small extension if ker p = (t), a principal ideal, such that [N, (t)] = (0). To say that (t) is a principal ideal means that the map

L(k) → (t)
1 ↦ t

is surjective. Note that any surjection can be be factorised as a composition of small extensions.

For F : N_{L,k} → Set, define F̂ : { \atop N_{L,k}} → Set by

F̂(L) = lim_{\overleftarrow{n}} F (L/Γ_{n}(L)),

where the Γ_{n}(L) are defined inductively by

Γ_{1}(L) = L, Γ_{n+1}(L) = [L, Γ_{n}(L)].

Note that F̂(L) ⟶^{∼} Hom(h_{L}, F), where

h_{L} : N_{L,k} → Set;
N ↦ Hom(L, N).

Definition 1.5. We will say a functor F : N_{L,k} → Set is pro-representable if it is isomorphic to h_{L}, for some L ∈ { \atop N_{L,k}}. By the above remark, this isomorphism is determined by an element ξ ∈ F̂(L). We say the pro-couple (L, ξ) pro-represents F. Definition 1.6. A natural transformation ϕ : F → G in [N_{L,k}, Set] is called:

1. unramified if ϕ : t_{F} → t_{G} is injective, where t_{F} = F (L(ϵ)), where

L(ϵ) := (L(ϵ), L ⟶^{0} L(ϵ)).

2. smooth if for every M ↠ N in N_{L,k}, we have F (M) ↠ G(M) ×_{G(N)} F (N)

3. étale if it is smooth and unramified. Definition 1.7. F : N_{L,k} → Set is smooth if and only if F → • is smooth. Theorem 1.8. 1. Let A → B be a morphism in N̂_{L,k}. Then h_{B} → h_{A} is smooth if

and only if B = A ∗ L(X_{1}, …, X_{n}).

2. If F → G and G → H are smooth morphisms of functors, then the composition F → H is smooth.

3. If u : F → G and v : G → H are morphisms of functors such that u is surjective

and vu is smooth, then v is smooth. Definition 1.9. A pro-couple (L, ξ) is a hull for F if the induced map h_{L} → F is étale. Theorem 1.10. Let (L, ξ), (L′, ξ′) be hull of F. Then there exists an isomorphism u : L → L′ such that F (u)(ξ) = ξ′. Lemma 1.11. Suppose F is a functor such that

F (V ⊕ W) ⟶^{∼} F (V) × F (W)

for vector spaces V and W over k, where V ∈ N_{L,k} is defined by [V, V] = 0, L ⟶^{0} V. Then F (V), and in particular t_{F}, has a canonical vector space structure, and F (V) ≅ t_{F} ⊗ V. Theorem 1.12. Given F : N_{L,k} → Set, let N′ → N and N′′ → N be morphisms in N_{L,k}, and consider the map: (†) F (N′ ×_{N} N′′) → F (N′) ×_{F (N)} F (N′′). Then

1. F has a hull if and only if F has properties (H1), (H2) and (H3) below: (H1) (†) is a surjection whenever N′′ → N is a small extension. (H2) (†) is a bijection when N = 0, N′′ = L(ϵ). (H3) dim_{k}(t_{F}) < ∞.

2. F is pro-representable if and only if F has the additional property (H4):

(H4) F (N′ ×_{N} N′′) ⟶^{∼} F (N′) ×_{F (N)} F (N′′).

for any small extension N′ → N. Proof. [Sch68], Theorem 2.11. Definition 1.13. F : N_{L,k} → Set is homogeneous if

η : F (N′ ×_{N} N′′) → F (N′) ×_{F (N)} F (N′′)

is an isomorphism for every N′ ↠ N.

Note that a homogeneous functor satisfies conditions (H1), (H2) and (H4). Definition 1.14. F : N_{L,k} → Set is a deformation functor if:

1. η is surjective whenever N′ ↠ N.

2. η is an isomorphism whenever N = 0.

Note that a deformation functor satisfies conditions (H1) and (H2).

The following results are adapted from[Man99]: Definition 1.15. Given F : N_{L,k} → Set, an obstruction theory (V, v_{e}) for F consists of: a k-vector space V, the obstruction space, together with obstruction maps v_{e} : F (N) → V ⊗_{k} I for each small extension

e : 0 → I → L → N → 0,

such that:

1. If ξ ∈ F (N) can be lifted to F (L) then v_{e}(ξ) = 0.

2. For every morphism α : e → e′ of small extensions, we have v_{e′}(α(ξ)) = (id_{V} ⊗ α)(v_{e}(ξ)), for all ξ ∈ F (N).

An obstruction theory (V, v_{e}) is called complete if ξ ∈ F (N) can be lifted to F (L) whenever v_{e}(ξ) = 0. Theorem 1.16. (Standard Smoothness Criterion) Given ϕ : F → G, with (V, v_{e}) ⟶^{ϕ′} (W, w_{e}) a compatible morphism of obstruction theories, if (V, v_{e}) is complete, V ⟶^{ϕ′} W injective, and t_{F} → t_{G} surjective, then ϕ is smooth. Proof. [Man99], Proposition 2.17.

For functors F : N_{L,k} → Set and G : N_{L,k} → Grp, we say that G acts on F if we have a functorial group action G(N) × F (N) ⟶^{∗} F (N), for each N in N_{k}. The quotient functor F/G is defined by (F/G)(N) = F (N)/G(N). Theorem 1.17. If F : N_{L,k} → Set, a deformation functor, and G : N_{L,k} → Grp a smooth deformation functor, with G acting on F, then D := F/G is a deformation functor, and if ν : t_{G} → t_{F} denotes h ↦ h ∗ 0, then t_{D} = coker ν, and the universal obstruction theories of D and F are isomorphic. Proof. [Man99], Lemma 2.20. Theorem 1.18. For F : N_{L,k} → Set homogeneous, and G : N_{L,k} → Grp a deformation functor, given a, b ∈ F (L), define Iso(a, b) : N_{L,k} → Set by

Iso(a, b)(L ⟶^{f} N) = {g ∈ G(N)|g ∗ f(a) = f(b)}.

Then Iso(a, b) is a deformation functor, with tangent space ker ν and, if G is moreover smooth, complete obstruction space coker ν = t_{D}. Proof. [Man99], Proposition 2.21. Theorem 1.19. If G, G′ smooth deformation functors, acting on F, F′ respectively, with F, F′ homogeneous, ker ν → ker ν′ surjective, and coker ν → coker ν′ injective, then F/G → F′/G′ is injective.

Proof. [Man99], Corollary 2.22.

This final result does not appear in [Man99], but proves extremely useful: Corollary 1.20. If F : N_{L,k} → Set and G : N_{L,k} → Grp are deformation functors, with G acting on F, let D := F/G, then:

1. If G is smooth, then η_{D} is surjective for every M ↠ N (i.e. D is a deformation functor).

2. If F is homogeneous and ker ν = 0, then η_{D} is injective for every M ↠ N. Thus, in particular, F/G will be homogeneous if F is homogeneous, G is a smooth deformation functor and ker ν = 0. Proof. As for [Pri04b] Theorem 0.15.

To summarise the results concerning the pro-representability of the quotient D = F/G, we have:

1. If F is a deformation functor and G a smooth deformation functor, with coker ν finite dimensional, then D has a hull.

2. If F is homogeneous and G a smooth deformation functor, with ker ν = 0 and

coker ν finite dimensional, then D is pro-representable. Example 1.21. There is one main class of examples of deformation functors on N_{k}. Let k be of characteristic zero. Then it is well-known (see e.g. [ABC^{+}96]) that there is an equivalence of categories

N_{k} ⟶^{exp} Uni_{k},

where Uni_{k} is the category of unipotent k-groups. Fix a finitely presented group G, and consider the functor

h_{G} : Uni_{k} → Set
U ↦ Hom(G, U).

This is equivalent to the functor

F : N_{k} → Set
N ↦ Hom(G, exp(N)).

In fact, F is pro-representable by a pro-nilpotent Lie algebra L, and by definition G ⊗_{ℤ} k := exp(L).

1.1 Differential graded algebras

Definition 1.22. A DGA over a field k of characteristic 0 is a graded vector space A = ⨁_{i∈ℤ} A^{i} over k, equipped with operators · : A × A → A bilinear and d : A → A linear, satisfying:

1. A^{i} · L^{j} ⊂ A^{i+j}.

2. a · b = (−1)^{āb̄}b · a.

3. (a · b) · c = ·(b · c).

4. d(A^{i}) ⊂ A^{i+1}.

5. d ∘ d = 0.

6. d(a · b) = da · b + (−1)āa · db

Here ā denotes the degree of a, mod 2, for a homogeneous. Note that our DGAs need not have a unit.

Fix a DGA A.

Definition 1.23. The Maurer-Cartan functor MC_{A} : N_{k} → Set is defined by

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

Observe that for ω ∈ A^{1} ⊗ N,

dω + \frac{1}{2}[ω, ω] = 0 ⇒ (d + ad_{ω}) ∘ (d + ad_{ω}) = 0,

so (A ⊗ N, [,], d + ad_{ω}) is a DGLA.

Definition 1.24. Define the gauge functor G_{A} : N_{k} → Grp by

G_{A}(N) = exp(A^{0} ⊗ N).

We may now define the DGLA (A ⊗ N)_{d} as in [Man99]:

(A ⊗ N)_{d}^{i} = \begin{cases}(A ⊗ N)^{1} ⊕ kd & i = 1 \\ (A ⊗ N)^{i} & i ≠ 1, \end{cases}

with

d_{d}(d) = 0, [d, d] = 0, [d, a]_{d} = da, ∀a ∈ (A ⊗ N).

Lemma 1.25. exp(A^{0} ⊗N) commutes with [,] when acting on (A⊗N)_{d} via the adjoint action.

Corollary 1.26. Since exp(A^{0} ⊗ N) preserves (A^{1} ⊗ N) + d ⊂ (A ⊗ N)_{d} under the adjoint action, and

x ∈ MC_{A}(N) ⇔ [x + d, x + d] = 0,

the adjoint action of exp(A^{0} ⊗ N) on A^{1} ⊗ N + d induces an action of G_{A}(N) on MC_{A}(N), which we will call the gauge action.

Definition 1.27. Def_{A} = MC_{A}/G_{A}, the quotient being given by the gauge action α(x) = ad_{α}(x + d) − d. Observe that G_{A} is homogeneous.

For a ∈ MC_{A}(N), define K_{a} : N_{N,k} → Grp by

K_{a}(M) = exp((d + ad_{a})A^{−1} ⊗ M).

Note that this makes sense, since (d + ad_{a})^{2} = 0, so

(d + ad_{a})A^{−1} ⊗ M ≤ A^{0} ⊗ M

is a Lie subalgebra. Note that K_{a} is then a subfunctor of Iso(a, a) ≤ G_{A}(A), so acts on Iso(a, b) by right multiplication.

Define the deformation groupoid 𝔇𝔢𝔣_{A} to have objects MC_{A}, and morphisms given by Iso(a, b)/K_{a}.

Now, t_{G_{A}} = A^{0}ϵ, and t_{MC_{A}} = Z^{1}(Aϵ), with action

t_{G_{A}} × t_{MC_{A}} → t_{MC_{A}};
(b, x) ↦ x + db, so
t_{Def_{A}} = H^{1}(A).

Lemma 1.28. H^{2}(A) is a complete obstruction space for MC_{A}.

Proof. Given a small extension

e : 0 → K → N → M → 0,

and x ∈ MC_{A}(M), lift x to x̃ ∈ A^{1} ⊗ N, and let

h = dx̃ + \frac{1}{2}[x̃, x̃] ∈ A^{2} ⊗ N.

In fact, h ∈ A^{2} ⊗ K, as dx + \frac{1}{2}[x, x] = 0.

Now,

dh = d^{2}x̃ + [dx̃, x̃] = [h − \frac{1}{2}[x̃, x̃], x̃] = [h, x̃] = 0,

since [[x̃, x̃], x̃] = 0 and [K, N] = 0. Let

v_{e}(x) = [h] ∈ H^{2}(A ⊗ K) = H^{2}(A) ⊗ K.

This is well-defined: if y = x̃ + z, for z ∈ A^{1} ⊗ K, then

dy + \frac{1}{2}[y, y] = dx̃ + dz + \frac{1}{2}[x̃, x̃] + \frac{1}{2}[z, z] + [x̃, z] = h + dz,

as [K, N] = 0.

This construction is clearly functorial, so it follows that (H^{2}(A), v_{e}) is a complete obstruction theory for MC_{A}.

Now Theorem 1.17 implies the following:

Theorem 1.29. Def_{A} is a deformation functor, t_{Def_{A}} ≅ H^{1}(A), and H^{2}(A) is a complete obstruction theory for Def_{A}.

The other theorems can be used to prove: Theorem 1.30. If ϕ : A → B is a morphism of DGAs, and

H^{i}(ϕ) : H^{i}(A) → H^{i}(B)

are the induced maps on cohomology, then:

1. If H^{1}(ϕ) is bijective, and H^{2}(ϕ) injective, then Def_{A} → Def_{B} is étale.

2. If also H^{0}(ϕ) is surjective, then Def_{A} → Def_{B} is an isomorphism.

3. Provided condition 1 holds, 𝔇𝔢𝔣_{A} → 𝔇𝔢𝔣_{B} is an equivalence of functors of

groupoids if and only if H^{0}(ϕ) is an isomorphism. Proof. [Man99], Theorem 3.1, mutatis mutandis. Theorem 1.31. If H^{0}(A) = 0, then Def_{A} is homogeneous. Proof. This is essentially Theorem 1.20, with some allowance made for K_{a}.

Thus, in particular, a quasi-isomorphism of DGAs gives an isomorphism of deformation functors and of deformation groupoids. Example 1.32. Fix a differentiable manifold X. The deformation functor

F : N_{ℝ} → Set
𝔤 ↦ Hom(π_{1}(X, x), exp(𝔤))/ exp(𝔤),

where exp(𝔤) acts by conjugation, is governed by the DGA

A^{•} := Γ(X, 𝒜_{ℝ}^{•}),

the correspondence being as follows. Given ω ∈ MC_{A}, let

𝔹_{ω} := D^{−1}(ω),

where

D : exp(𝔤 ⊗ 𝒜^{0}) → 𝔤 ⊗ 𝒜^{1}
α ↦ dα · α^{−1}.

Then 𝔹_{ω} is a principal exp(𝔤)-sheaf on X. There is a monodromy action of π_{1}(X, x) on the stalk 𝔹_{x}, and this gives us a representation π_{1}(X, x) → G.

That this gives rise to an isomorphism of functors follows from the flabbiness of 𝒜^{•} (compare tangent and obstruction spaces). Of course, Def_{A} is not pro-representable, but its hull is π_{1}(X, x) ⊗ ℝ.

2 SDCs over N_{k}

The formal definitions, properties and constructions of SDCs over N_{k} hold in exactly the same manner as those defined in [Pri04b] over C_{Λ}. A summary follows. Definition 2.1. A simplicial deformation complex S^{•} consists of smooth homogeneous functors S^{n} : N_{k} → Set for each n ≥ 0, together with maps

\begin{matrix}∂^{i} : S^{n} → S^{n+1} & 1 ≤ i ≤ n \\ σ^{i} : S^{n} → S^{n−1} & 0 ≤ i < n \end{matrix},

an associative product ∗ : S^{m} × S^{n} → S^{m+n}, with identity 1 : • → S^{0}, where • is the constant functor •(𝔤) = • on N_{k}, such that:

1. ∂^{j}∂^{i} = ∂^{i}∂^{j−1} i < j.

2. σ^{j}σ^{i} = σ^{i}σ^{j+1} i ≤ j.

3. σ^{j}∂^{i} = \begin{cases}∂^{i}σ^{j−1} & i < j \\ id & i = j, i = j + 1 \\ ∂^{i−1}σ^{j} & i > j + 1 \end{cases} .

4. ∂^{i}(s) ∗ t = ∂^{i}(s ∗ t).

5. s ∗ ∂^{i}(t) = ∂^{i+m}(s ∗ t), for s ∈ S^{m}.

6. σ^{i}(s) ∗ t = σ^{i}(s ∗ t).

7. s ∗ σ^{i}(t) = σ^{i+m}(s ∗ t), for s ∈ S^{m}. Remark 2.2. If we set ω_{0} to be the unique element of S^{1}(0), then, since 0 is the initial object in N_{k}, we may set ∂^{0}(s) = ω_{0} ∗ s, and ∂^{n+1}(s) = s ∗ ω_{0}. S^{•} then becomes a cosimplicial complex. Definition 2.3. Let t_{S}^{•} be the tangent space of S^{•}, i.e. t_{S}^{n} = S^{n}(L(ϵ)). Definition 2.4. Define the Maurer-Cartan functor MC_{S} by

MC_{S}(𝔤) = {ω ∈ S^{1}(𝔤): ω ∗ ω = ∂^{1}(ω)}.

Lemma 2.5. S^{0} is a group under multiplication.

Now, if ω ∈ MC_{S}(𝔤) and g ∈ S^{0}(𝔤), then g ∗ ω ∗ g^{−1} ∈ MC_{S}(𝔤). We may therefore make the following definition: Definition 2.6.

Def_{S} = MC_{S}/S^{0},

the quotient being with respect to the adjoint action. The deformation groupoid

𝔇𝔢𝔣_{S}

has objects MC_{S}, and morphisms given by S^{0}.

Lemma 2.7. The action S^{0} × S^{n} → S^{n} is faithful (i.e. s ∗ h = t for some t only if s = 1).

This implies that, for all ω ∈ MC_{S}(𝔤), σ^{0}(ω) = 1. Definition 2.8. Define the cohomology groups of S to be

H^{i}(S) := H^{i}(t_{S}^{•}),

the cohomology groups of the cosimplicial complex t_{S}^{•}. Lemma 2.9. The tangent space of MC_{S} is Z^{1}(t_{S}^{•}), with the action of S^{0} giving ν(s) = ∂^{1}(s) − ∂^{0}(s). Lemma 2.10. H^{2}(S) is a complete obstruction space for MC_{S}.

Theorems 1.16 to 1.20 now imply: Theorem 2.11. Def_{S} is a deformation functor, with tangent space H^{1}(S) and complete obstruction space H^{2}(S). Moreover, if H^{0}(S) = 0, then Def_{S} is homogeneous. Proof. Theorem 1.17 and Corollary 1.20. Theorem 2.12. If ϕ : S → T is a morphism of SDCs, and

H^{i}(ϕ) : H^{i}(S) → H^{i}(T)

are the induced maps on cohomology, then:

1. If H^{1}(ϕ) is bijective, and H^{2}(ϕ) injective, then Def_{S} → Def_{T} is étale.

2. If also H^{0}(ϕ) is surjective, then Def_{S} → Def_{T} is an isomorphism.

3. Provided condition 1 holds, 𝔇𝔢𝔣_{S} → 𝔇𝔢𝔣_{T} is an equivalence of functors of groupoids if and only if H^{0}(ϕ) is an isomorphism.

Call a morphism ϕ : S → T a quasi-isomorphism if the H^{i}(ϕ) : H^{i}(S) → H^{i}(T) are all isomorphisms. Definition 2.13. Given a morphism ϕ : S → T of SDCs, define the groupoid

𝔇𝔢𝔣_{ϕ}

to be the fibre of the morphism

𝔇𝔢𝔣_{S} → 𝔇𝔢𝔣_{T}

over the unique point x_{0} ∈ MC_{T} (0).

Explicitly, 𝔇𝔢𝔣_{ϕ}(𝔤) has objects

{(ω, h) ∈ MC_{S}(𝔤) × T^{0}(𝔤) : hϕ(ω)h^{−1} = x_{0}},

and morphisms

S^{0}(𝔤), where g(ω, h) = (gωg^{−1}, hϕ(g)^{−1}).

Theorem 2.14. Let S_{ϕ}^{n}(A) be the fibre of S^{n}(A) ⟶^{ϕ} T^{n}(A) over x_{0}^{n}. Then S_{x}^{•} is an SDC, and the canonical map

𝔇𝔢𝔣_{S_{ϕ}} → 𝔇𝔢𝔣_{ϕ}

is an equivalence of groupoids.

2.1 Computing SDCs

Throughout this section, we will consider functors D : N_{k} → Cat. We will not require that these functors satisfy (H0) (the condition that F (0) = •).

Definition 2.15. Given a functor D : N_{k} → Cat, and an object D ∈ Ob D(0), define 𝔇𝔢𝔣_{D,D} : N_{k} → Grpd by setting 𝔇𝔢𝔣_{D,D}(𝔤) to be the fibre of D(𝔤) → D(0) over (D, id).

Definition 2.16. We say a functor E : N_{k} → Cat has uniformly trivial deformation theory if the functor Mor E is smooth and homogeneous, and the functor Cmpts E is constant, i.e for 𝔤 ↠ 𝔥 in N_{k}, E(𝔤) → E(𝔥) is full and essentially surjective.

Assume that we are given functors A, B, D, E : N_{k} → Cat, and a diagram

\xymatrix{D \ar[r]^{U}_{⊤} \ar[d]_{V} & A \ar[l]^{F} \ar[d]_{V} \\ B \ar[u]_{G}^{⊣} \ar[r]^{U}_{⊤} & E, \ar[l]^{F} \ar[u]_{G}^{⊣}}

where E has uniformly trivial deformation theory, the horizontal adjunctions are monadic and the vertical adjunctions comonadic. Let

⊤_{h} = UF  ⊥_{h} = F U
⊥_{v} = V G  ⊤_{v} = GV,

with

η : 1 → ⊤_{h}, γ : ⊥_{v} → 1, ε : ⊥_{h} → 1 and α : 1 → ⊤_{v}.

Assume moreover that the following identities hold:

GU = UG
F V = V F
UV = V U
V ε = εV  Uα = αU
V η = ηV  Uγ = γU.

Theorem 2.17. For D ∈ Ob D(0), let E = UV D ∈ Ob E(0), and write E(𝔤) for the image of E under the map Ob E(0) → Ob E(𝔤). Set S^{n}(𝔤) to be the fibre

S^{n}(𝔤) = Hom_{E(𝔤)}(⊤_{h}^{n}E(𝔤), ⊥_{v}^{n}E(𝔤))_{UV (α_{D}^{n}∘ε_{D}^{n})}

of

Hom_{E(𝔤)}(⊤_{h}^{n}E(𝔤), ⊥_{v}^{n}E(𝔤)) → Hom_{E(0)}(⊤_{h}^{n}E, ⊥_{v}^{n}E)

over UV (α_{D}^{n} ∘ ε_{D}^{n}).

We give S^{n} the structure of an SDC as in [Pri04b] Theorem 2.5.

Then

𝔇𝔢𝔣_{D,D} ≃ 𝔇𝔢𝔣_{S}.

Proof. As for [Pri04b] Theorem 2.5.

2.2 The pro-unipotent fundamental group

Definition 2.18. Take a pro-finite group Γ. Consider the functor on N_{ℚ_{l}} given by

𝔤 ↦ Hom(Γ, exp(𝔤)).

This has tangent space H^{1}(Γ, ℚ_{l}), and universal obstruction space H^{2}(Γ, ℚ_{l}). If dim H^{1}(Γ, ℚ_{l}) < ∞, it is pro-representable, by L ∈ N̂_{ℚ_{l}} say. Define the pro-ℚ_{l}-unipotent completion of Γ by

Γ ⊗ℤ̂ ℚ_{l} := exp(L(Γ, ℚ_{l})),

and the ℚ_{l} Malcev Lie algebra by

L(Γ, ℚ_{l}) := L.

Remark 2.19. It follows that L is a hull for the quotient functor

𝔤 ↦ Hom(Γ, exp(𝔤))/ exp(𝔤).

Fix a connected scheme X and a geometric point x̄ → X. We wish to study the structure of the group

π_{1}(X, x̄) ⊗ ℚ_{l}.

Consider the functor

Hom(π_{1}(X, x̄), −) : N_{ℚ_{l}} → Grpd,

which associates to 𝔤 ∈ N_{ℚ_{l}} the discrete groupoid

Hom(π_{1}(X, x̄), exp(𝔤)).

Recall the following definition from [Pri04a] Section 2.2:

Definition 2.20. Given a pro-l group K, define a constructible principal K-sheaf to be a principal K-sheaf 𝔻, such that

𝔻 = lim_{\overleftarrow{K→F finite}} F ×^{K} 𝔻.

Given an l-adic Lie group G, a constructible principal G-sheaf is a G-sheaf 𝔹 for which there exists a constructible principal K-sheaf 𝔻, for some K ≤ G compact, with 𝔹 = G ×^{K} 𝔻 (observe that compact and totally disconnected is equivalent to pro-finite).

Definition 2.21. Define the functor

𝔓𝔅(X) : N_{ℚ_{l}} → Grpd

to associate to each 𝔤 ∈ N_{ℚ_{l}}, the groupoid of principal constructible exp(𝔤)-sheaves on X.

Define the groupoid

𝔓𝔅x̄(X)(𝔤)

to be the fibre of

𝔓𝔅(X)(𝔤) → 𝔓𝔅(x̄)(𝔤)

over (exp(𝔤), id).

Alternatively, we may think of 𝔓𝔅x̄(X)(𝔤) as consisting of pairs (𝔹, θ), for

𝔹 ∈ 𝔓𝔅(X)(𝔤),

and θ an isomorphism

θ : 𝔹x̄ → exp(𝔤).

Thus the crucial difference between the groupoids 𝔓𝔅x̄(X)(𝔤) and 𝔓𝔅(X)(𝔤) is that only those morphisms which give the identity on 𝔹x̄ are permitted in the former.

Theorem 2.22. As in [Pri04b] Section 4.2, we have a functorial equivalence of groupoids

Hom(π_{1}(X, x̄), exp(𝔤))/ exp(𝔤) → 𝔓𝔅(X)(𝔤).

In fact, the same construction gives us an equivalence

Hom(π_{1}(X, x̄), exp(𝔤)) → 𝔓𝔅x̄(X)(𝔤).

Remark 2.23. If we have an automorphism σ of the pair x̄ → X, it yields automorphisms of both the groupoids above. The construction of the equivalence ensures that it is σequivariant.

Definition 2.24. Let

𝔉𝔅(X) : N_{ℚ_{l}} → Cat

associate to 𝔤 the category of faithful constructible exp(𝔤)-sheaves on X. Observe that 𝔓𝔅(X) is the fibre of 𝔉𝔅(X) over the trivial exp(0)-sheaf 1 (the constant sheaf of singleton sets).

As in [Pri04b], we have a functorial comonadic adjunction

\xymatrix{𝔉𝔅(X) \ar[d]_{u^{∗}} \\ 𝔉𝔅(X′), \ar[u]_{u_{∗}}^{⊣}}

where

X′ = ∐_{x∈X} x̄,

where for each x ∈ X, a geometric point x̄ → X has been chosen. We have maps u_{x} : x̄ → X, giving a map u : X′ → X.

If X is a variety, we may instead take

X′ = ∐_{x∈|X|} x̄.

Observe that the second category has uniformly trivial deformation theory, exp(𝔤) being smooth, so that deformations are described by the SDC

S_{X}^{n}(𝔤) = Hom_{𝔉𝔅(X′)(𝔤)}(𝕋(𝔤), (u^{∗}u_{∗})^{n}𝕋(𝔤)),

where 𝕋(𝔤) = exp(𝔤), the trivial exp(𝔤)-sheaf. Note that (u^{∗}u_{∗})^{n}𝕋(0) = 1, and S_{X}^{n}(0) is one point, so we need not worry about taking the fibre over UV (α_{D}^{n} ∘ ε_{D}^{n}) (in the notation of Theorem 2.17).

Now, write

C^{n}(X, ℱ) := Γ(X′, (u^{∗}u_{∗})^{n}u^{∗}ℱ),

for sheaves ℱ on X, and

𝒞^{n}(ℱ) := (u_{∗}u^{∗})^{n+1}ℱ ,

so that C^{n}(X, ℱ) = Γ(X, 𝒞^{n}(ℱ)).

We then have isomorphisms

S^{n}(𝔤) ≅ Γ(X′, (u^{∗}u_{∗})^{n}𝕋(𝔤)) ≅ C^{n}(X, exp(𝔤)),
(b ↦ g · b) ↤ g,

the latter being a cosimplicial complex of groups, becoming an SDC via the Alexander-Whitney cup product

g ∗ h = (∂^{m+n} … ∂^{m+2}∂^{m+1}g) · (∂^{0})^{m}h,

for g ∈ C^{m}, h ∈ C^{n}.

Lemma 2.25. If H^{i}(X, ℤ_{l}) and H^{i−1}(X, ℤ_{l}) are finitely generated, then the cohomology groups of S_{X}^{•} are isomorphic to

H^{∗}(X, ℚ_{l}).

Proof. Observe that the tangent space of S^{•} is

(lim_{\overleftarrow{n}} C^{•}(X, ℤ/l^{n})) ⊗_{ℤ_{l}} ℚ_{l},

where

C^{•}(X, ℤ/l^{n}))

is a complex computing

H_{ét}^{∗}(X, ℤ/l^{n}).

Now,

H^{i}(S_{X}) = H^{i}(lim_{\overleftarrow{n}} C^{•}(X, ℤ/l^{n}) ⊗_{ℤ_{l}} ℚ_{l})
= H^{i}(lim_{\overleftarrow{n}} C^{•}(X, ℤ/l^{n})) ⊗_{ℤ_{l}} ℚ_{l}
= H^{i}(lim_{\overleftarrow{n}} C^{•}(X, ℤ_{l}) ⊗ ℤ/l^{n}) ⊗_{ℤ_{l}} ℚ_{l},

where

C^{•}(X, ℤ_{l}) = lim_{\overleftarrow{n}} C^{•}(X, ℤ/l^{n})),

noting that this is a flat l-adic system.

Now, the tower

⋯ → C^{•}(X, ℤ_{l}) ⊗ ℤ/l^{n+1} → C^{•}(X, ℤ_{l}) ⊗ ℤ/l^{n} → ⋯

clearly satisfies the Mittag-Leffler condition, so, by [Wei94] Theorem 3.5.8, we have the exact sequence:

[figure: 0; \underleftarrow{lim}^{1} H^{i−1}(C^{•}(X, ℤ_{l})_{n}); H^{i}(\underleftarrow{lim}_{n} C^{•}(X, ℤ_{l})_{n}); \underleftarrow{lim}_{n} H^{i}(C^{•}(X, ℤ_{l})_{n}); 0; 0; \underleftarrow{lim}^{1} H^{i−1}(X, ℤ/l^{n}); H^{i}(\underleftarrow{lim}_{n} C^{•}(X, ℤ_{l})_{n}); H^{i}(X, ℤ_{l}); 0.]

From [Mil80] Lemma V 1.11, it follows that the H^{i−1}(X, ℤ/l^{n}) are finitely generated ℤ/l^{n}-modules, so satisfy DCC on submodules, so this inverse system satisfies the Mittag-Leffler condition, making the \underleftarrow{lim}^{1} on the left vanish.

Thus

H^{i}(S) = H^{i}(X, ℤ_{l}) ⊗_{ℤ_{l}} ℚ_{l} = H^{i}(X, ℚ_{l}).

Lemma 2.26. Given a map f : X → Y of schemes, there is a canonical morphism

f^{∗} : S_{X}^{•} → S_{Y}^{•}

of SDCs, inducing the usual map

f^{∗} : H^{i}(Y, ℚ_{l}) → H^{i}(X, ℚ_{l})

on cohomology.

Proof. The map f induces a map f′ : X′ → Y′, making the following diagram commute:

X′ ⟶^{f′} Y′
u↓  ↓^{v}
X ⟶^{f} Y.

We then have

S_{Y}^{n} (𝔤) = Hom_{Y′}(𝔤, (v^{∗}v_{∗})^{n}𝔤) ⟶^{(f′)^{♯}} Hom_{X′}((f′)^{∗}𝔤, (f′)^{∗}(v^{∗}v_{∗})^{n}𝔤).

Now, using the the canonical maps

(f′)^{∗}v∗ ≅ u^{∗}f^{∗}
f^{∗}v_{∗} → u_{∗}(f′)^{∗},

we get a map

Hom_{X′}((f′)^{∗}𝔤, (f′)^{∗}(v^{∗}v_{∗})^{n}𝔤) → Hom_{X′}((f′)^{∗}𝔤, (u^{∗}u_{∗})^{n}(f′)^{∗}𝔤),

but 𝔤 is just the constant sheaf, so there is a canonical isomorphism (f′)^{∗}𝔤 ≅ 𝔤. Putting these maps together gives us the morphism

f^{∗} : S_{X}^{•} → S_{Y}^{•}

of SDCs.

Remark 2.27. As in [Pri04a] Remark 2.14, we may replace ℚ_{l} by any finite extension E, or indeed by ℚ̄_{l}.

In fact, we can go further than this in describing π_{1}(X, x̄) ⊗ ℚ_{l}:

Theorem 2.28. If we consider the morphism

S_{X}^{•} ⟶^{u_{x}^{∗}} S• ; x̄

of SDCs, define the SDC

S_{X,x̄}^{n} ⊂ S_{X}^{n}

to be the fibre of u_{x}^{∗} over the unique 0-valued point of Sn ; x̄. Then

𝔇𝔢𝔣_{S_{X,x̄}}

is equivalent to the discrete groupoid pro-represented by L(π_{1}(X, x̄), ℚ_{l}).

The cohomology of S_{X,x̄}^{•} is

H^{i}(X, x̄; ℚ_{l}) ≅ \begin{cases}0 & i = 0 \\ H^{i}(X, ℚ_{l}) & i ≠ 0, \end{cases}

provided that the H^{i}(X, ℤ_{l}) are finitely generated.

Proof. Recall, from Theorem 2.14, that 𝔇𝔢𝔣_{S_{X,x̄}}(𝔤) is equivalent to 𝔇𝔢𝔣_{u_{x}^{∗}}(𝔤), the fibre of

𝔇𝔢𝔣_{S_{X}} (𝔤) ⟶^{u_{x}^{∗}} 𝔇𝔢𝔣_{Sx̄}(𝔤)

over (a_{0}, id), where a_{0} ∈ S1 ; x̄(0) is the unique 0-valued object. From the above discussion, it follows that this groupoid is equivalent to the fibre of

𝔓𝔅(X)(𝔤) → 𝔓𝔅(x̄)(𝔤)

over (exp(𝔤), id), which is precisely

𝔓𝔅x̄(X)(𝔤).

Hence, by Theorem 2.22, we have a canonical equivalence

Hom(π_{1}(X, x̄), 𝔤) → 𝔇𝔢𝔣_{S_{X,x̄}}(𝔤).

The statement about cohomology follows from Theorem 2.14.

Remark 2.29. Observe that, from Lemma 2.26, an automorphism σ of the pair x̄ ⟶^{u_{x}} X will naturally give rise to an automorphism of

S_{X}^{•} ⟶^{u_{x}^{∗}} S• ; x̄,

and hence of 𝔇𝔢𝔣_{u_{x}^{∗}}(𝔤). It is clear that the equivalence in the theorem will be σequivariant.

Remark 2.30. Tangential base points.

If X = Y − Z, for Y smooth and Z a closed subscheme, there is a definition of the fundamental group π_{1}(X, v̄), where v is a non-zero tangent vector at a point on the boundary of Z. This makes use of a morphism of sites:

Ét(X) → Ét(v̄).

In order to define an SDC governing this fundamental group, we want to replace X′ by X′ ⊔ v̄. Instead, we make use of the morphism

Ét(X) → Ét(X′) × Ét(v̄) = Ét(v̄) × ∏_{x∈X} Ét(x̄),

which is sufficient. In fact, working with the sites is implicitly what we were doing before, since infinite coproducts are not defined in the category of schemes.

We use X′ ⊔ v̄ to define S_{X}^{•}, and the morphism v̄ → X′ ⊔ v̄ enables the construction of a map S_{X}^{•} → S_{v}^{•}, which is what we require.

2.3 Equivalence between SDCs and DGAs in characteristic 0

The reasoning of [Pri04b] Section 5 carries over directly to this context. We define the category N_{k}^{ℕ_{0}} in the obvious manner. The only result we need to check is the analogue of [Pri04b] Lemma 5.6:

Lemma 2.31. Given a smooth homogeneous functor G : N_{k}^{ℕ_{0}} → Grp, the tangent space A^{∗} has the structure of a GA (graded-commutative associative algebra without unit). Moreover, there is a canonical isomorphism (functorial in 𝔤_{∗} ∈ N_{k}^{ℕ_{0}}):

exp(⨁_{n} A^{n} ⊗ 𝔤_{n}) ≅ G(𝔤_{∗}).

Proof. Since G is smooth and homogeneous, it will be “pro-represented” by a “pro-Artinian” topological free graded Lie algebra L_{∗}. The product on G gives rise to a co-associative coproduct

ρ : L_{∗} → L_{∗} ∗ L_{∗},

for which L_{∗} → 0 is the co-identity (existence of a co-inverse is an automatic consequence of pro-nilpotence).

G(𝔤_{∗}) = Hom_{Lie}(L^{∗}, 𝔤^{∗})
⨁_{n} A^{n} ⊗ 𝔤_{n} = Der(L^{∗}, 𝔤^{∗}).

Hence both embed into Hom_{Vect}(L_{∗}, 𝔤_{∗}). Observe the isomorphism

Hom_{Vect}(L_{∗}, 𝔤_{∗}) ≅ End_{𝔤_{∗}−Mod}^{ρ}(L_{∗} ∗ 𝔤_{∗}),
f ↦ L_{∗} ∗ 𝔤_{∗} ⟶^{ρ∗1_{𝔤∗}} L_{∗} ∗ L_{∗} ∗ 𝔤_{∗} ⟶^{1_{L∗}∗[f,1_{𝔤∗}]} L_{∗} ∗ 𝔤_{∗},
L_{∗} ↪ L_{∗} ∗ 𝔤_{∗} ⟶^{α} L_{∗} ∗ 𝔤_{∗} ⟶^{0∗1_{𝔤∗}} 𝔤_{∗} ↤ α,

where the latter set is defined to consist of those α ∈ End_{𝔤_{∗}−Mod}(L_{∗} ∗ 𝔤_{∗}) for which the following diagram commutes:

L_{∗} ∗ 𝔤_{∗}  ⟶^{α} L_{∗} ∗ 𝔤_{∗}
ρ∗𝔤_{∗}↓  ^{ρ∗𝔤_{∗}}↓
L_{∗} ∗ L_{∗} ∗ 𝔤_{∗} ⟶^{L_{∗}∗α} L_{∗} ∗ L_{∗} ∗ 𝔤_{∗}.

This is an associative algebra, with product given by composition. Then G(𝔤_{∗}) is the subset consisting of Lie algebra homomorphisms, composition corresponding to the original product. ⨁_{n} A^{n} ⊗ 𝔤_{n} is the subset of derivations, with Lie bracket given by the usual formula [α, β] = α ∘ β ∓ β ∘ α.

Consider the maps

log(g) = ∑_{n≥1}(−1)^{n−1} \frac{(g − id)^{∘n}}{n}

exp(α) = ∑_{n≥0} \frac{α^{∘n}}{n!}

on End_{𝔤_{∗}−Mod}^{ρ}(L_{∗} ∗ 𝔤_{∗}). exp takes derivations to homomorphisms, and log is its inverse, taking homomorphisms to derivations.

Hence, for α ∈ Der(L^{∗}, 𝔤^{∗}) = ⨁_{n} A^{n} ⊗ 𝔤_{n},

exp(α) ∈ G(𝔤_{∗}).

Similarly, for g ∈ Hom_{Lie}(L^{∗}, 𝔤^{∗}) = G(𝔤_{∗}),

log(g) ∈ ⨁_{n} A^{n} ⊗ 𝔤_{n}.

It remains only to show that A^{∗} is a graded-commutative algebra. This follows from the functorial (in 𝔤_{∗}) isomorphism

⨁_{n} A^{n} ⊗ 𝔤_{n} = Der(L^{∗}, 𝔤^{∗}),

the right hand object being a Lie algebra, since it is log of a group.

As in [Pri04b] Section 5, we then obtain functors

\xymatrix{DGA \ar[r]^{S} & SDC \ar[l]^{D}}

with functorial quasi-isomorphisms 1 → SD, 1 → DS inducing an equivalence of the localised categories.

3 Structure of the fundamental group

3.1 The topological fundamental group

Fix a connected differentiable manifold X, and a point x ∈ X. Let 𝒜^{n} be the sheaf of real-valued C^{∞} n-forms on X.

As in Section 2.2, the functor

ℜ(π_{1}(X, x)) : N_{ℝ} → Grpd,

which associates to 𝔤 ∈ N_{ℝ} the groupoid with objects

Hom(π_{1}(X, x), exp(𝔤)),

and morphisms exp(𝔤) (acting by conjugation), is governed by the SDC

S^{n}(𝔤) = C^{n}(X, exp(𝔤)) = C^{n}(X, S^{n}(ℝ))(𝔤).

As in [Pri04a] Section 1, we have the following quasi-isomorphisms of SDCs (over N_{ℝ}):

C^{n}(X, S^{n}(ℝ))
↓
C^{n}(X, S(𝒜^{•})^{n})
↑
S(Γ(X, 𝒜^{•}))^{n},

since on cohomology we have:

H^{n}(X, ℝ)
↓
ℍ^{n}(X, 𝒜^{•})
↑
H^{n}(Γ(X, 𝒜^{•})),

the first quasi-isomorphism arising because ℝ → 𝒜^{•} is a resolution, and the second from the flabbiness of 𝒜^{•}.

Hence ℜ(π_{1}(X, x)) is governed by the DGA

A^{•} := Γ(X, 𝒜^{•}),

whose quasi-isomorphism class, also known as the real homotopy type of X, is an invariant which has been studied in great detail.

In [Sul77], the minimal model of a connected DGA is defined, from which the de Rham fundamental group π_{1}(X) ⊗ ℝ is recovered as the dual Lie algebra. We can, however, recover the de Rham fundamental group as the hull of the functor Def_{A}, so do not have to involve minimal models.

We can now adapt the results of [Pri04a] Section 1 to this context. In fact the proofs are slightly simpler, since we are working with the constant sheaf ℝ, rather than the twisted sheaf ad𝔹_{0}. [Pri04a] Section 1.1 then becomes the main result of [DGMS75],

[Pri04a] Section 1.2 becomes the main result of [Mor78], and the only new result is from adapting [Pri04a] Section 1.3:

Theorem 3.1. Let X be a complex variety. Then there is a mixed Hodge structure (hence a weight decomposition) on the rational minimal model for X. In particular, the Malcev Lie algebra

L(π_{1}(X) ⊗ ℝ)) ≅ ℝ[[H^{1}(X, ℝ)^{∨}]]/(f(H^{2}(X, ℝ)^{∨})),

where f preserves the mixed Hodge structures.

Proof. Identical to [Pri04a] Theorem 1.7.

This answers a question posed in [ABC^{+}96] as to whether Hodge theory can provide any information about the homotopy type (and hence the structure of the fundamental group) for singular varieties. Since zero weights are permitted in the cohomology of arbitrary varieties, we can, in general, say nothing about the function f, vindicating the observation in [Del75] that zero weights are unremarkable. However, if in a particular case H^{1} is known to have no component of weight zero, then conclusions can be drawn.

3.2 The algebraic fundamental group

Let k = 𝔽_{q}, take a connected variety X_{k}/k, and let X = X_{k} ⊗_{k} k̄. Fix a closed point x of X, and denote the associated geometric point x → X by x̄. Without loss of generality (increasing q if necessary), we assume that k(x) ⊂ 𝔽_{q}.

From Section 2.2, it follows that the Malcev Lie algebra (see Definition 2.18)

L(π_{1}(X, x̄), ℚ_{l})

pro-represents the groupoid 𝔇𝔢𝔣_{S_{X,x̄}}, and hence the functor Def_{S_{X,x̄}}.

Observe that, since k(x) ⊂ 𝔽_{q}, there is a natural action of Gal(𝔽̄_{q}/𝔽_{q}) on π_{1}(X, x̄). Let F be the geometric Frobenius element. It follows from the remarks in Section 2.2 that the action of F on L(π_{1}(X, x̄), ℚ_{l}) will correspond to the action of F^{∗} on the SDC above.

For X either smooth or proper, [Del80] shows that all the eigenvalues of Frobenius acting on the cohomology group H^{i}(X, ℚ_{l}) are algebraic numbers α, and for each α, there exists a weight n, such that all complex conjugates of α have norm q^{n/2}. This provides us with a weight decomposition

H^{i}(X, ℚ_{l}) = ⨁_{n} W_{n}H^{i}(X, ℚ_{l}).

Theorem 3.2. There is an isomorphism

L(π_{1}(X, x̄), ℚ_{l}) ≅ L(H^{1}(X, ℚ_{l})^{∨})/(f(H^{2}(X, ℚ_{l})^{∨})),

where L(V) is the free pro-nilpotent Lie algebra (as in 1.1), and

f : H^{2}(X, ℚ_{l})^{∨} → Γ_{2}L(H^{1}(X, ℚ_{l})^{∨})

preserves the (Frobenius) weight decompositions of [Del80] 3.3.7.

Moreover, the quotient map

f : H^{2}(X, ℚ_{l})^{∨} → Γ_{2}/Γ_{3} ≅ ⋀^{2}(H^{1}(X, ℚ_{l})^{∨})

is dual to half the cup product

H^{1}(X, ℚ_{l}) × H^{1}(X, ℚ_{l}) ⟶^{\frac{1}{2} ∪} H^{2}(X, ℚ_{l}).

Proof. This will essentially be the same as [Pri04a] Theorem 2.11. From the remarks in Section 2.2, it follows that the isomorphism

Hom(Lπ_{1}(X, x̄), 𝔤) ≅ Def_{S_{X,x̄}}(𝔤)

is Frobenius equivariant. Write

L := Lπ_{1}(X, x̄).

Since L/Γ_{n}(L) is a finite dimensional vector space over ℚ_{l} for all n, we may use the Jordan decomposition (over ℚ̄_{l}) successively to lift the cotangent space L/Γ_{2}(L) to L/Γ_{n}(L), obtaining a space of generators V ⊂ L, with the map V → L/Γ_{2}(L) an isomorphism preserving the weight decompositions.

Now look at the map

L(V) ↠ L;

let its kernel be J ⊂ [V, V]. (J/[V, J])^{∨} is then a universal obstruction space for h_{L}. Note that we may lift the Frobenius action on L to an action on L(V).

Since H^{2}(X, ℚ_{l}) is an obstruction space for Defx̄, there is a unique injective map of obstruction spaces ([Man99] Proposition 2.18)

(J/V J)^{∨} ↪ H^{2}(X, ℚ_{l}).

Observe that both of these obstruction spaces are Frobenius equivariant. Given a functor D on which F (Frobenius) acts, we say an obstruction theory (W, w_{e}) is Frobenius equivariant if there is a Frobenius action on the space W such that for every small extension

e : 0 → I → 𝔥 → 𝔤 → 0,

and every d ∈ D(𝔤), we have w_{e}(F d) = F w_{e}(d).

Since h_{L} → Defx̄ commutes with Frobenius, the corresponding map of obstruction spaces does (by uniqueness).

As before, we may use the Jordan decomposition on the successive quotients to obtain a space of generators W ⊂ J, such that W → J/[V, J] is an isomorphism preserving the weight decompositions.

Hence we have maps

H^{2}(X, ℚ_{l})^{∨} ↠ J/[V, J] ≅ W ↪ L(V),

and

V ≅ L/[L, L] ≅ H^{1}(X, ℚ_{l})^{∨},

preserving the Frobenius decompositions. Let

f : H^{2}(X, ℚ_{l})^{∨} → L(H^{1}(X, ℚ_{l})^{∨})

be the composition. Then

L = L(V)/J ≅ L(H^{1}(X, ℚ_{l})^{∨})/(f(H^{2}(X, ℚ_{l})^{∨})),

as required.

It only remains to prove the statement about the cup product:

Lemma 3.3. Let F : N_{K} → Set be a deformation functor. If V is an obstruction theory for F, then there is a canonically defined obstruction map t_{F} × t_{F} → V.

Proof. (Of Lemma.) Since F is a deformation functor, t_{F} × t_{F} ≅ F (L(ϵ) ⊕ L(ν)). Consider the small extension

e : 0 → K[ϵ, ν] → L(ϵ, ν)/([ϵ, [ϵ, ν]], [ν, [ν, ϵ]]) → L(ϵ) ⊕ L(ν) → 0.

Sending f ∈ F (L(ϵ)⊕L(ν)) to v_{e}(f) ∈ V [ϵ, ν] defines the obstruction map t_{F} ×t_{F} → V. Standard arguments show that this is bilinear and anti-symmetric.

Now, the obstruction map

V^{∨} ⊗ V^{∨} → W^{∨}

is the dual map to f : W → Γ_{2}(V)/Γ_{3}(V), while the obstruction map

H^{1}(X, ℚ_{l}) ⊗ H^{1}(X, ℚ_{l}) → H^{2}(X, ℚ_{l})

is the usual cup product. The result now follows from uniqueness of the obstruction map.

Corollary 3.4. If X is smooth and proper, then

L(π_{1}(X, x̄), ℚ_{l})

is quadratically presented.

In fact,

L(π_{1}(X, x̄), ℚ_{l}) ≅ L(H^{1}(X, ℚ_{l})^{∨})/(ˇ∪ H^{2}(X, ℚ_{l})^{∨})),

where ˇ∪ is dual to the cup product.

Proof. This follows since, under these hypotheses, [Del80] Corollaries 3.3.4–3.3.6 imply that H^{1}(X, ℚ_{l}) is pure of weight 1, and H^{2}(X, ℚ_{l}) is pure of weight 2.

Example 3.5. This implies that, for X smooth and proper, the pro-l quotient π_{1}^{l}(X, x̄) of π_{1}(X, x̄) cannnot be the Heisenberg group

H_{3}(ℤ_{l}) = \{(\begin{matrix}1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix}) ∈ GL_{3}(ℤ_{l})\} ,

since this is not of quadratic presentation (in particular, this can be inferred from the non-vanishing of the Massey triple product on H^{1}(π_{1}(X, x̄), ℚ_{l}) — see [ABC^{+}96] Ch.3 §3 for criteria for a Lie algebra to be quadratically presented).

Corollary 3.6. If X is smooth, then

L(π_{1}(X, x̄), ℚ_{l})

is a quotient of the free Lie algebra L(H^{1}(X, ℚ_{l})^{∨}) by an ideal which is finitely generated by elements of bracket length 2, 3, 4.

Proof. This follows since, under these hypotheses, [Del80] Corollaries 3.3.4–3.3.6 imply that H^{1}(X, ℚ_{l}) is of weights 1 and 2, while H^{2}(X, ℚ_{l}) is of weights 2, 3 and 4. Hence the image of f : H^{2}(X, ℚ_{l})^{∨} → L(H^{1}(X, ℚ_{l})^{∨}) gives equations:

(W_{−1})^{2} weight − 2
(W_{−1})^{3} + (W_{−1})(W_{−2}) weight − 3
(W_{−1})^{4} + (W_{−1})^{2}(W_{−2}) + (W_{−2})^{2} weight − 4,

hence equations of degree at most 4.

Example 3.7. Thus π_{1}^{l}(X, x̄) cannot be the group

\{(\begin{matrix}1 & ∗ & ∗ \\ 0 & ... & ∗ \\ 0 & 0 & 1 \end{matrix}) ∈ GL_{5}(ℤ_{l})\} .

Remark 3.8. If X is singular and proper, weights tell us nothing about the structure of the fundamental group, since zero weights are permitted, so any equations may arise.

Finally, a result which does not have an analogue in [Pri04a].

Theorem 3.9. Let X be smooth and projective. The cup product

H^{1}(π_{1}(X, x̄), ℚ_{l}) × H^{1}(π_{1}(X, x̄), ℚ_{l}) ⟶^{∪} H^{2}(π_{1}(X, x̄), ℚ_{l})

is non-degenerate.

Proof. Recall that H^{1}(π_{1}(X, x̄), ℚ_{l}) is a tangent space for our deformation functor, and H^{2}(π_{1}(X, x̄), ℚ_{l}) the universal obstruction space. This gives an isomorphism

H^{1}(π_{1}(X, x̄), ℚ_{l}) ⟶^{≅} H^{1}(X, ℚ_{l}),

and an injection

H^{2}(π_{1}(X, x̄), ℚ_{l}) ↪ H^{2}(X, ℚ_{l}),

the maps being compatible with cup products. This yields:

H^{1}(π_{1}(X, x̄), ℚ_{l}) ⊗ H^{1}(π_{1}(X, x̄), ℚ_{l}) ⟶^{∪} H^{2}(π_{1}(X, x̄), ℚ_{l})
↓≅  ↓
H^{1}(X, ℚ_{l}) ⊗ H^{1}(X, ℚ_{l})  ⟶^{∪}  H^{2}(X, ℚ_{l})
η^{n−1}∪↓≅  ^{η^{n−1}∪}↓
H^{2n−1}(X, ℚ_{l}) ⊗ H^{1}(X, ℚ_{l})  ⟶^{∪} H^{2n}(X, ℚ_{l}) ≅ ℚ_{l},

the isomorphism on the bottom left being the Hard Lefschetz Theorem ([Del80] Theorem 4.1), η being the Chern class of an ample line bundle. The cup product along the bottom is non-degenerate by Poincaré duality, hence result.

Since the relations defining the hull arise from the cup product, this indicates that π_{1}(X, x̄) must have some defining relations. More precisely:

Corollary 3.10. If X is smooth and proper, then

L(π_{1}(X, x̄))/Γ_{3}(L(π_{1}(X, x̄))) ≇ L(V)/Γ_{3}(L(V)),

for any free Lie algebra L(V).

Proof. As for [ABC^{+}96] Proposition 3.25.

Remarks 3.11. Homotopy Theory

1. Classically, the DGA Γ(X, 𝒜^{•}) (or, rather, its quasi-isomorphism class) is regarded as being the real homotopy type of a differentiable manifold X. It would therefore make sense to think of the canonical SDC

S^{n}(𝔤) = C^{n}(X, exp(𝔤)),

as capturing the ℚ_{l}-homotopy theory of a scheme X, since it yields a DGA D(S). In fact, S is not merely an SDC, but a cosimplicial complex of group-valued functors. Under the duality of Lemma 2.31, this cosimplicial complex of functors is equivalent to the cosimplicial complex

C^{∗}(X, ℚ_{l})

of algebras. It seems reasonable to regard the quasi-isomorphism class of this cosimplicial complex as being the ℚ_{l}-homotopy type of X. The corresponding differential graded algebra will be quasi-isomorphic to the ℚ_{l}-homotopy type defined in [Del80] §5, but the approach used there did not allow any practical consequences to be drawn from the formality of the homotopy type.

If we recall from [Sul77] that the correct notion of automorphisms of a homotopy type is that of outer automorphisms, i.e. automorphisms modulo homotopy, then the reasoning of [Del80] Corollary 5.3.7 shows that the homotopy type is formal, and that the quasi-isomorphism is Galois equivariant.

2. We may think of the complex C^{∗}(X, x̄; ℚ_{l}) as being the ℚ_{l}-homotopy type of the pair x̄ → X. This will be also be formal when X is smooth and proper.

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