Let G be a simply connected solvable Lie group

93(2013) 269-297

MINIMAL MODELS, FORMALITY, AND HARD LEFSCHETZ PROPERTIES OF SOLVMANIFOLDS WITH LOCAL SYSTEMS

Hisashi Kasuya

Abstract

For a simply connected solvable Lie groupGwith a cocompact discrete subgroup Γ, we consider the space of differential forms on the solvmanifoldG/Γ with values in a certain flat bundle so that this space has a structure of a differential graded algebra (DGA).

We construct Sullivan’s minimal model of this DGA. This result is an extension of Nomizu’s theorem for ordinary coefficients in the nilpotent case. By using this result, we refine Hasegawa’s result of formality of nilmanifolds and Benson-Gordon’s result of hard Lefschetz properties of nilmanifolds.

1. Introduction

The main purpose of this paper is to compute the de Rham cohomol- ogy of solvmanifolds with values in local coefficients associated to some diagonal representations by using the invariant forms and the unipotent hulls. The computations are natural extensions of Nomizu’s computa- tions of untwisted de Rham cohomology of nilmanifolds by the invariant forms in [23]. The computations give natural extensions of Hasegawa’s result of formality of nilmanifolds ([15]) and Benson and Gordon’s result of hard Lefschetz properties of nilmanifolds ([6]).

First we explain the central tools of this paper, called the unipotent hulls and algebraic hulls. Let G be a simply connected solvable Lie group. There exists a unique algebraic group H_G called the algebraic hull of Gwith an injection ψ:G→H_G so that:

1) ψ(G) is Zariski-dense in H_G;

2) the centralizer ZH_G(U(HG)) of U(HG) is contained in U(HG);

3) dimU(H_G) = dimG;

where we denote U(H) the unipotent radical of an algebraic group H.

We denote U_G =U(H_G) and call it the unipotent hull of G.

We consider Hain’s DGAs in [14] which are expected to be effec- tive techniques for studying de Rham homotopy theory of non-nilpotent

Received 8/11/2011.

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spaces. Let M be a C^∞-manifold, ρ :π₁(M, x) → (C^∗)ⁿ a representa- tion; and T the Zariski-closure of ρ(π₁(M, x)) in (C^∗)ⁿ. Let {V_α} be the set of one-dimensional representations for all characters α of T, let (E_α, D_α) be a rank one flat bundle with the monodromy α◦ρ and with A^∗(M, E_α) the space of E_ρ-valued C^∞-differential forms. Denote A^∗(M,O_ρ) =L

αA^∗(M, E_α) andD=L

αD_α. Then (A^∗(M,O_ρ), D) is a cohomologically connected (i.e. the 0-th cohomology is isomorphic to the ground field) DGA. In this paper we construct Sullivan’s minimal model ([29]) of such DGAs on solvmanifolds.

On simply connected solvable Lie groups, we consider DGAs of left- invariant differential forms with local systems which are analogues of Hain’s DGAs. SupposeGis a simply connected solvable Lie group and g is the Lie algebra ofG. Consider the adjoint representation Ad :G→ Aut(g) and its derivation ad :g→D(g), whereD(g) is the Lie algebra of the derivations ofg. We construct representations ofgandGas follows.

Construction 1.1. Let n be the nilradical of g. There exists a sub- vector space (not necessarily Lie algebra)V ofgso thatg=V⊕nas the direct sum of vector spaces and for any A, B ∈V (ad_A)_s(B) = 0where (ad_A)_s is the semi-simple part of ad_A(see[12, Proposition III.1.1]). We define the map ads : g → D(g) as adsA+X = (adA)s for A ∈ V and X ∈ n. Then we have [ad_s(g),ad_s(g)] = 0 and ad_s is linear (see [12, Proposition III.1.1]). Since we have [g,g]⊂n, the map ad_s :g → D(g) is a representation and the imagead_s(g)is abelian and consists of semi- simple elements. We denote by Ad_s:G→Aut(g) the extension of ad_s. Then Ad_s(G) is diagonalizable.

LetTbe the Zariski-closure of Ads(G) in Aut(gC). Then Tis diago- nalizable. Let{V_α}be the set of one-dimensional representations for all characters α ofT. We consider V_α the representation of gwhich is the derivation ofα◦Ad_s. Then we have the cochain complex of Lie algebra (V

g^∗_C⊗V_α, d_α). Denote A^∗(g_C,ad_s) =L

α

Vg^∗_C⊗V_α and d=L

αd_α. Then (A^∗(gC,ad_s), d) is a cohomologically connected DGA. In this pa- per we compute the cohomology of this DGA by the unipotent hullU_G of G. Letu be the Lie algebra ofU_G and V

u^∗ be the cochain complex of the dual space u^∗ of u. We prove the following theorem.

Theorem 1.1 (Theorem 5.4). We have a quasi-isomorphism (i.e. a morphism which induces a cohomology isomorphism) of DGAs

^u^∗→A^∗(gC,ad_s).

Thus V

u^∗ is Sullivan’s minimal model of A^∗(gC,ads).

Suppose G has a lattice Γ, i.e. a cocompact discrete subgroup ofG.

We call a compact homogeneous space G/Γ a solvmanifold. We have π₁(G/Γ)∼= Γ. For the restriction of the semi-simple part of the adjoint

representation Ad_s|Γ on Γ, we consider Hain’s DGA A^∗(G/Γ,O_Ads|Γ).

By using Theorem 1.1, we prove:

Theorem 1.2 (Corollary 7.5). Let G be a simply connected solvable Lie group with a lattice Γ, and let U_G be the unipotent hull of G. Let u be the Lie algebra of U_G. Then we have a quasi-isomorphism

^u^∗ →A^∗(G/Γ,O_Ads|Γ).

Thus V

u^∗ is Sullivan’s minimal model of A^∗(G/Γ,O_Ads|Γ).

IfGis nilpotent, the adjoint operator Ad is a unipotent representation and hence A^∗(G/Γ,O_Ads|

Γ) = A^∗_C(G/Γ) and A^∗(gC,ads) = V gC = Vu^∗. In this case, Theorem 1.2 reduces to the classical theorem proved by Nomizu in [23]. Moreover, this result gives more refined computations of untwisted de Rham cohomology of solvmanifolds than the results of Mostow and Hattori (see Corollary 7.4 and Section 10).

We call a DGAAformal if there exists a finite diagram of morphisms A→C1 ←C2· ·· ←H^∗(A)

such that all morphisms are quasi-isomorphisms; and we call manifolds M formal if the de Rham complex A^∗(M) is formal. In [15] Hasegawa showed that formal nilmanifolds are tori. By the results of this paper, we have a natural extension of Hasegawa’s theorem for solvmanifolds.

Theorem 1.3 (Theorem 8.2). Let G be a simply connected solvable Lie group. Then the following conditions are equivalent:

(A) The DGAA^∗(gC,ads) is formal.

(B) U_G is abelian.

(C) G = Rⁿ⋉_φR^m such that the action φ : Rⁿ → Aut(R^m) is semi- simple.

Moreover, suppose G has a lattice Γ. Then the above three conditions are equivalent to the following condition:

(D) A^∗(G/Γ,O_Ads|Γ) is formal.

In [6] Benson and Gordon showed that symplectic nilmanifolds with the hard Lefschetz properties are tori. We can also have an extension of Benson and Gordon’s theorem.

Theorem 1.4 (Theorem 8.4). Let G be a simply connected solvable Lie group. Suppose dimG = 2n and G has an G-invariant symplectic form ω. Then the following conditions are equivalent:

(A)

[ω]ⁿ⁻ⁱ∧:Hⁱ(A^∗(gC,ad_s))→H²ⁿ⁻ⁱ(A^∗(gC,ad_s)) is an isomorphic for any i≤n.

(B) U_G is abelian.

(C) G = Rⁿ⋉_φR^m such that the action φ : Rⁿ → Aut(R^m) is semi- simple.

Suppose Ghas a latticeΓand G/Γ has a symplectic form (not neces- sarily G-invariant) ω. Then the conditions (B) and (C) are equivalent to the following condition:

(D)

[ω]ⁿ⁻ⁱ∧:Hⁱ(A^∗(G/Γ,O_Ads|

Γ))→H²ⁿ⁻ⁱ(A^∗(G/Γ,O_Ads|

Γ)) is an isomorphism for any i≤n.

Remark 1.1. As a representation in an algebraic group, Ad_s is in- dependent of the choice of a subvector space V in Construction 1.1 (see Lemma 2.5). By this, the structures of DGAs A^∗(g_C,ad_s) and A^∗(G/Γ,O_Ads|

Γ) are independent of the choice of a subvector spaceV. Finally we consider relations with K¨ahler geometries. We review stud- ies of K¨ahler structures on solvmanifolds briefly. See [5] and [17] for more details. In [7] Benson and Gordon conjectured that for a com- pletely solvable simply connected Lie group G with a lattice Γ, G/Γ has a K¨ahler metric if and only if G/Γ is a torus. In [16] Hasegawa studied K¨ahler structures on some classes of solvmanifolds which are not only the completely solvable type, and suggested a generalized ver- sion of Benson-Gordon’s conjecture: A compact solvmanifold can have a Kahler structure if and only if it is a finite quotient of a complex torus that is a holomorphic fiber bundle over a complex torus with its fiber a complex torus. In [1] Arapura showed Benson-Gordon’s conjecture and also showed that the fundamental group of a K¨ahler solvmanifold is virtually abelian by the result in [2]. In [1] a proof of Hasegawa’s con- jecture was also written, but we notice that this proof contains a gap and Hasegawa complement in [17]. We also notice that Baues and Cort´es showed a more generalized version of Benson-Gordon’s conjecture for aspherical manifolds with polycyclic fundamental groups in [5].

By the theory of Higgs bundle studied by Simpson, we have a twisted analogue of formality (see [11]) and the hard Lefschetz properties of compact K¨ahler manifolds. We have:

Theorem 1.5 (Special case of Thoerem 4.1). Suppose M is a com- pact K¨ahler manifold with a K¨ahler formω, and thatρ:π₁(M)→(C^∗)ⁿ is a representation. Then the following conditions hold:

(A) (formality) The DGAA^∗(M,Oρ) is formal.

(B)(hard Lefschetz) For any 0≤i≤n the linear operator [ω]ⁿ⁻ⁱ∧:Hⁱ(A^∗(M,O_ρ))→H²ⁿ⁻ⁱ(A^∗(M,O_ρ)) is an isomorphism where dim_RM = 2n.

Now by Theorem 1.5, formality and the hard Lefschetz property of DGAA^∗(G/Γ,O_Ads|Γ) are criteria forG/Γ to have a K¨ahler metric. We will see such conditions are stronger than untwisted formality and the hard Lefschetz property.

Remark 1.2. There exist examples of solvmanifoldsG/Γ which sat- isfy formality and the hard Lefschetz property of the untwisted de Rham complex A^∗(G/Γ),but do not satisfy formality and the hard Lefschetz property of A^∗(G/Γ,O_Ads|

Γ).

However, we will see that these criteria cannot classify K¨ahler solv- manifolds completely.

Remark 1.3. There exist examples of non-K¨ahler solvmanifolds which satisfy formality and the hard Lefschetz property of A^∗(G/Γ,O_Ads|

Γ).

Acknowledgments.The author would like to express his gratitude to Toshitake Kohno for helpful suggestions and stimulating discussions.

He would also like to thank Oliver Baues, Keizo Hasegawa, and Takumi Yamada for their active interest in this paper. This research is supported by JSPS Research Fellowships for Young Scientists. The author would like to express many thanks to the referee for his careful reading of the earlier version of manuscript with several important remarks, which lead to many improvements in the revised version.

2. Preliminaries on algebraic hulls

LetGbe a discrete group (resp. a Lie group). We call a mapρ:G→ GL_n(C) a representation, if ρ is a homomorphism of groups (resp. Lie groups).

2.1. Algebraic groups.In this paper an algebraic group means an affine algebraic variety G over C with a group structure such that the multiplication and inverse are morphisms of varieties. All algebraic groups in this paper arise as Zariski-closed subgroups ofGLn(C). Letk be a subfield of C. We call G k-algebraic ifG is defined by polynomi- als with coefficient in k. We denote as G(k) thek-points of G. We say that an algebraic group is diagonalizable if it is isomorphic to a closed subgroup of (C^∗)ⁿfor somen.

2.2. Algebraic hulls.A group Γ is polycyclic if it admits a sequence Γ = Γ₀⊃Γ₁ ⊃ · · · ⊃Γ_k={e}

of subgroups such that each Γ_i is normal in Γ_i−1 and Γ_i−1/Γ_i is cyclic.

For a polycyclic group Γ, we denote rank Γ = P_i=k

i=1rank Γi−1/Γi. Let G be a simply connected solvable Lie group, and let Γ to be a lattice in G. Then Γ is torsion-free polycyclic and dimG = rank Γ (see [26, Proposition 3.7]). Let ρ:G→ GL_n(C), for g∈G be a representation.

LetGandG^′ be the Zariski-closures ofρ(G) andρ(Γ) inGL_n(C). Then we have U(G) =U(G^′) (see [26, Theorem 3.2]).

We review the algebraic hulls.

Proposition 2.1. ([26, Proposition 4.40]) Let G be a simply con- nected solvable Lie group (resp. torsion-free polycyclic group). Then there exists a unique R-algebraic group H_G with an injective group ho- momorphism ψ:G→H_G(R) so that:

(1) ψ(G) is Zariski-dense in HG. (2) ZH_G(U(H_G))⊂U(H_G).

(3) dimU(HG)=dimG(resp. rankG).

Such H_G is called the algebraic hull of G.

We denote U_G =U(H_G) and call U_G the unipotent hull of G.

2.3. Direct constructions of algebraic hulls.Letgbe a solvable Lie algebra, and n={X ∈g|adX is nilpotent}. nis the maximal nilpotent ideal ofgand called the nilradical ofg. Then we have [g,g]⊂n. Consider the adjoint representation ad : g → D(g) and the representation ad_s : g→D(g) as Construction 1.1.

Let ¯g= Im ad_s⋉gand

¯

n={X−ad_sX ∈¯g|X∈g}.

Then we have [g,g] ⊂ n ⊂ ¯n, and ¯n is the nilradical of ¯g (see [12]).

Hence we have ¯g= Im ad_s⋉¯n.

Lemma 2.2. Supposeg=R^k⋉_φnsuch thatφis a semi-simple action and nis nilpotent. Then n¯=R^k⊕n.

Proof. By assumption, forX+Y ∈R^k⋉_φn, we have ad_sX+Y = ad_X. Hence we have

[X₁+Y₁−ad_sX1+Y1, X₂+Y₂−ad_sX2+Y2] = [X₂, Y₂]

forX₁+Y₁, X₂+Y₂ ∈R^k⋉_φn. Hence the lemma follows. q.e.d.

Let G be a simply connected solvable Lie group and g be the Lie algebra of G. Let N be the subgroup of G which corresponds to the nilradical n of g. We consider the exponential map exp : g → G. In general exp is not a diffeomorphism. But we have the useful property of exp as following.

Lemma 2.3. ([10, Lemma 3.3]) Let V be a subvector space (not necessarily Lie algebra) V of g so that g= V ⊕n as the direct sum of vector spaces. We define the map F :g =V ⊕n→ G as F(A+X) = exp(A) exp(X) for A∈V, X ∈n. Then F is a diffeomorphism and we have the commutative diagram

1 ^//N ^//G ^//G/N ∼=R^k //1

0 ^//n ^//

exp

OO

g ^//

F

OO

g/n∼=R^k

exp=idRk

OO //0

where dimG/N =k.

By this lemma, forA ∈V,X ∈n, the extension Ad_s :G→Aut(gC) is given by

Ad_s(exp(A) exp(X)) = exp((ad_A)_s) = (exp(ad_A))_s

and we have Ad_s(G) = {(exp(ad_A))_s ∈ Aut(gC)|A ∈ V}. Let ¯G = Ad_s(G)⋉G. Then the Lie algebra of ¯Gis ¯g. For the nilradical ¯N of ¯G, by the spritting ¯g = Im ad_s⋉¯n we have ¯G = Ad_s(G)⋉N¯ such that we can regard Ad_s(G)⊂Aut( ¯N) and Ad_s(G) to consist of semi-simple automorphisms of ¯N. By the construction of ¯nwe have ¯G=G·N¯.

A simply connected nilpotent Lie group is considered to be the real points of a unipotentR-algebraic group (see [24, p. 43]) by the exponen- tial map. We have the unipotent R-algebraic groupN¯ withN(R) = ¯¯ N. We identify Aut_a(N) with Aut(n¯ _C), and Aut_a(N) has the¯ R-algebraic group structure with Aut_a(N)(R) = Aut( ¯¯ N). So we have theR-algebraic group Aut_a(N)⋉¯ N. By Ad¯ _s(G)⋉G= Ad_s(G)⋉N¯, we have the injection I :G→Aut( ¯N)⋉N¯ = Aut_a(N)¯ ⋉N(R). Let¯ Gbe the Zariski-closure of I(G) in Aut_a(N)¯ ⋉N.¯

Proposition 2.4. Let T be the Zariski-closure of Ad_s(G) in Aut ¯N.

Then we have G = T⋉N, and¯ G is the algebraic hull of G with the unipotent hull U_G=N. Hence the Lie algebra of the unipotent hull¯ U_G of G is

¯

n_C={X−ad_sX ∈¯g_C|X∈g_C}.

Proof. The algebraic groupT⋉N¯ is the Zariski-closure of Ads(G)⋉N¯ in Aut( ¯N)⋉N¯. By Ad_s(G)·I(G) = Ad_s(G)⋉N¯, we haveT·G=T⋉N.¯ SinceTis a diagonalizable algebraic group, we have ¯N⊂G. Otherwise, since G ⊂ T ⋉N¯ is a connected solvable algebraic group, we have U(G) = ¯N∩G= ¯N. Since we have Ad_s(G)⋉N¯ =G·N¯,Gis identified with the Zariski-closure of Ads(G)⋉N¯. Hence we have G=T⋉N. By¯ dimG= dim ¯N, we can easily check thatT⋉N¯ is the algebraic hull of

G. q.e.d.

By this proposition, the Zariski-closureTof Ad_s(G) is a maximal torus of the algebraic hull of G. By the uniqueness of the algebraic hull (see [26, Lemma 4.41]), we have:

Lemma 2.5. Let H_G be the algebraic hull of G and q : H_G → H_G/U_G the quotient map. Then for any injection ψ : G → H_G(R) as in Proposition 2.1, there exists an isomorphism ϕ : H_G/UG → T such that the diagram

H_G/UG ϕ

//T

G

q◦ψ

OO

Ads

;;

vv v vv vv vv v

commutes.

Lemma 2.6. LetH_G =H_G(R) be the real points of the algebraic hull of G. LetTbe the Zariski-closure of Ad_s(G) in Aut(g_C) and T =T(R) its real points. Then we have a semi-direct product

HG=T ⋉G.

Proof. By Im(ad_s)⋉¯n= Im(ad_s)⋉g, we have Ad_s(G)⋉N¯ = Ad_s(G)⋉

I(G). Hence the lemma follows from Proposition 2.4. q.e.d.

Proposition 2.7. ([19]) Let G be a simply connected solvable Lie group. Then U_G is abelian if and only if G=Rⁿ⋉_φR^m such that the action φ:Rⁿ→Aut(R^m) is semi-simple.

Proof. SupposeU_Gis abelian. Then by Proposition 2.4, the Lie alge- bra ¯nis abelian. By n⊂n, the nilradical¯ nof gis abelian. By [g,g]⊂n, g is two-step solvable. We consider the lower central seriesgⁱ asg⁰=g and gⁱ = [g,gⁱ⁻¹] for i ≥ 1. We denote n^′ = ∩^∞_i=0gⁱ. Then by [9, Lemma 4.1], we haveg=g/n^′⋉n^′. For this decomposition, the subspace {X−ad_sX|X∈g/n^′} ⊂¯nis a Lie subalgebra of ¯n. Sinceg/n^′ is a nilpo- tent subalgebra of g, this space is identified with g/n^′. Thus since ¯n is abelian,g/n^′ is also abelian. We show that the action ofg/nonnis semi- simple. Suppose for some X ∈g/n^′, adX on nis not semi-simple. Then ad_X−ad_sX onnis not trivial. Since we have ¯n={X−ad_sX|X∈g}, we have [¯n,n]6= {0}. This contradicts that ¯n is abelian. Hence the action of g/n on n is semi-simple, and hence the first half of the proposition follows. The converse follows from Lemma 2.2. q.e.d.

3. Left-invariant forms and the cohomology of solvmanifolds LetGbe a simply connected solvable Lie group, gthe Lie algebra of G, and ρ :G → GL(V_ρ) a representation on a C-vector space V_ρ. We consider the cochain complex V

g^∗ with the derivation d which is the dual to the Lie bracket of g. ThenV

g^∗_C⊗V_ρis a cochain complex with the derivationdρ=d+ρ_∗, whereρ_∗ is the derivation ofρ, and consider ρ_∗∈g^∗_C⊗gl(V_ρ). We can consider the cochain complex (V

g^∗_C⊗V_ρ, d_ρ) to be the twistedG-invariant differential forms onG. Consider the cochain complex A^∗_C(G)⊗V_ρ with the derivationdsuch that

d(ω⊗v) = (dω)⊗v ω∈A^∗_C(G), v∈V_ρ.

By the left action ofG(given by (g·f)(x) =f(g⁻¹x), f ∈C^∞(G),g∈ G) andρ, we have the action ofGonA^∗_C(G)⊗V_ρ. Denote (A^∗_C(G)⊗V_ρ)^G theG-invariant elements ofA^∗_C(G)⊗V_ρ. Then we have an isomorphism

(A^∗_C(G)⊗V_ρ)^G∼=^

g^∗_C⊗V_ρ.

SupposeG has a lattice Γ. Sinceπ₁(G/Γ) = Γ, we have a flat vector bundleE_ρ|Γ with flat connectionD_ρ|Γ onG/Γ whose monodromy isρ_|Γ. Let A^∗(G/Γ, E_ρ|Γ) be the cochain complex of E_ρ|Γ-valued differential

forms with the derivation D_ρ|Γ. Consider the cochain complexA^∗_C(G)⊗ V_ρ with derivation dsuch that

d(ω⊗v) = (dω)⊗v ω∈A^∗_C(G), v∈Vρ.

Then we have the G-action on A^∗_C(G)⊗V_ρ and denote (A^∗_C(G)⊗V_ρ)^Γ the subcomplex of Γ-invariant elements of A^∗_C(G)⊗V_ρ. We have the isomorphism (A^∗_C(G)⊗V_ρ)^Γ∼=A^∗(G/Γ, E_ρ|

Γ). Thus we have

^g^∗_C⊗Vρ∼= (A^∗_C(G)⊗Vρ)^G ⊂(A^∗_C(G)⊗Vρ)^Γ∼=A^∗(G/Γ, Eρ_|Γ) and we have the inclusionV

g^∗_C⊗Vρ→A^∗(G/Γ, Eρ_|Γ).

We call a representation ρ Γ-admissible if for the representation ρ⊕ Ad : G → GL_n(C)×Aut(gC), (ρ⊕Ad)(G) and (ρ⊕Ad)(Γ) have the same Zariski-closure inGL_n(C)×Aut(g_C).

Theorem 3.1. ([20], [26, Theorem 7.26]) If ρ isΓ-admissible, then the inclusion

^g^∗_C⊗V_ρ→A^∗(G/Γ, E_ρ|

Γ) induces a cohomology isomorphism.

Proposition 3.2. Let G be a simply connected solvable Lie group with a lattice Γ. We suppose Ad(G) and Ad(Γ) have the same Zariski- closure in Aut(gC). We consider the diagonalizable representation Ads: G→Aut(G). Let T be the Zariski-closure of Ad_s(G) and α be a char- acter of T. Then α◦Ad_s is Γ-admissible.

Proof. Let G be the Zariski-closure of Ad(G) in Aut(gC). We first show that T is a maximal torus of G. For the direct sum g = V ⊕n as Construction 1.1, the map F :V ⊕n→ G defined by F(A+X) = exp(A) exp(X) forA ∈V,X∈nis a diffeomorphism (see [10, Lemma 3.3]). ForA∈V, we consider the Jordan decomposition Ad(exp(A)) = exp((ad_A)_s) exp((ad_A)_n). Then we have exp((ad_A)_s),exp((ad_A)_n) ∈G.

For X ∈ n, we have exp(ad_X) ∈ U(G). Hence we have Ad(G) ⊂ TU(G) ⊂ G. Since G is a Zariski-closure of Ad(G), G = TU(G).

Thus Tis a maximal torus of G.

We take a sprittingG=T⋉U(G). We consider the algebraic group G^′ ={(α(t),(t, u))∈C^∗×G|(t, u)∈T⋉U(G)}.

Then we have (α◦Ad_s⊕Ad)(G)

={(α(exp((ad_A)_s)),exp(ad_A) exp(ad_X))|A+X ∈V ⊕n}

⊂G^′. Since G is a Zariski-closure of Ad(G), (α ◦Ad_s⊕Ad)(G) is Zariski- dense in G^′. Since Ad(G) and Ad(Γ) have the same Zariski-closure,

(α◦Ad_s⊕Ad)(G) and (α◦Ad_s⊕Ad)(Γ) have the same Zariski-closure

G^′. q.e.d.

4. Hain’s DGAs

4.1. Constructions.Let M be a C^∞-manifold, S be a reductive al- gebraic group, and ρ : π₁(M, x) → S be a representation. We assume the image of ρ is Zariski-dense inS. Let {Vα} be the set of irreducible representations of S, (E_α, D_α) a flat bundle with the monodromyα◦ρ, and A^∗(M, Eα) the space of Eρ-valuedC^∞-differential forms. Then we have an algebra isomorphism of L

αV_α⊗V_α^∗ and the coordinate ring C[S] of S(see [14, Section 3]). Denote

A^∗(M,O_ρ) =M

α

A^∗(M, E_α)⊗V_α^∗ and D =L

αD_α. Then by the wedge product, (A(M,O_ρ), D) is a co- homologically connected DGA with coefficients in C.

Suppose S is a diagonal algebraic group. Then {V_α} is the set of one-dimensional representations for all algebraic charactersα ofT, and (E_α, D_α) are rank one flat bundles with the monodromy α◦ρ. In this case, for characters α and β, we have the wedge productA^∗(M, E_α)⊗ A^∗(M, E_β)→A^∗(M, E_αβ) andD_αβ(ψα∧ψ_β) =Dαψα∧ψ_β+ (−1)^pψα∧ D_βψ_β forψ_α ∈A^p(M, E_α), ψ_β ∈A^q(M, E_β) (see [22] for details in this case).

4.2. Formality and the hard Lefschetz properties of compact K¨ahler manifolds.In this subsection we will prove the following the- orem by theories of Higgs bundles studied by Simpson.

Theorem 4.1. Let M be a compact K¨ahler manifold with a K¨ahler form ω and ρ : π₁(M) → S a representation to a reductive algebraic group S with the Zariski-dense image. Then the following conditions hold:

(A) (formality) The DGAA^∗(M,O_ρ) is formal.

(B)(hard Lefschetz) For any 0≤i≤n, the linear operator [ω]ⁱ∧:Hⁿ⁻ⁱ(A^∗(M,O_ρ))→Hⁿ⁺ⁱ(A^∗(M,O_ρ)) is an isomorphism where dimRM = 2n.

Let M be a compact K¨ahler manifold and E a holomorphic vector bundle on M with the Dolbeault operator ¯∂. For an End(E)-valued holomorphic form θ, we denote D^′′ = ¯∂+θ. We call (E, D^′′) a Higgs bundle if it satisfies the Leibniz rule: D^′′(ae) = ¯∂(a)e+ (−1)^pD^′′(e) for a ∈ A^p(M), e ∈ A⁰(E) and the integrability (D^′′)² = 0. Let h be a Hermitian metric on E. For a Higgs bundle (E, D^′′= ¯∂+θ), we define D^′_h=∂_h+ ¯θ_h as follows:∂_h is the unique operator which satisfies

h( ¯∂e, f) +h(e, ∂_hf) = ¯∂h(e, f)

and ¯θ_h is defined by (θe, f) = (e,θ¯_hf). Let D_h =D^′_h+D^′′. Then D_h is a connection. We call a Higgs bundle (E, D^′′, h) with a metric harmonic if D_h is flat, i.e. (D_h)² = 0.

For two Higgs bundles (E, D^′′), (F, D^′′) with metrich_E,h_F, the tensor product (E⊗F, D^′′⊗1+1⊗D^′′) is also a Higgs bundle, andh_E⊗h_F gives the connection D_hE⊗hF =D_hE⊗1 + 1⊗D_hF on E⊗F. If (E, D^′′, h_E) and (F, D^′′, hF) are harmonic, (E⊗F, D^′′⊗1+1⊗D^′′) is also a harmonic Higgs bundle with the flat connection D_hE ⊗1 + 1⊗D_hF.

Theorem 4.2. ([28, Theorem 1]) Let (E, D) be a flat bundle on M whose monodromy is semi-simple. ThenDis given by a harmonic Higgs bundle (E, D^′′, h), that is, D=D_h.

Theorem 4.3. ([28, Lemma 2.2])Let(E, D^′′, h)be a harmonic Higgs bundle with the flat connection D=D^′+D^′′. Then the inclusion

(KerD^′, D^′′)→(A^∗(E), D) and the quotient

(KerD^′, D^′′)→(H_D^′(A^∗(E)), D^′′) = (H_D^∗(A^∗(E)),0) induce the cohomology isomorphisms.

Theorem 4.4. ([28, Lemma 2.6])Let(E, D^′′, h)be a harmonic Higgs bundle with the flat connection D=D^′+D^′′. Then for any 0≤i≤n the linear operator

[ω]ⁿ⁻ⁱ∧:H_Dⁱ (A^∗(E_ρ))→H_D²ⁿ⁻ⁱ(A^∗(E_ρ)) is an isomorphism.

Proof of Theorem 4.1. By Theorems 4.2 and 4.4, the condition (B) holds.

By Theorem 4.2, for (A^∗(E_α), D_α), we have D_α = D_α^′ +D_α^′′ such that D^′′_αis a harmonic Higgs bundle. DenoteD^′ =L

αD^′_αandD^′′=L

αD_α^′′. Then by properties of the Higgs bundle, (KerD^′, D^′′) is a DGA, and the maps

(KerD^′, D^′′)→(A^∗(M,O_ρ)), D) and

(KerD^′, D^′′)→(H_D^∗(A^∗(M,O_ρ)),0)

are DGA homomorphisms, and thus quasi-isomorphisms by Theorem

4.3. Hence the condition (A) holds. q.e.d.

5. Minimal models of invariant forms on solvable Lie groups with local systems

LetGbe a simply connected solvable Lie group andgthe Lie algebra of G. Consider the diagonal representation Ad_s as in Section 1 and the derivation ad_s of Ad_s. For some basis {X₁, . . . , X_n} of gC, Ad_s is represented by diagonal matrices. LetTbe the Zariski-closure of Ad_s(G)

in Aut(gC). Let {V_α} be the set of one-dimensional representations for all characters α of T. We consider V_α the representation of g which is the derivation of α◦Ad_s. Then we have the cochain complex of Lie algebra (V

g^∗_C⊗V_α, d_α). Denote d = L

αd_α. Then (L

α

Vg^∗_C⊗V_α, d) is a cohomologically connected DGA with coefficients in C as the last section. By Ad_s(G) ⊂ Aut(g_C) we have T ⊂ Aut(g_C), and hence we have the action of T on L

α

Vg^∗_C⊗V_α. Denote (L

α

Vg^∗_C⊗V_α)^T the sub-DGA of L

α

Vg^∗_C⊗V_α which consists of the T-invariant elements of L

α

Vg^∗_C⊗V_α.

Lemma 5.1. We have an isomorphism H^∗((M

α

^g^∗_C⊗V_α)^T)∼=H^∗(M

α

^g^∗_C⊗V_α).

Proof. We show that the action of Ad_s(G) ⊂ T on the cohomology H^∗(V

g^∗_C⊗V_α) is trivial. Consider the direct sum g = V ⊕n as Con- struction 1.1. Then we have Ad_s(G) = Ad_s(exp(V)) by Lemma 2.3. For A∈V, the action Ad_s(exp(A)) on the cochain complex V

g^∗_C⊗V_α is a semi-simple part of the action of exp(A) onV

g^∗_C⊗V_α via Ad⊗α◦Ad_s. Since the action ofGon the cohomologyH^∗(V

g^∗_C⊗V_α) via Ad⊗α◦Ad_s is the extension of the Lie derivation onH^∗(V

g^∗_C⊗Vα), thisG-action on H^∗(V

g^∗_C⊗V_α) is trivial. Hence forA∈V the action of Ad_s(exp(A)) = (exp(adA))s on the cohomology H^∗(V

g^∗_C⊗Vα) is trivial.

SinceTis the Zariski-closure of Ad_s(G) in Aut(gC) and the action of T on V

g^∗_C⊗V_α is algebraic, the action ofT on H^∗(V

g^∗_C⊗V_α) is also trivial. Since the action ofTonV

g^∗_C⊗V_α is diagonalizable, we have an isomorphism

H^∗(^

g^∗_C⊗V_α)∼=H^∗(^

g^∗_C⊗V_α)^T ∼=H^∗((^

g^∗_C⊗V_α)^T).

Hence we have the lemma. q.e.d.

Consider the unipotent hullU_GofG. Letube theC-Lie algebra of U_G and u^∗ the C-dual space. We consider the DGA V

u^∗ with coefficients inC.

Lemma 5.2. We have an isomorphism of DGA

^u^∗∼= (M

α

^g^∗_C⊗V_α)^T.

Proof. Let{x₁, . . . , x_n}be the dual of a basis{X₁, . . . , X_n}ofgsuch that Ads is represented by diagonal matrices. We define characters αi

ast·X_i=α_i(t)X_i fort∈T. Then we have t·x_i =α⁻_i ¹(t)x_i. Hence the vector space (L

α

V1

g^∗_C⊗Vα)^T is spanned by {x1⊗vα1, . . . , xn⊗vαn} where V_αi ∋v_αi 6= 0. For

ω= X

i1,...,ip,α

a_i1,...,ip,αx_i1∧ · · · ∧x_ipv_α∈(M

α p

^g^∗_C⊗V_α)^T,

since any x_i1 ∧ · · · ∧ x_ipv_α is an eigenvector of the action of T, if a_i1,...,ip,α6= 0 then x_i1 ∧ · · · ∧x_ipv_α is also a T-invariant element. Since we have

t·xi1 ∧ · · · ∧xip =α⁻_i1¹(t)· · ·α⁻_ip¹(t)xi1∧ · · · ∧xip

fort∈T, we have

x_i1 ∧ · · · ∧x_ip⊗v_α =x_i1v_αi1 ∧ · · · ∧x_ipv_αip. Thus the DGA (L

α

Vg^∗_C⊗V_α)^Tis generated by{x₁⊗v_α1, . . . , x_n⊗v_αn}.

Consider the Maurer-Cartan equations dx_k=−X

ij

c^k_ijx_i∧x_j

and denote adsXi(Xj) = aijXj. Since Adsg(X_k) =αi(Adsg)X_k for g ∈ G, we have dv_αk = P_n

i=1ad_sXi(X_k)x_iv_αk = P_n

i=1a_ikx_iv_αk. Then we have

d_αk(x_k⊗v_αk) =−X

ij

(c^k_ijx_i∧x_j⊗v_αk −a_ikx_i∧x_k⊗v_αk).

Hence the DGA (L

α

Vg^∗_C⊗V_α)^Tis isomorphic to a free DGA generated by degree 1 elements {y₁, . . . , y_n}such that

d(y_k) =−X

ij

(c^k_ijy_i∧y_j−a_iky_i∧y_k).

Lethbe the Lie algebra which is the dual of the free DGA (L

α

Vg^∗_C⊗ Vα)^T and {Y1, . . . , Yn} the dual basis of {y1, . . . , yn}. It is sufficient to show h∼=u. Then the bracket ofhis given by

[Y_i, Y_j] =X

k

c^k_ijY_k−a_ijY_j+a_jiY_i.

Otherwise, by Section 2.3, we haveu∼={X−ad_sX|X∈gC} ⊂D(gC)⋉ g_C. For the basis{X₁−ad_sX1, . . . , X_n−ad_sXn} of u, we have

[X_i−ad_sXi, X_j−ad_sXj] =X

k

c^k_ijX_k−a_ijX_j+a_jiX_i.

By [g,g]⊂n, we have [u,u]⊂n_C, wherenis the nilradical ofg. By this we have

X

k

c^k_ijX_k−aijXj+ajiXi ∈nC, and hence we have

ad_s^P

kc^k_ijXk−aijXj+ajiXi = 0.

This gives

[X_i−ad_sXi, X_j−ad_sXj]

=X

k

c^k_ij(X_k−ad_sXk)−a_ij(X_j−ad_sXj) +a_ji(X_i−ad_sXi).

This gives an isomorphismh∼=u. Hence the lemma follows. q.e.d.

Since Ads(G) is Zariski-dense inT, Ads(G)-invariant elements are also T-invariant. In particular, we have the following lemma.

Lemma 5.3. Let T =T(R) be the real points of T. Then we have (M

α

^g^∗_C⊗V_α)^T ∼= (M

α

^g^∗_C⊗V_α)^T ∼=^ u^∗. Later we use this lemma.

Denote A^∗(g_C,ad_s) = L

α

Vg^∗_C⊗V_α. By Lemmas 5.1 and 5.2 we have:

Theorem 5.4. We have a quasi-isomorphism of DGAs

^u^∗→A^∗(g_C,ad_s).

Thus V

u^∗ is the minimal model of A^∗(g_C,ad_s).

6. Cohomology of A^∗(G/Γ,O_Ads|

Γ) Consider the two DGAs A^∗(gC,ads) and A^∗(G/Γ,O_Ads|

Γ). For any character α of an algebraic group T which is the Zariski-closure of Ad_s(G) in Aut(g_C), we have the inclusion

^g^∗_C⊗V_α∼= (A^∗_C(G)⊗V_α)^G⊂(A^∗_C(G)⊗V_α)^Γ∼=A^∗(G/Γ, E_α◦Ads|Γ).

Thus we have the morphism of DGAs

φ:A^∗(g_C,ad_s)→A^∗(G/Γ,O_Ads|Γ).

Proposition 6.1. The morphismφ:A^∗(gC,ad_s)→A^∗(G/Γ,O_Ads|Γ) is injective and the induced map

φ^∗:H^∗(A^∗(gC,ads))→ H^∗(A^∗(G/Γ,O_Ads|

Γ)) is also injective.

Proof. Since G has a lattice Γ, G is unimodular (see [26, Remark 1.9]). Choose a Haar measure dµ such that the volume ofG/Γ is 1. We define a mapϕ_α : (A^∗_C(G)⊗V_α)^Γ→V

g^∗_C⊗V_α as ϕ_α(ω⊗v_α)(X₁, . . . , X_p) =

Z

G/Γ

ω_x

α(x)(X₁, . . . , X_p)dµ·v_α

for ω ⊗v_α ∈ (A^p_C(G) ⊗ V_α)^Γ, X₁, . . . , X_p ∈ g_C. Then each ϕ_α is a morphism of cochain complexes and we haveϕ_α◦φ_|V

g∗

C⊗Vα = id_|V g∗

C⊗Vα

(see [26, Remark 7.30]). Thus the restriction φ^∗:H^∗(^

g^∗_C⊗Vα)→H^∗(A^∗(G/Γ, Eα))

is injective. By this it is sufficient to show that two distinct characters α, β withα◦Ad_s|Γ =β◦Ad_s|Γ satisfyϕ_β◦φ_|V

g∗

C⊗Vα = 0. For ω⊗v_α ∈ Vg^∗_C⊗V_α, we have

ϕ_β◦φ_|V g∗

C⊗Vα(ω⊗v_α) = Z

G/Γ

α(x)

β(x)ω_x(X₁, . . . , X_p)dµ·v_α. Since ω ∈ V

g^∗_C, ωx(X1, . . . , Xp) is constant on G/Γ. Let λ = ^β_αd(^α_β).

Then λis a G-invariant form. Choose η ∈V

g^∗_C such that λ∧η =dµ.

Then we have

d α

βη

= α

βλ∧η= α βdµ.

By α◦Ad_s|Γ =β◦Ad_s|Γ, ^α_βη is Γ-invariant and we can consider ^α_βη a differential form onG/Γ. Hence by Stokes’ theorem, we have

Z

G/Γ

α(x)

β(x)ω_x(X₁, . . . , X_p)dµ=ω(X₁, . . . , X_p) Z

G/Γ

α(x) β(x)dµ

=ω(X₁, . . . , X_p) Z

G/Γ

d α

βη

= 0.

This proves the proposition. q.e.d.

Corollary 6.2. Let Gbe a simply connected solvable Lie group with a latticeΓ. We supposeAd(G)andAd(Γ)have the same Zariski-closure in Aut(g_C). Then we have an isomorphism

H^∗(A^∗(G/Γ,O_Ads|Γ))∼=H^∗(A^∗(gC,ad_s)).

Proof. LetTbe the Zariski-closure of Ad_s(G). For any 1-dimensional representation V_α of T given by a character α of T, we consider a flat bundle E_α on G/Γ given by the representation α◦Ad_s and the two cochain complexA^∗(G/Γ, E_α) andV

g^∗_C⊗V_αas above. Then sinceα◦Ad_s is Γ-admissible, by Theorem 3.1 we have an isomorphism

H^∗(^

g^∗⊗V_α)∼=H^∗(A^∗(G/Γ, E_α)).

By the definitions of A^∗(G/Γ,O_Ads|Γ) and A^∗(gC,ad_s), the corollary

follows. q.e.d.

7. Extensions

In this section we extend Corollary 6.2 to the case of general solv- manifolds. To do this we consider infra-solvmanifolds which are gener- alizations of solvmanifolds.

7.1. Infra-solvmanifold.Let G be a simply connected solvable Lie group. We consider the affine transformation group Aut(G)⋉Gand the projectionp: Aut(G)⋉G→Aut(G). Let Γ⊂Aut(G)⋉Gbe a discrete subgroup such thatp(Γ) is contained in a compact subgroup of Aut(G) and the quotient G/Γ is compact. We callG/Γ an infra-solvmanifold.

Theorem 7.1. [4, Theorem 1.5] For two infra-solvmanifolds G1/Γ1

and G₂/Γ₂, if Γ₁ is isomorphic to Γ₂, then G₁/Γ₁ is diffeomorphic to G2/Γ2.

7.2. Extensions for infra-solvmanifolds.Let Γ be a torsion-free polycyclic group and H_Γ be the algebraic hull. Then there exists a fi- nite index normal subgroup ∆ of Γ and a simply connected solvable subgroupGofH_Γ such that ∆ is a lattice of G, andGand ∆ have the same Zariski-closure inH_Γ (see [4, Proposition 2.9]). Since the Zariski- closure of ∆ in H_Γ is finite index normal subgroup of H_Γ, this group is the algebraic hull H_∆ of ∆ by the properties in Proposition 2.1. By rank Γ = dimG,H_∆ is also the algebraic hullH_G of G. Hence we have the commutative diagram

G ^//H_∆(=H_G) ^//H_Γ

∆

OO 99s

ss ss ss ss s

//Γ

99

rr rr rr rr rr rr

Since ∆ is a finite index normal subgroup of Γ, by this diagram H_∆ is a finite index normal subgroup of H_Γ. We supposeH_Γ/U_Γ is diagonal- izable. Let T and T^′ be maximal diagonalizable subgroups of H_Γ and H_∆. Then we have decompositionsH_Γ=T⋉U_Γ,H_∆=T^′⋉U_Γ. Since T/T^′ = H_Γ/H_∆ is a finite group, we have a finite subgroup T^′′ of T such that T=T^′′T^′(see [8, Proposition 8.7]).

Lemma 7.2. H_Γ = ΓH∆.

Proof. Consider the quotient q :H_Γ → H_Γ/H_∆. Since Γ is Zariski- dense in H_Γ,q(Γ) is Zariski-dense in H_Γ/H_∆. SinceH_Γ/H_∆is a finite group,q(Γ) =H_Γ/H_∆. Thus we have

ΓH_∆= ΓT^′⋉U_Γ =T^′′T^′⋉U_Γ =H_Γ.

q.e.d.

Let H_Γ = H_Γ(R), T^′ = T^′(R), and T^′′ = T^′′(R). Then by Lemma 2.6 and H_G = H_∆, we have H_Γ = T^′T^′′⋉G. Hence we have Γ ⊂ H_Γ ⊂

Aut(G)⋉G. Since ∆ is a lattice ofGand a finite index normal subgroup of Γ, Γ is a discrete subgroup of Aut(G)⋉G and G/Γ is compact and hence an infra-solvmanifold.

Theorem 7.3. Let Γ be a torsion-free polycyclic group and Γ→H_Γ be the algebraic hull of Γ. Suppose H_Γ/UΓ is diagonalizable. Let u be the Lie algebra of U_Γ. Let ρ be the composition

Γ→H_Γ→H_Γ/U_Γ. Then we have a quasi-isomorphism

^u^∗→A^∗(G/Γ,O_ρ).

Proof. In this proof for a DGA A with a groupG-action, we denote (A)^G the sub-DGA which consists of G-invariant elements of A. Con- sider decompositionsH_Γ=T⋉U_Γ,H_∆=T^′⋉U_Γ as above. Let{V_α} be the set of 1-dimensional representations of T for all charactersα of T. Consider the DGA L

αA^∗(G)⊗V_α with the derivationdgiven by d(ω⊗v_α) = (dω)⊗v_α ω∈A^∗(G), v_α ∈V_α

and the products given by

(ω1⊗vα)∧(ω2⊗v_β) = (ω1∧ω2)⊗(vα⊗v_β).

Then by the definition, we have A^∗(G/Γ,O_ρ) = (M

α

A^∗(G)⊗V_α)^Γ.

Let {V_α^′} and {V_α^′′}be the sets of 1-dimensional representations of T^′ and T^′′for all charactersα^′ ofT^′ and α^′′ ofT^′′. ByT=T^′T^′′, we have {V_α}={V_α^′⊗V_α^′′}. Then we have

H^∗(A^∗(G/Γ,O_ρ)) =H^∗









 M

α^′,α^′′

A^∗(G)⊗(V_α^′ ⊗V_α^′′)





Γ



. Since ∆ is a finite index normal subgroup of Γ, we have

H^∗









 M

α^′,α^′′

A^∗(G)⊗(V_α^′ ⊗V_α^′′)





Γ





∼=H^∗









 M

α^′,α^′′

A^∗(G)⊗(V_α^′ ⊗V_α^′′)





∆





Γ/∆

.