aa r X i v : . [ m a t h . AG ] F e b ON THE TEICHM ¨ULLER STACK OF HOMOGENEOUSSPACE OF SL ( C ) TH´EO JAMIN
Abstract.
Let Γ be a discrete torsion-free co-compact subgroup ofSL ( C ). E. Ghys has shown in [7] that the Kuranishi space of M =SL ( C ) / Γ is given by the germ of the representation variety Hom(Γ , SL ( C ))at the trivial morphism and gave a description of the complex structuresgiven by representations. In this note, we prove that for all admissible representation, i.e. which allow to construct compact complex manifoldby this description, the representation variety (pointed at this represen-tation), leads to a complete family even at singular points. Hence, wewill consider the (admissible) character stack [ R (Γ) a / SL ( C )], where R (Γ) a stands for the open subset formed by admissible representationswith SL ( C ) acting by conjugation on it and show that this quotientstack is an open substack of the Teichm¨uller stack of M . Introduction
Let Γ be a discrete co-compact subgroup of SL ( C ) and let R (Γ) be theassociated SL ( C )-representation variety Hom(Γ , SL ( C )). Take a represen-tation ρ ∈ R (Γ) and consider the following right actionΓ × SL ( C ) SL ( C )( γ, x ) γ • ρ x = ρ ( γ ) − xγ (1)When this action is free and properly discontinuous we say that ρ is ad-missible and we denote by M ρ the corresponding quotient manifold and by R (Γ) a the set of admissible representations. One can show [7, Lemme 2.1,p.115] that R (Γ) a and R (Γ) coincide on a open neigborhood of the trivialmorphism ρ : Γ → Id. Theorem A of [7, p.115] states that the Kuranishispace of SL ( C ) / Γ is the analytic germ of algebraic variety R (Γ) at the triv-ial morphism ρ : Γ → Id. We will show that this result can be extended ina global version:
Theorem 1.
The quotient stack [ R (Γ) a / SL ( C )] where SL ( C ) act by conjugaison is an open substack of the Teichm¨ullerstack of SL ( C ) / Γ . This theorem basically follows from two results, the completeness of thetautological family over the representation and the computation of somegroup of automorphisms of M ρ (which give the isotropy group of a point inthe Teichm¨uller stack). More rigorously Date : February 25, 2021.
Key words and phrases. representation variety, Teichm¨uller space and analytic stacks.
Theorem 2.
For any admissible representation ρ , the deformation {M ρ | ρ ∈ R (Γ) a } → R (Γ) a pointed at ρ is complete. The plan of this article is to review some notions about the geometry ofthe M ρ , such as ( G, X )-structure and the admissibility condition on repre-sentations given by a work of Gu´eritaud, Guichard, Kassel and Wienhard[11] and completed by Tholozan [25]. We will conclude the first part withsome computations of automorphisms groups, in particular Aut ( M ρ ) whichleads to the isotropy group of a point in the character stack and prove Proposition 1.
For any admissible representation ρ , the group Aut ( M ρ ) :=Aut( M ρ ) ∩ Diff ( M ρ ) is equal to the quotient of centralizer of ρ (Γ) in SL ( C ) by {± Id } . Then, in a second part, after some cohomological considerations we demon-strate theorems 2 and 1 and we briefly discuss the differences between thecharacter stack and character variety, as a GIT quotient. We will also givesome local informations through the computation of the Kodaira-Spencermap and results about equivariant transversal slices which, as germs, givesthe Kuranishi space of M ρ . To conclude this paper, we give an example ofapplication. 2. Geometry of M ρ G, X ) -structure. In this section, we recall some general ideas of (
G, X )-structure inspired by Ehresmann and developped by Thurston.A (
G, X )-structure on a manifold M is an atlas of charts with values in themodel space X and whose transition functions are restrictions of elementsof G . A ( G, X )-manifold is a manifold endowed with this structure. Notethat every G -invariant geometric structure g on X , in the sense of Gromov[10], defines a structure (locally isomorphic to g ) on M . For example, aholomorphic metric G -invariant on X defines a holomorphic metric on M .In the case of M = SL ( C ) / Γ, we have an obvious (SL ( C ) × SL ( C ) , SL ( C ))-structure given by left/right translations on SL ( C ) and the Killing formon sl ( C ), which is bi-invariant and non-degenerate, induces a holomorphicmetric on M with constant negative curvature, computed in [7]. We call a( G, X )-morphism between two (
G, X )-manifold, a morphism between mani-folds which is a local diffeomorphism given in charts of the (
G, X )-structureby an element of G . When dealing with the natural morphism from theuniversal covering f M of a ( G, X )-manifold M to X , one recover the usualnotion of developping and holonomy maps: D : f M → X, h : π ( M ) → G which satisfies D ( γ.x ) = h ( γ ) .D ( x ) for γ ∈ π ( M ) and x ∈ f M . The well-known Ehresmann-Thurston principle [26] states that this holonomy mapdefines a local homeomorphism from the set of marked ( G, X )-structures on M to the topological quotient Hom( π ( M ) , G ) /G (see also [8]). In otherword, if M is a ( G, X )-manifold and h ′ a representation close to the ho-lonomy h of M , there exists a ( G, X )-structure on M with holonomy N THE TEICHM ¨ULLER STACK OF HOMOGENEOUS SPACE OF SL ( C ) 3 given by h ′ and two ( G, X )-structures are equivalent if their correspond-ing holonomies are conjugated by a small element in G . But the topologicalquotient Hom( π ( M ) , G ) /G can be quite bad, even non-reduced [19] and wewant to consider it as the stack for the global point of view, see section 4.3.When the developping map is a diffeomorphism, we say that the ( G, X )-structure is complete and we can recover it by taking the quotient of thewhole X by h ( π ( M )). The completeness of such a structure on M isequivalent to the completeness of the holomorphic metric on M , in thesense that all local geodesics can be extended in global geodesics.A result of Tholozan [25, Theorem 3, p.1923] state, in the particular caseof SL ( C ), that the set of complete (SL ( C ) × SL ( C ) , SL ( C ))-structure forma union of connected component in the set of deformation of this structure.Hence, we cannot have a continuous deformation of a complete (SL ( C ) × SL ( C ) , SL ( C ))-structure with non-complete fibers.2.2. Admissibility condition.
We refer to [25], [15] or [11] for details onproperness condition.In order to construct the Kuranishi space of M , Ghys show that theaction (1) is free and properly discontinuous for, at least, representationsthat are close to the trivial one (see [7, Lemma 2.1, p.115]). This result waswidely improved:[15, Theorem 1.3, p.3] Assume that Γ is residually finite and not a torsiongroup. Then ρ ∈ R (Γ) is admissible if, and only if, for all R > , µ ( γ ) − µ ( ρ ( γ )) > R (2) for almost all γ ∈ Γ . where µ : SL ( C ) → R + is the projection of a fixedCartan decomposition of SL ( C ) given by SU A + SU on A + ≃ R + . Thismeans that ρ is admissible if its image ”drift away at infinity” from Γ.In this note, Γ is the fundamental group of a hyperbolic 3-manifold thus itis residually finite and without torsion. This theorem state for example thateach representation with image contained in a compact subset of SL ( C ) isadmissible.Moreover, we have the following key result for this note: Proposition 2. [11, Corollary 1.18]
The set of admissible representations R (Γ) a is a (classical) open in R (Γ) . Remark.
Actually, Kassel’s results are more precise and in particular onecan show that R (Γ) a is not, in general, a Zariski open. It only happen inthe ”rigid case” that is to say when all admissible representations are rigids(i.e. they corresponds to isolated points in R (Γ)), see example 7.2.3. Automorphisms groups.
Let φ be an automorphism of M ρ and e φ its lifted application to the universal cover. We will denote by L g ( resp. R g )the left ( resp. right) translation by g and by ι g the conjugation by g . Lemma 1.
Let φ be an automorphism of M ρ . Then there exists g and δ in SL ( C ) such that e φ = L g ◦ R δ . Note that this is not a diffeomorphism, in opposition to the ”classical” Cartan de-composition, due to the non-unicity in this decomposition. Only the projection on A + isuniquely determined. See [12, Chapitre 9, Theorem 1.1] TH´EO JAMIN
Proof.
This is using a particular case of theorem B in [7].Let ρ ∈ R (Γ) and let φ an automorphism of M ρ . This automorphism φ liftsto a biholomorphism e φ of SL ( C ) such that there exists θ ∈ Aut(Γ) suchthat the Γ-equivariance of e φ is e φ ( γ • ρ x ) = θ ( γ ) • ρ e φ ( x ) , ∀ γ ∈ Γ . (3)Because SL ( C ) has non-trivial center {± Id } , we apply Mostow’s rigidityto PSL ( C ) and lift it to SL ( C ). Hence, we know that there exists Θ acontinuous group automorphism of SL ( C ) and ǫ ∈ Hom(Γ , {± Id } ) suchthat θ = ǫ. Θ | Γ . Since φ is a holomorphic function, Θ as to be so. But, upto conjugation, the only continuous automorphism of SL ( C ) is either theidentity or the complex conjugation. Hence, Θ is an inner automorphism.Consider another representation morphism η ∈ R ( θ (Γ)) such thatΘ( ρ ( γ )) = ǫ ( γ ) .η ( θ ( γ )) , ∀ γ ∈ ΓIt is easy to see that Θ go down to a biholomorphism between M ρ and M η .In fact, Θ( γ • ρ x ) = θ ( γ ) • η Θ( x ) , ∀ γ ∈ ΓNow, let ψ = e φ ◦ Θ − , we get: ψ ( γ • η x ) = γ • ρ ψ ( x ) , ∀ γ ∈ ΓE. Ghys has proved that such biholomorphism has to be a left translationby some element g of SL ( C ) such that η and ρ are conjugate by g .As Θ = ι δ and ψ = L h , for some δ and h in SL ( C ), we have e φ ( x ) = ψ ◦ Θ( x ) = hδxδ − . Back to the equivariance condition (3), we get successively e φ ( γ • ρ x ) = (cid:0) ǫ ( γ ) ι δ ( γ ) (cid:1) • ρ e φ ( x ) hδ (cid:0) ρ ( γ ) − xγ (cid:1) δ − = ρ ( ǫ ( γ ) ι δ ( γ )) − (cid:0) hδxδ − (cid:1) ǫ ( γ ) δγδ − Which simplify in ρ ( ǫ ( γ )) .ρ ( ι δ ( γ )) = ǫ ( γ ) .ι g ( ρ ( γ )) , ∀ γ ∈ Γ(4)where g = hδ . (cid:3) Denote by G ρ the set of pairs ( g, δ ) ∈ SL ( C ) × SL ( C ) for which x L g ◦ R δ ( x ) descends to an automorphism of M ρ , i.e. pairs ( g, δ ) whichsatisfies (4) for some ǫ ∈ Hom(Γ , {± Id } ). As L g ◦ R δ = L − g ◦ R − δ , we willconsider the quotient P G ρ := G ρ / {± Id } Lemma 2.
Let ρ ∈ R (Γ) a then, we have a surjective morphism of group P G ρ → Aut( M ρ ) with kernel given by Deck transformations, i.e. isomorphic to Γ .Proof. The morphism
P G ρ ∋ ( g, δ ) φ ∈ Aut( M ρ ) such that e φ = L g ◦ R δ is surjective by definition of P G ρ and by previous lemma. As SL ( C ) issimply connected, φ is the identity in Aut( M ρ ) if, and only if, e φ is a Decktransformation. That is, e φ ( x ) = γ • ρ x for some γ ∈ Γ, or equivalently( g, δ ) = ( ρ ( γ ) − , γ ). (cid:3) N THE TEICHM ¨ULLER STACK OF HOMOGENEOUS SPACE OF SL ( C ) 5 Lemma 3.
Let ρ ∈ R (Γ) a , then the connected component of the auto-morphism group of M ρ is the projection on PSL ( C ) of the centralizer C SL ( C ) ( ρ (Γ)) of ρ (Γ) in SL ( C ) .Proof. By Mostow’s theorem, Aut(Γ) is discrete and so is the projection onthe second factor of
P G ρ . Hence, we get a injection P G ρ → PSL ( C ) ×{ Id } . It is straighforward to check that the action of Γ on P G ρ induced bycomposition of automorphism is given byΓ × P G ρ → P G ρ , ( γ, ( g, δ )) ( gρ ( γ ) , γ − δ )thus, there is no element of Γ \ { Id } fixing the connected component of P G ρ .Moreover, the condition (4) apply to ( g, Id) is equivalent to require g to bein centralizer of ρ (Γ).Finally, by previous lemma we haveAut ( M ρ ) ≃ ( P G ρ / Γ) ≃ P G ρ ≃ (cid:0) C SL ( C ) ( ρ (Γ)) / {± Id } (cid:1) And one can check that the centralizer C SL ( C ) ( ρ (Γ)) is always connected. (cid:3) As in [17], we denote by Aut ( M ρ ) the group of automorphisms isotopicto identity through C ∞ -diffeomorphisms (eventually not through biholomor-phisms), that is Aut ( M ρ ) = Aut( M ρ ) ∩ Diff ( M ρ ). This group will beused in the next section as it is the isotropy group of a point in the Te-ichm¨uller stack. Note that there exists examples of manifolds X for whichAut ( X ) = Aut ( X ), see [17]. Proposition 3.
Let ρ ∈ R (Γ) a , then Aut ( M ρ ) = Aut ( M ρ ) .Proof. Let φ ∈ Aut ( M ρ ) then by lemma 1, there exists g and δ in SL ( C )such that e φ = L g ◦ R δ . Suppose we have an isotopyΦ : SL ( C ) × [0 , → SL ( C )with Φ( − , t ) ∈ Diff( M ρ ) , Φ( − ,
1) = φ and Φ( − ,
0) is the identity. Obvi-ously, Φ t := Φ( − , t ) have to preserve fibers (and also its inverse) so that thereexists for each t ∈ [0 ,
1] a corresponding automorphism θ t of Γ ≃ π ( M ρ )such that the fibers-preserving condition isΦ t ( γ • ρ x ) = θ t ( γ ) • ρ Φ t ( x ) , ∀ t ∈ [0 , , ∀ γ ∈ ΓBy discretness of Aut(Γ), general continuity argument shows that θ t is con-stant and by assumption on Φ , it is the identity. Hence, with the samenotations as in lemma 1, the lifted continuous automorphisms Θ (such that θ = ǫ Θ) is also the identity and Θ = ι δ = Id. We conclude that δ = Id andthe fiber-preserving condition applied on g gives the same constraint on itthat Aut ( M ρ ) does. (cid:3) representation variety. As Γ arises as a fundamental group of an hyperbolic compact manifold itis finitely presented. Let h γ , · · · , γ n | R , · · · , R m i TH´EO JAMIN be a presentation of Γ. Thus, we define the representation variety R (Γ) := { ( g , · · · , g n ) ∈ SL ( C ) n | R i ( g , · · · , g n ) = Id , ∀ ≤ i ≤ m } This set has a structure of algebraic variety (since SL ( C ) is algebraic) andcarries two topologies, the Zariski topology and the classical one. Notethat, up to isomorphism, the presentation does not change the structure ofalgebraic variety of R (Γ).Let review the Weyl’s constuction. Let ρ t be a smooth path of represen-tation with extremity ρ in R (Γ). By setting c ( γ ) := d ρ t ( γ ) dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ρ ( γ ) − we obtain a cocycle c ∈ Z (Γ , sl ρ ), where sl ρ stands for the Lie algebra sl ( C ) with the structure of Γ-module given by the adjoint representationcomposed by ρ . If ρ t is given by the conjugaison of ρ by a path of matrices A t emanating from Id the corresponding cocycle is a coboundary, i.e. givenby γ X − Ad ρ ( γ ) X with X = d A t dt (cid:12)(cid:12) t =0 . This construction leads to: We have the following isomorphism [16, Proposition 2.2] T Zarρ R (Γ) ≃ Z (Γ , sl ρ ) and the inclusion I ⊂ √I induces an injection T Zarρ R (Γ) red ֒ → Z (Γ , sl ρ )where R (Γ) red is the reduction of the affine scheme R (Γ) and I is the idealdefining the variety R (Γ). This inclusion can be strict, see [13, Example2.18]. As we are interested in compute Kuranishi spaces, which can benon-reduced, it is very important to deal with the scheme R (Γ) and not itsreduction. Remark.
Actually, Kapovich and Millson [19] proved that there are no“local” restrictions on geometry of the SL ( C )-representation schemes of3-manifold groups. 4. Teichm¨uller stack
Preliminary results.
Let ρ ∈ R (Γ) be admissible. The tangent bun-dle of M ρ is identified with the adjoint bundle associated to the SL ( C )-principal bundle π ρ : SL ( C ) → M ρ , the universal cover, that is T M ρ ≃ Ad ρ (SL ( C )) := SL ( C ) × Ad ρ sl ( C ) where the action is given byΓ × SL ( C ) × sl ( C ) −→ SL ( C ) × sl ( C )( γ, ( x, v )) (cid:0) ρ ( γ ) − xγ, Ad ρ ( γ ) − ( v ) (cid:1) (5)Consider the sheaf Θ ρ given by germs of its holomorphic sections. Remarkthat holomorphic sections of this bundle corresponds to holomorphic vectorfields on M ρ , it follows that Θ ρ is exactly the sheaf of germs of holomorphicvector fields on M ρ .It is well known that this tangent bundle, as it is constructed by a repre-sentation of the fundamental group, carries a flat connection (see for example[9]). We also denote by F ρ the sheaf of germs of its flat sections. The inter-est of these sheaves is that H ( M ρ , Θ ρ ) is identified to the Zariski tangent N THE TEICHM ¨ULLER STACK OF HOMOGENEOUS SPACE OF SL ( C ) 7 of the Kuranishi space at the base point and the elements of H ( M ρ , F ρ )corresponds to infinitesimal deformations of the (SL ( C ) × SL ( C ) , SL ( C ))-structure of M ρ . Proposition 4.
Let ρ be an admissible representation. Then, the embeddingof F ρ in Θ ρ induces an isomorphism H ( M ρ , F ρ ) ≃ H ( M ρ , Θ ρ ) and an injection H ( M ρ , F ρ ) ֒ → H ( M ρ , Θ ρ )We will show the successive maps: H i ( M ρ , F ρ ) ≃ H i (Γ , sl ρ ) ֒ → H i (Γ , H ρ ) ≃ H i ( M ρ , Θ ρ ) , i ≥ H ρ is the set of global holomorphic functions with values in sl ρ . Here, sl ρ stands for sl ( C ) endowed with the structure of Γ-module induced by(1), i.e. given by Ad ◦ ρ . Then we will prove that the embedding is actuallyan isomorphism for i = 0 and 1. Lemma 4.
Let ρ ∈ R (Γ) a , then H i ( M ρ , F ρ ) ≃ H i (Γ , sl ρ ) , ֒ → H i (Γ , H ρ ) ≃ H i ( M ρ , Θ ρ ) , i ≥ Proof.
The way to go from ˇCech coholomogy to group cohomology is givenby a well known result in [21, Appendix to §
2, p.22]. Consider the case π ρ : SL ( C ) → M ρ and F is F ρ or Θ ρ . As both sheaves are obtained assheaves of germs of sections of fiber bundles, the pullback sheaves are simplythe corresponding sheaves of germs of sections of the pullback bundles:SL ( C ) × sl ( C ) SL ( C ) × Ad ρ sl ( C )SL ( C ) M ρdπp π ρ π Therefore, the global holomorphic sections ( resp. flat sections) of thetrivial bundle SL ( C ) × sl ( C ) → SL ( C ) is the set of holomorphic ( resp. constant) functions from SL ( C ) to sl ( C ), which we denoted by H ρ ( resp. sl ρ ). The Γ-structure of both sets is given by precomposition by the actionof Γ via • ρ and postcomposition by adjoint representation of ρ , that is H ρ ∋ f (cid:0) γ.f : x Ad ρ ( γ ) − f ( ρ ( γ − ) xγ ) (cid:1) (6)The Cartan’s theorem B states that for any Stein manifold X and anycoherent sheaf F , H p ( X, F ) vanish for p ≥
1. In our context, SL ( C ) is aStein manifold as it is isomorphic to the affine variety ad − bc = 1 in C and the sheaves Θ ρ and F ρ are locally free. We finally end up with theisomorphisms H i (Γ , sl ρ ) ≃ H i ( M ρ , F ρ ) , H i (Γ , H ρ ) ≃ H i ( M ρ , Θ ρ ) , ∀ i ∈ N Finally, as the embedding of sl ρ in H ρ is SL ( C )-equivariant, by generalarguments in group cohomology [4], the applications H i (Γ , sl ρ ) → H i (Γ , H ρ )are injective. (cid:3) TH´EO JAMIN
Proof of proposition 4.
Let as always ρ be an admissible representation.Consider the short exact sequence of Γ-modules0 → sl ρ → H ρ → Ξ ρ := H ρ / (1 ⊗ sl ρ ) → H (Γ , Ξ ρ ) H (Γ , sl ρ ) H (Γ , H ρ ) H (Γ , Ξ ρ ) H (Γ , sl ρ ) H (Γ , H ρ ) δ f δ f By lemma 4, f and f are injective maps, so the coboundary maps δ i arethe zero maps. We end up with the short exact sequence0 → H (Γ , sl ρ ) → H (Γ , H ρ ) → H (Γ , Ξ ρ ) → ρ and Ξ ρ underlying the same abeliangroup (identify with global holomorphic functions from SL ( C ) to sl ( C )with vanishing constant term) but with Γ-module structures induced re-spectively by ρ and ρ , the trivial morphism. From the short exact sequenceof groups: 1 Γ := ker( ρ ) Γ ρ (Γ) ρ and Ξ ρ :0 H ( ρ (Γ) , Ξ Γ • ) H (Γ , Ξ • ) H (Γ , Ξ • ) ρ (Γ) H ( ρ (Γ) , Ξ Γ • ) H (Γ , Ξ • ) res for • = ρ or ρ . Obviously, the action of Γ = ker( ρ ) on Ξ • is the same for • = ρ or ρ , which is given by precomposition by right multiplication by γ ∈ Γ (see (6)). Moreover, E. Ghys showed [7, p.131-132] that a holomorphicfunction invariant by Γ is also invariant by its Zariski closure, which isSL ( C ) by [7, Lemma 5.6]. Hence, these functions are constant and bydefinition of Ξ • , equal to zero. We end up with Ξ Γ ρ = Ξ Γ ρ = 0 and it followsthat H ( ρ (Γ) , Ξ Γ • ) = H ( ρ (Γ) , Ξ Γ • ) = 0. In other words, the restrictionmap, which is the curvy arrow in the previous inflation-restriction exactsequence, is an isomorphism either for ρ and ρ . to summarize, we have thefollowing isomorphisms H (Γ , Ξ ρ ) ≃ H (Γ , Ξ ρ ) res : H (Γ , Ξ ρ ) ∼ −→ H (Γ , Ξ ρ ) ρ (Γ) res : H (Γ , Ξ ρ ) ∼ −→ H (Γ , Ξ ρ ) ρ (Γ) = H (Γ , Ξ ρ )Theorem 4 . H (Γ , Ξ ρ ) = 0. Withthe previous isomorphisms and this result we have that H (Γ , Ξ ρ ) = 0 asannounced and (7) gives the desired isomorphism. (cid:3) N THE TEICHM ¨ULLER STACK OF HOMOGENEOUS SPACE OF SL ( C ) 9 Higher obstructions.
We want to describe deformations of M ρ over( C , U in M ρ , we consider biholomorphisms f : W → W ′ where W, W ′ ⊂ M ρ × C are open which contains U × { } . We consider theset of such biholomorphisms which preserves the fibers M × { p } and suchthat f | M ρ ×{ } = Id. We define the sheaf Λ ρ by Λ ρ ( U ) as the quotient of thisset by identify two biholomorphisms which coincides on a neighborhood of U × { } . The important fact is that The space H ( M ρ , Λ ρ ) is identify to the set of classes of germs of deforma-tions of M ρ parametrized by ( C ,
0) [6].This sheaf is naturally filtered:For each open U , we consider the set of biholomorphisms of Λ nρ ( U ) whichare tangent to the identity up to the order n − nρ thecorresponding sheaf. For all n ≥
1, we denote by Q nρ the quotient sheafΛ ρ / Λ n +1 ρ . It is well know that (see [20])ker (cid:0) Q n +1 ρ → Q nρ (cid:1) ≃ Θ ρ Thus, we get the following exact sequence of sheaves0 → Θ ρ → Q n +1 ρ → Q nρ → H ( M ρ , Q n ) are called n -th order deformation of M ρ . Proof of Theorem 2.
Assume that up to order n , the set of classes of germsof deformation of M ρ over C is given by germs of deformations of the rep-resentation ρ by cochains { c i } ni =1 via ρ n := ρ ( c , ··· ,c n ) : γ exp n X i =1 c i ( γ ) t i ! ρ ( γ )Then, we can equip g n := sl ( C [ t ] / ( t n +1 )) with the Γ-structure given byAd ρ n . We denote g ρ n n the Lie algebra with its Γ-structure.Interpreting B nρ := H (SL ( C ) , π ∗ Q nρ ) as a set of global sections of n -jets,we get an injection of Γ-modules g ρ n n → B nρ . These maps induce a morphismbetween exact sequences0 sl ρ g ρ n n g ρ n − n − H ρ B nρ B n − ρ H (Γ , sl ρ ) H (Γ , g ρ n n ) H (Γ , g ρ n − n − ) H (Γ , sl ρ ) H (Γ , H ρ ) H (Γ , B nρ ) H (Γ , B n − ρ ) H (Γ , H ρ ) H ( M ρ , Θ ρ ) H ( M ρ , Q nρ ) H ( M ρ , Q n − ρ ) H ( M ρ , Θ ρ ) i i i δ i ≀ ≀ ≀ ≀ ˇ δ Proposition 4 says that i is an isomorphism and i is a monomorphism.By assumption, i is an isomorphism. The four-lemma states that i issurjective, thus it is an isomorphism.Let U n ( γ ) := (cid:0) ddt ρ n ( γ ) (cid:1) ρ n ( γ ) − and f n the corresponding element in H ( M ρ , Q n − ρ ). If f n can be extended to order ( n + 1) then the class ofˇ δ ( f n ) in H ( M ρ , Θ ρ ) is zero. The class of [ δU n ] ∈ H ( γ, sl ρ ) is then alsozero which is an equivalent condition to the existence of a cochain c n +1 suchthat ρ n +1 := ρ ( c , ··· ,c n +1 ) is a morphism up to order n +1 (see [14, Proposition3.1]).Inductively, a deformation of the complex structure of M ρ parametrizedby ( C ,
0) is given by a formal deformation of the representation ρ : ρ ∞ : γ exp ∞ X i =1 c i ( γ ) t i ! ρ ( γ )The existence of a convergent solution follows directly from a result of Artin[1], as in [14, Proposition 3.6]. This show us that the representation varietyis complete at each point that corresponds to an admissible representationand therefore this conclude the proof of theorem 2. (cid:3) Teichm¨uller stack.
The Newlander-Nirenberg Theorem [22] says thata structure of a complex manifold on M is equivalent to a a C ∞ bundleoperator J on the tangent bundle of M such that J = − Id , and [ T , , T , ] ⊂ T , Where T , = { v + iJ v | v ∈ T M ⊗ C } is the subbundle given by the eigen-vectors of J with eigenvalue − i of the complexified tangent bundle of M .We denote by I ( M ) the set of complex structure on the C ∞ manifold M diff (forgetting its natural complex structure). Note that the group Diff( M ρ ) of C ∞ -diffeomorphisms of M act on I ( M ) asDiff( M ) × I ( M ) → I ( M ) , ( f, J ) → ( df ) − ◦ J ◦ df The Teichm¨uller space of M , denoted T ( M ), is given by the quotient of I ( M ) by the action of the subgroup Diff ( M ) of Diff( M ) formed by dif-feomorphisms isotopic to the identity. There exists example of manifold M for which (see [18, Example 12.3]), this topological space does not admit astructure of analytic space. But, under some assumption on the dimensionof the group of automorphisms of M , the Teichm¨uller has a structure ofArtin stack and we shall review some definitions of its construction. N THE TEICHM ¨ULLER STACK OF HOMOGENEOUS SPACE OF SL ( C ) 11 Teichm¨uller and character stack.
Let An C be the category of com-plex analytic space. In this note, a stack is a stack in groupoids over the site An C in the sense of [24, Definition 8.5.1]. Let M be a C ∞ manifold whichadmits a complex structure J . We construct the Teichm¨uller stack T ( M )of M as the category whose • objects are deformations of M which are Diff ( M )-bundle when con-sidered in the C ∞ categorie.That is, smooth and proper morphism π : X → B , between objects X, B ∈ ( An C ), which is diffeomorphic, when considered as real ana-lytic spaces, to a bundle E → B with fiber M and structural groupreduced to Diff ( M ). • morphisms are cartesian diagrams X ′ XB ′ B π ′ πf where the isomorphism f ∗ X ≃ X ′ induced a Diff ( M )-isomorphismof the smooth bundle structure in the category of real analytic spaces.If V is an open subset of I ( M ), we can define in the same way T V ( M )the Teichm¨uller stack of M for complex structures belonging to V , that isobjects are smooth morphisms π : X → B as well but the complex structureson fibers of π belongs to V . For more details see [18].We want to define a map i : R (Γ) a → I ( M ) which sends an admissiblerepresentation ρ to the bundle operator corresponding to the natural com-plex structure of M ρ . So we can define T R (Γ) a ( M ) the Teichm¨uller stack of M for complex structures arising as M ρ for some ρ ∈ R (Γ) a . The way toconstruct i is the following. Take ρ ∈ R (Γ) a and consider the frame bundle F ( M diff ρ ) of M diff ρ the C ∞ manifold underlying M ρ . Points in this bundleover x ∈ M ρ are identified with linear isomorphisms R → T x M diff ρ . Notethat the tangent bundle Ad ρ (SL ( C )) (see (5)) gives a natural subbundleof F ( M diff ρ ) by C -linear isomorphisms C → T x M diff ρ and the correspond-ing reduction of the structural group is exactly the C ∞ bundle operator J ρ corresponding to the complex structure of M ρ . Hence, we define i by i : ρ J ρ .Naturally, we define Definition.
The character stack ( resp. admissible character stack ) is thequotient stack [ R (Γ) / SL ( C )] , ( resp. [ R (Γ) a / SL ( C )])over the site An C .Obviously the admissible character stack is a substack of the characterstack in the sense of [2, Definition 6.9, p.112], that is a full saturated sub-category of the character stack which is also a stack. Remark.
It is important to notice that the character stack see as a stackover the site
Sch of schemes is algebraic but the admissible character stackis not since R (Γ) a is not a Zariski open in R (Γ) (see remark 2.2). However, both of them are analytic stacks and this explains why we have to work onthe analytic site rather than an algebraic one. Theorem 3.
The admissible character stack is an open substack of theTeichm¨uller stack of M .Proof. The completeness theorem 2 implies that there exists an open V a ⊂I ( M ) of complex structures M ρ given by representations ρ ∈ R (Γ) a . Hence,locally we know that any deformation X → B in the Teichm¨uller stack T V a ( M ) can be seen as a SL ( C )-principal bundle P → B with an SL ( C )-equivariant map p : P → R (Γ) a , that is an element of the (admissible)character stack.Denote by X → R (Γ) a the tautological family above R (Γ) a , that is X isobtained as the quotient of SL ( C ) × R (Γ) a by the action of Γ :Γ × SL ( C ) × R (Γ) a → SL ( C ) × R (Γ) a ( γ, x, ρ ) ( ρ ( γ ) − xγ, ρ )We restrict our attention on isomorphism between SL ( C )-torsors so we onlylook at tautological families. Let B be an analytic space and φ, ψ : B →R (Γ) a analytic maps such that the induced tautological families φ ∗ X → B and ψ ∗ X → B are isomorphic in the Teichm¨uller stack. So there exists ananalytic map F : φ ∗ X → ψ ∗ X such that φ ∗ X ψ ∗ X B B Fπ φ π ψ Id is a cartesian diagram and F is a Diff -bundle isomorphism. Lifting F to ananalytic map e F : SL ( C ) × R (Γ) a → SL ( C ) × R (Γ) a , we see that on eachfibers e F (cid:12)(cid:12)(cid:12) π − φ ( b ) ( x, ρ ) = ( ι g ( x ) , ι g ◦ ρ ), where g ∈ Aut ( M ρ ) ≃ C SL ( C ) ( ρ (Γ))by proposition 3. Doing this on each fibers, we obtain a map f : R (Γ) a → SL ( C )such that e F ( x, ρ ) = ( ι f ( ρ ) ( x ) , ι f ( ρ ) ◦ ρ ). This application obviously satisfies s ( f ( ρ ) , ρ ) = ρ and t ( x, f ) = ι f ( ρ ) ( ρ ), where the map s and t are the sourceand the target map of the Lie groupoidSL ( C ) × R (Γ) a ι ⇒ p R (Γ) a that is s is the projection on the second factor and t is the SL ( C )-action ofconjugation on R (Γ) a . (cid:3) We easily deduce the following corollary, which is a reformulation of thetheorem 1:
Corollary 1.
The Lie groupoidSL ( C ) × R (Γ) a ι ⇒ p R (Γ) a is an atlas for T R (Γ) a ( M ). N THE TEICHM ¨ULLER STACK OF HOMOGENEOUS SPACE OF SL ( C ) 13 Remark.
However, it is an open question to know if this open substack isa union of connected components of the Teichm¨uller stack, or if it is not,what is the boundary of this substack.5.
Characters stack versus character variety
We want to emphasize the use of stack instead of GIT quotient. Asremarked before (see remark 4.4), R (Γ) a is not a Zariski open when b (Γ) =0, hence it is not possible to form the quotient (in the sense of geometricinvariant theory) nor the algebraic stack associated. Actually, the situationseems to be even worse, for instance, let π : R (Γ) → R (Γ) // SL ( C ) be theaffine quotient and take ρ ∈ R (Γ) a such that its orbit is not closed, there isno reason for π − ( π ( ρ )) to contains only admissible representations, even ifwe don’t have any example of such situation.We want to underline the fact that the character stack contains moreinformations as the character variety. To do so, we can look at fibers of themorphism φ : [ R (Γ) a / SL ( C )] → [ R (Γ) // SL ( C )]where X (Γ) := [ R (Γ) // SL ( C )] stands for the stack associated to the affinequotient R (Γ) // SL ( C ), over the site An C . Let χ be a point in X (Γ), thenthe it is easy to see that the preimage of χ by φ is formed by all SL ( C )-principal bundles O ( ρ ) → { ρ } such that π ( ρ ) = χ . Whenever χ is obtainedas the trace character of two non conjugated representations ρ and η , thepreimage of φ contains two non biholomorphic families in the Teichm¨ullerstack. In other words, there are points in the Teichm¨uller stack which areidentified in the character variety.6. Kodaira-Spencer map
In this section, we will show that the Kodaira-Spencer map associated tothe natural deformation over R (Γ) a is surjective at each point.Consider the variety e X := SL ( C ) × R (Γ) a and its quotient X by Γ givenby the action Γ × e X → e X ( γ, ( x, ρ )) ( ρ ( γ − ) xγ, ρ )The natural projection p : X → R (Γ) a is a deformation of complex struc-tures with M ρ ⊂ X above ρ ∈ R (Γ) a . Let ρ be an admissible representationand V a Stein open neigborhood in R (Γ) a containing ρ . One can considerthe fundamental exact sequence of this deformation restricted to V → Θ | p − ( V ) → Π | p − ( V ) → Υ | p − ( V ) → p ,Π is the sheaf of projectable vector fields and Υ the sheaf of germs of vectorfields on R (Γ).And for the infinitesimal neigborhood of ρ , this sequence tends to0 → Θ | M ρ → Π | M ρ → Υ | M ρ → KS ρ of this deformation isthe connecting homorphism of the long exact sequence associated to it. Let V be an small Stein neigborhood of ρ in R (Γ). Since e X | V = SL ( C ) × V is a product of Stein manifolds it is also Stein. The pullback sheaf is notnecesseraly coherent, but e X | V is an open in affine and thus noetherian so thepullback of any coherent sheaf if also coherent. Then by Cartan’s theorem H i ( e X | V , π ∗ Θ | V ) = { } for i >
0, in particular for i = 1. Hence, on theinfinitesimal neighborhood of ρ , the sequence0 → H (SL ( C ) , π ∗ Θ | M ρ ) → H (SL ( C ) , π ∗ Π | M ρ ) → H (SL ( C ) , π ∗ Υ | M ρ ) → −→ H Γ ρ −→ H (Γ , H (SL ( C ) , π ∗ Π | M ρ )) −→ Z (Γ , sl ρ ) g KS ρ −→ H (Γ , H ρ )(11)where Z (Γ , sl ρ ) ≃ T ρ R (Γ) ≃ H (Γ , H (SL ( C ) , π ∗ Υ | M ρ )). This exact se-quence is isomorphic to the long exact sequence associated to (8) in ˇCechcohomology by [21]. All diagrams formed by this isomorphism is commu-tative and this is why we called g KS ρ the map above the Kodaira-Spencermap: · · · Z (Γ , sl ρ ) H (Γ , sl ρ ) · · ·· · · H ( M ρ , Υ M ρ ) H ( M ρ , Θ ρ ) · · · ∼ g KS ρ ∼ KS Proposition 5.
Let ρ be an admissible representation. Then H ( M ρ , Π | M ρ ) ≃ H (SL ( C ) , π ∗ Π | M ρ ) Γ ≃ sl ( C ) Proof.
We show that the vector space of projectable vector fields, i.e. vectorfields that descends to the quotient, is isomorphic to sl ( C ). Let G γ : e X → e X ( x, ρ ) ( ρ ( γ − ) xγ, ρ )Then a vector field V on e X | ρ is projectable if, and only if, ( G γ ) ∗ V = V, ∀ γ ∈ Γ. We decompose V in ( v, c ) where v is a vector field on SL ( C ) and c is acocycle in Z (Γ , sl ρ ) ≃ T ρ R (Γ) a . From( G γ ) ∗ (cid:18) v ( x )0 (cid:19) = (cid:18) Ad ρ ( γ ) − ( v ( x ))0 (cid:19) and ( G γ ) ∗ (cid:18) c (cid:19) = (cid:18) c ( γ − ) c (cid:19) it follows that V is Γ-invariant, or projectable, if c ( γ − ) + Ad ρ ( γ ) − ( v ( x )) = v ( ρ ( γ ) − xγ )(12)Or equivalently c ( γ ) = v ( ρ ( γ ) xγ − ) − Ad ρ ( γ ) ( v ( x )) N THE TEICHM ¨ULLER STACK OF HOMOGENEOUS SPACE OF SL ( C ) 15 Let us remark that if v is constant, then c has to be a coboundary to forma projectable vector field. Hence, for γ, δ ∈ ker( ρ ), we get c (( γδ ) − ) = v ( xγδ ) − v ( x )= c ( γ − ) + c ( δ − )= v ( xγ ) − v ( x ) + v ( xδ ) − v ( x )Putting these lignes together, we obtain v ( x ) = v ( xγ ) + v ( xδ ) − v ( xγδ )(13)If v is fixed by the action described by this formula, see as an action ofthe subgroup ker( ρ ) × ker( ρ ) in SL ( C ) × SL ( C ), then it is fixed by theZariski closure of it. By lemma [7, Lemme 5.6], the Zariski closure of ker( ρ )is SL ( C ) and the Zariski closure of a product is the product of the Zariskiclosure (true for general topology), so thatker( ρ ) × ker( ρ ) Zar = SL ( C ) × SL ( C )As constant vector fields together with the corresponding coboundary form asuitable projetable vector field, one can suppose that v (Id) = 0 sl ( C ) . Thus,(13) implies (for x = Id) that v is an holomorphic morphism from SL ( C )to its Lie algebra.Finally, as sl ( C ) is an abelian group, v factorizes throughSL ( C ) ab = SL ( C ) / [SL ( C ) , SL ( C )] = { Id } and v is globally constant and by assumption equal to 0. (cid:3) Proposition 6.
For any ρ ∈ R (Γ) a , the Kodaira-Spencer map KS ρ is sur-jective.Proof. By (11), it is equivalent to prove that ] KS ρ : H ( π ∗ Υ | M ρ ) Γ → H (Γ , H ρ )is surjective.By proposition 5, we know that H ( π ∗ Π | M ρ ) Γ is isomorphic to the image of sl ( C ) under the following map f : sl ( C ) → sl ( C ) ⊕ B (Γ , sl ρ ) X → ( X, φ : γ X − Ad ρ ( γ ) X ) . We obtain the following diagram of exact sequences0 sl ( C ) ρ (Γ) sl ( C ) ⊕ B (Γ , sl ρ ) Z (Γ , sl ρ ) H (Γ , sl ρ )0 H Γ ρ H ( π ∗ Π | M ρ ) Γ H ( π ∗ Υ | M ρ ) Γ H (Γ , H ρ ) ≀ ≀ ≀ g KS ρ ≀KS ρ We conclude that the Kodaira-Spencer map is nothing else than the naturalprojection Z (Γ , sl ρ ) → H (Γ , sl ρ ) which is surjective. (cid:3) Equivariant slices and Kuranishi spaces.
The geometry of a rep-resentation variety is in general very complicated and it is a hard game tofind an SL ( C )-equivariant slice which, as a germ at ρ ∈ R (Γ), gives byproposition 2 the Kuranishi space of ρ . However, few words can be said ingeneral. Corollary 2.
Let ρ ∈ R (Γ) a , then any complex analytic space transverseto the orbits passing through ρ gives (as a germ) the Kuranishi space of M ρ . Proof.
The Kodaira-Spencer is surjective and two conjugated representa-tions gives the same manifold M ρ up to biholomorphism. (cid:3) In opposition to this imprecise result, more acurate results can be statein some particular cases. For instance, assume that the first betti numberof Γ is one and let h γ , · · · , γ n | Ri be a presentation of Γ such that the natural projection p : Γ → Γ ab ≃ Z sends γ to 1. Proposition 7.
For all representation ρ such that ρ ( γ ) is semi-simple in SL ( C ) then there exists an ´etale slice V at ρ in R (Γ) such that T ρ V isisomorphic to H (Γ , sl ρ ) . Since the orbit of such representation is closed [23, Theorem 30], thisproposition is given by the Luna’s slice theorem as in [3, Proposition 2.8].7.
Example.
We give an example which emphasize the main contribution of this thiswork compare to [7]. The Weeks manifold M W is known to has the smallestvolume among hyperbolic 3-manifold with first Betti number 0. Among allits properties, it is compact, closed, oriented, arithmetic which turn it intoa particular case of interest in this note. We also have a presentation of itsfundamental group π ( M W ) = h a, b | a b a b − ab − , a b a − ba − b i and one discrete and faithful SL ( C )-representation of this group is given by η ( a ) = (cid:18) x x − (cid:19) , η ( b ) = (cid:18) x r x − (cid:19) with r = 2 − x − x − and 1 + 2 x − x + 2 x + x = 0 (see [5]). Up to thechoice of x we fix Γ W to be the image of π ( M W ) under this representation.With this presentation, it is clear that this manifold has betti numberequal to 0 and it follows that the trivial representation is an isolated pointin representation variety and that SL ( C ) / Γ W is rigid (in the sense that theKuranishi space is a point). But it is not globally rigid.Consider the case of a representation of Γ W with abelian image. Directcomputations gives that, up to conjugation, it is given by ρ n,m ( a ) = (cid:18) ω m ω − m (cid:19) , ρ n,m ( b ) = (cid:18) ω n ω − n (cid:19) N THE TEICHM ¨ULLER STACK OF HOMOGENEOUS SPACE OF SL ( C ) 17 for ω k = e ikπ and m, n = 0 , · · · ,
4. Those representations ρ n,m have obvi-ously images in a compact subgroup of SL ( C ) and the properness criterion(2) implies that the induced action is admissible.Let treat the non abelian case. It is a general fact that ρ ( a ) and ρ ( b ) canbe of the form ρ ( a ) = (cid:18) x x − (cid:19) , ρ ( b ) = (cid:18) y r y − (cid:19) with x, y = 0. According to [5, p. 24-25], the second relation in the presen-tation of π ( M W ) implies that y is either equal to x or x − . Moreover, if r = 0 and x is not a root of unity, the representation is faithful and it followsthat it is not admissible (as remarked in the proof of proposition 5). Fol-lowing the computations of [5], none of these cases have a solution (except x = 1, y = 1 and r = 0 which leads to an abelian representation) and weconclude that the admissible locus of R (Γ W ) contains only representationswith abelian images.By the open criterion of R (Γ W ) in the Teichm¨uller stack, we know that |T (SL ( C ) / Γ W ) | (the underlying topological space) contains at least 25 iso-lated points as connected components, corresponding to the representations ρ n,m , n, m = 0 , · · · , Remark.
There are plenty of examples, but few of them are actually ”com-putable” since the complexity in finding a discrete and faithful representa-tion increase with the complexity of the relations. However, we can mentionanother example: the manifold v ,
1) (in the notations of SnapPy).This manifold has a first Betti number of 2. It is a closed, oriented, compactand hyperbolic 3-fold which fiber over the circle. We have a presentation ofits fundamental group with SnapPy and by same arguments as above, onecan show that the quotient stack { A, B ∈ SU × SU | [ A, B ] = Id } / SL ( C )is in the Teichm¨uller space and corresponds to a subset of the set of abelianrepresentations. But it is a hard game to find other admissible representa-tions since the numerical criterion (2) not allow us to do real computations. Acknowledgement
I would like to express my deep gratitude to Laurent
Meersseman andMarcel
Nicolau , my research supervisors, for their patient guidance, en-thusiastic encouragement and useful critics of this research work. I wouldalso like to thank Alberto
Verjovsky , Joan
Porti and Nicolas
Tholozan for interesting discussions.
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Th´eo JAMIN, Laboratoire Angevin de REcherhe en MAth´ematiques, Uni-versit´e d’Angers, Universit´e d’Angers, F-49045 Angers Cedex, France
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