Symplectic coordinates on the deformation spaces of convex projective structures on 2-orbifolds
SSYMPLECTIC COORDINATES ON THE DEFORMATIONSPACES OF CONVEX PROJECTIVE STRUCTURES ON2-ORBIFOLDS
SUHYOUNG CHOI AND HONGTAEK JUNG
Abstract.
Let O be a closed orientable 2-orbifold of negative Euler charac-teristic that has only cone singularities. Huebschmann constructed the Atiyah-Bott-Goldman type symplectic form ω on the deformation space C p O q of con-vex projective structures on O . We show that the deformation space C p O q ofconvex projective structures on O admits a global Darboux coordinates sys-tem with respect to ω . To this end, we show that C p O q can be decomposedinto smaller symplectic spaces. In the course of the proof, we also study thedeformation space C p O q for an orbifold O with boundary and construct thesymplectic form on the deformation space of convex projective structures on O with fixed boundary holonomy. Introduction
Motivation.
Let O be a closed orientable 2-orbifold of negative Euler char-acteristic that has only cone singularities. In this paper we study the deformationspace C p O q of convex projective structures on O and its symplectic nature.It is a classical result that the Teichm¨uller space T p S q of a given orientableclosed hyperbolic surface S with genus g admits a global Darboux coordinatessystem [33]. In this case the Fenchel-Nielsen coordinates system is the one wherethe Weil-Petersson symplectic form ω W P can be written in the standard form: ω W P “ g ´ ÿ i “ d (cid:96) i ^ d θ i . The Teichm¨uller space T p S q for a closed surface S can be generalized in manydirections. One way to do this starts from regarding T p S q as the set of conjugacyclasses of discrete faithful representations of π p S q into PSL p R q . Since there is aunique (up to conjugation) irreducible representation PSL p R q Ñ PSL n p R q for each n , we can canonically embed T p S q into Hom p π p S q , PSL n p R qq{ PSL n p R q . We callthe image of this embedding the Fuchsian locus and each element in the Fuchsianlocus is called a Fuchsian character. The Hitchin component Hit n p S q is then de-fined by a connected component of Hom p π p S q , PSL n p R qq{ PSL n p R q that containsa Fuchsian character. This “higher Teichm¨uller space” has been studied by manypeople including Hitchin [15], Goldman [12, 10], Choi-Goldman [5], Labourie [23],and Guichard-Wienhard [13].Due to Goldman [10], we know that the Hitchin components also carry thesymplectic form so called Atiyah-Bott-Goldman form ω ABG which is a naturalgeneralization of ω W P . Their symplectic nature has been extensively studied bySun-Zhang [28], Sun-Zhang-Wienhard [27] and they proved that p Hit n p S q , ω ABG q admits a global Darboux coordinates system. Using a different method, the authors[7] also obtained the same result for Hit p S q . Suhyonug Choi is supported in part by NRF grant 2019R1A2C108454412.Hongtaek Jung is supported by IBS-R003-D1. a r X i v : . [ m a t h . G T ] J u l SUHYOUNG CHOI AND HONGTAEK JUNG
Another direction is to generalize a surface S to an orbifold O . Thurston’s lec-ture note [29], for instance, contains some results about the Teichm¨uller spaces oforbifolds.In this paper we proceed in both directions and study symplectic nature of the“Hitchin components” of orbifolds via convex projective geometry. Our approach isgeometric and works mostly for PSL p R q . Recently Alessandrini-Lee-Schaffhauser[1] gave a generalization of Hitchin components of orbifolds by using equivariantHiggs bundle theory.1.2. Statement of results.
Let O be a compact orientable hyperbolic cone 2-orbifold. One can think of O as a quotient H of the hyperbolic plane by a discretesubgroup π orb1 p O q “ Γ Ă PSL p R q . Denote by C p O q the deformation space of convexprojective structures on O . If B O has b ą B “ t B , ¨ ¨ ¨ , B b u of hyperbolic elements in PSL p R q anddefine C B p O q to be a subspace of C p O q whose i th boundary holonomy is in B i . SeeSection 2 for the precise definitions.We can relate C p O q to an algebraic object. For this, we defineHom s p Γ , PSL n p R qq : “ t ρ P Hom p Γ , PSL n p R qq | Z p ρ q “ t u and ρ is irreducible u . Our target object is the character variety X n p Γ q , which defined by X n p Γ q : “ Hom s p Γ , PSL n p R qq{ PSL n p R q . It is known that X n p Γ q is a smooth manifold and X p Γ q contains C p O q as an opensubspace.If O has boundary components ζ , ¨ ¨ ¨ , ζ b , we choose B as above and defineHom B s p Γ , PSL n p R qq : “ t ρ P Hom s p Γ , PSL n p R qq | ρ p ζ i q P B i u and X B n p Γ q : “ Hom B s p Γ , PSL n p R qq{ PSL n p R q . It is also known that, for a suitably chosen B , X B n p Γ q is an embedded submanifoldof X n p Γ q and C B p O q is an open subspace of X B p Γ q .In our first main theorem, we construct a symplectic form on X B n p Γ q . In fact weconstruct an equivariantly closed 2-form on a neighborhood of Hom B s p Γ , PSL n p R qq which is reduced to a symplectic form on X B n p Γ q . Theorem 1.1.
Let O be a compact oriented hyperbolic 2-orbifold that has only conesingularities. Let G “ PSL n p R q . Choose a set B of conjugacy classes of hyperbolicelements in SL n p R q . Then there is a neighborhood of Hom B s p π orb1 p O q , G q and anequivariantly closed 2-form ω such that the restriction of ω to Hom B s p π orb1 p O q , G q descends to a symplectic form ω O on X B n p π orb1 p O qq . In particular for n “ , ω O isa symplectic form on C B p O q . Goldman [10] studied the symplectic structure on X n p π p S qq for a closed hyper-bolic surface S . After his work, Huebschmann [18], Karshon [20] also gave theirown constructions of the symplectic form on X n p π p S qq . Huebschmann [17] ex-tended his work to X n p π orb1 p O qq where O is a closed orientable hyperbolic orbifold.Guruprasad-Huebschmann-Jeffrey-Weinstein [14] generalized the construction to X B n p π p S qq for a compact orientable hyperbolic surface S . Our work is mainlybased on [10], [17] and [14] and covers all previous cases.We first construct ω O using the Poincar´e duality for the group pair p π orb1 p O q , S q where S “ tx z y , ¨ ¨ ¨ , x z b yu is the set of cyclic subgroups of π orb1 p O q each of which isgenerated by a boundary component z i P π orb1 p O q . We show that p π orb1 p O q , S q is a P D R -pair, meaning that there is a fundamental class r O , B O s P H p π orb1 p O q , S ; R q YMPLECTIC COORDINATES 3 such that the cap product H p π orb1 p O q , S ; R q Ñ H p π orb1 p O q ; R q “ R is an isomor-phism. This 2-form is the pull-back of the well-known Atiyah-Bott symplectic formon X B n p π p X O qq for some compact surface X O .To find the equivariantly closed 2-form on a neighborhood of Hom B s p π orb1 p O q , G q ,we use the equivariant de Rham complex and Cartan-Maurer calculus. Then weshow that after taking the reduction ω O coincides up to sign with the symplecticform on X B n p π orb1 p O qq constructed as above.Choose pairwise disjoint essential simple closed curves or full 1-suborbifolds t ξ , ¨ ¨ ¨ , ξ m u such that the completion of each connected component O i of O z Ť mi “ ξ i has also negative Euler characteristic. Say ξ , ¨ ¨ ¨ , ξ m are simple closed curvesand ξ m ` , ¨ ¨ ¨ , ξ m ` m are full 1-suborbifolds, so that m “ m ` m . By usingour previous work [7], we show that there is an Hamiltonian R M -action, M “ m ` m , on C B p O q with a moment map µ such that the Marsden-Weinsteinquotient µ ´ p y q{ R M exists for each y in the image of µ .For suitably chosen B , ¨ ¨ ¨ , B l , there is a map SP y : µ ´ p y q{ R M Ñ C B p O q ˆ ¨ ¨ ¨ ˆ C B l p O l q that is induced from restrictions to each subgroup π orb1 p O i q Ă π orb1 p O q . Theorem1.1 implies that the right hand side admits the symplectic form ω O ‘ ¨ ¨ ¨ ‘ ω O l .Then we have the following symplectic decomposition theorem: Theorem 1.2.
Let O be a compact oriented 2-orbifold of negative Euler character-istic that has only cone singularities. Let t ξ , ¨ ¨ ¨ , ξ m u be a set of pairwise disjointessential simple closed curves or full 1-suborbifolds such that the completions of con-nected components O i of O z Ť mi “ ξ i have also negative Euler characteristic. Thenthe map SP y : µ ´ p y q{ R M Ñ C B p O q ˆ ¨ ¨ ¨ ˆ C B l p O l q defined above is a symplectomorphism. The essential part of Theorem 1.2 is that the splitting map also preserves somesymplectic information. This is a consequence of the following more general localdecomposition theorem, whose proof is given in section 5.1.
Theorem 1.3.
Let O be a compact oriented 2-orbifold of negative Euler character-istic that has only cone singularities. Let t ξ , ¨ ¨ ¨ , ξ m u be pairwise disjoint essentialsimple closed curves or full 1-suborbifolds such that the completions of connectedcomponents O , ¨ ¨ ¨ , O l of O z Ť mi “ ξ i have also negative Euler characteristic. Foreach i , let ι i : π orb1 p O i q Ñ π orb1 p O q be the inclusion. Let r ρ s P X B n p π orb1 p O qq be suchthat, with a nice choice of B i , r ρ ˝ ι i s P X B i n p O i q for each i . Then ω O pr u s , r v sq “ l ÿ i “ ω O i p ι ˚ i r u s , ι ˚ i r v sq for all r u s , r v s P H p π orb1 p O q , S ; g q where S “ tx z y , ¨ ¨ ¨ , x z b y , x ξ y , ¨ ¨ ¨ , x ξ m yu . Strategy for proving Theorem 1.3 is the same as that of our previous paper [7].The essential difference is that the splitting can happen along a full 1-suborbifold.We will exhibit how the fundamental cycle of our orbifold is decomposed when wedo splitting along a full 1-suborbifold.By following the framework of [7], we can show our main theorem:
Theorem 1.4.
Let O be a closed oriented 2-orbifold of negative Euler character-istic that has only cone singularities. Then the deformation space C p O q of convexprojective structures admits a global Darboux coordinates system with respect to ω O . SUHYOUNG CHOI AND HONGTAEK JUNG
Organization of the paper.
Section 2 is about basic concepts related to orb-ifolds, convex projective structures, deformation spaces and character varieties. Wealso review geometric splitting operations which play an important role throughoutthis paper.In section 3, we review the theory of group cohomology and parabolic groupcohomology. To compute the parabolic group cohomology we introduce variousprojective resolutions. By using these resolutions and their relations, we show thatthe orbifold fundamental groups have a finite length projective resolution. This factwill be intensively used in the next section.In section 4, we construct the Atiyah-Bott-Goldman type symplectic form ω onthe deformation space of convex projective structures on a given compact oriented2-orbifold O of negative Euler characteristic that has only cone singularities as wellas an equivariant closed 2-form on a neighborhood of Hom B s p π orb1 p O q , G q . Using thePoincar´e duality we construct the 2-form ω on X B n p π orb1 p O qq and show that this2-form is the pull-back of some symplectic form. Then we utilizes the equivariantde Rham complex to find an equivariantly closed 2-form on a neighborhood ofHom B s p π orb1 p O q , G q . We argue that this 2-form restricts on Hom B s p π orb1 p O q , G q to ´ ω after the reduction.Section 5 is devoted to prove our main theorem. The proof goes almost parallelto that of our previous paper [7] for closed surface. We prove the local and globaldecomposition theorems. By using this decomposition theorem we can find a fiberbundle structure whose fibers are Lagrangian. Then apply our version of action-angle principle to get the main result. Acknowledgements.
The authors would like to appreciate J. Huebschmann forhelpful communication. The second author also wishes to give special thanks toKyeongro Kim.2.
Orbifolds and their convex projective structures
In this section, we introduce our two main objects of interest; orbifolds andconvex projective structures.We give a concise introduction to orbifolds in section 2.1. We also summarizeimportant facts on 2-orbifolds, which include the Euler characteristic, orbifold fun-damental groups, classification of singularities.In section 2.2, we introduce convex projective structures and their deformationspace. We also introduce the space of representations and character varieties. Atheorem of Choi-Goldman, Theorem 2.6, provides the link between the deformationspaces and the character varieties.In the last section 2.3 we introduce splitting operations that split a given convexprojective orbifold into smaller convex projective orbifolds. One should note thatthis operations are not only topological but also geometric. The splitting operationsplay an essential role throughout this paper.Most material in this section is based on Thurston [29], Choi [4], Goldman [12],and Choi-Goldman [6].2.1.
Orbifolds.
An orbifold is a generalization of a manifold that allows somesingular points. We adopt the chart and atlas style approach to orbifolds. Anothermore abstract definition using ´etale groupoid can be found, for instance, in [3].Let X be a paracompact Hausdorff topological space. An orbifold chart p U, Γ , ϕ q of x P X consists of an open neighborhood U of x , a finite group Γ acting on anopen subspace r U of R n as diffeomorphisms and a Γ-equivariant map ϕ : r U Ñ U that induces the homeomorphism ϕ : r U { Γ Ñ U . In this case, we say that p r U , Γ q isa model pair of x . Two orbifold charts p U , Γ , ϕ q and p U , Γ , ϕ q are said to be YMPLECTIC COORDINATES 5 compatible if each point x P U X U has a chart p V, Γ , ϕ q with, for each i “ ,
2, aninjective homomorphism f i : Γ Ñ Γ i and a f i -equivariant embedding ψ i : r V Ñ r U i such that the following diagram r V ψ i (cid:47) (cid:47) (cid:15) (cid:15) r U i (cid:15) (cid:15) r U i { f i p Γ q (cid:15) (cid:15) r V { Γ (cid:47) (cid:47) ϕ – (cid:15) (cid:15) r U i { Γ iϕ i – (cid:15) (cid:15) V inclusion (cid:47) (cid:47) U i commutes for each i “ ,
2. An orbifold atlas on X is a set of compatible orbifoldcharts that covers X . An n -orbifold is a paracompact Hausdorff topological spaceequipped with a maximal orbifold atlas. Unless otherwise stated, we assume thatthe underlying topological space is connected.We can also define an orbifold with boundary by allowing model pairs of the form p r U , Γ q where r U is an open subset of R n ` “ tp x , ¨ ¨ ¨ , x n q P R n | x n ě u . In this casewe define the boundary B O of an orbifold O with boundary by the set of x P O that has a model pair of the form p R n ` , t uq . Note that, in general, B O is not thesame as B X O where X O is the underlying space of O .We say that an orbifold is compact, orientable, or closed if the underlying space X O has the corresponding properties.Suppose that we are given an orbifold O . An orbifold O with an orbifold map f : O Ñ O is called a covering of O if each point x P O has a orbifold chart p U, Γ , ϕ q such that for each component U i of f ´ p U q , there is a subgroup Γ i of Γand an orbifold diffeomorphism ψ : r U { Γ i Ñ U i induced from a map ψ : r U Ñ U i .We also request that ψ is compatible with f in the sense that f ˝ ψ “ ϕ . A covering r O Ñ O of O that has the universal lifting property is called a universal covering orbifold of O . It is known that every orbifold has a universal covering orbifold r O .We call the group of deck transformations of the universal cover r O Ñ O the orbifoldfundamental group and denote it by π orb1 p O q .Finally, we present some facts on 2-orbifolds. Let O be a 2-orbifold. ‚ Each singular point x of O falls into three types. We say that x is – a cone point of order r if x has a model pair p R , Z { r q where Z { r actson R as rotations. – a mirror if x has a model pair p R , Z { q where Z { R asreflections. – a corner-reflector of order n if x has a model pair p R , D n q where thedihedral group D n of order 2 n . ‚ We can define the
Euler characteristic of O with cone points of order r , ¨ ¨ ¨ , r c and corner-reflectors of order n , ¨ ¨ ¨ , n k r by χ p O q “ χ p X O q ´ k r ÿ i “ ˆ ´ n i ˙ ´ c ÿ i “ ˆ ´ r i ˙ where X O is the underlying topological space of O . ‚ The genus of O is the genus of its underlying space. SUHYOUNG CHOI AND HONGTAEK JUNG ‚ We say that O is hyperbolic if O can be realized as a quotient orbifold H { Γ for some discrete subgroup Γ Ă PSL p R q . Compact orientable O ishyperbolic if and only χ p O q ă Definition 2.1. A cone 2-orbifold is a 2-orbifold that has only cone type singular-ities.2.2. Convex projective structures.
We can equip some additional structure ona given orbifold. Our main interest is the real projective structures on 2-orbifold andtheir deformation space. This deformation space is closely related to the Hitchincomponent of the character variety. Throughout this section O denotes a compact2-orbifold.The real projective plane RP is the quotient space of p R zt uq{ „ where theequivalence relation is given by x „ y if and only if x “ λy for some nonzero realnumber λ . Hence each point of RP represents a 1-dimensional subspace in R . Ageodesic line in RP represents a 2-dimensional subspace in R .Recall that the projective linear group PSL p R q acts on RP . This action inducesa simply transitive action on p RP q p q “ tp x , x , x , x q P p RP q | x i ‰ x j for all i ‰ j u . An element g P SL p R q is hyperbolic if g is conjugate to a matrix of the form ¨˝ λ λ
00 0 λ ˛‚ with λ ą λ ą λ ą
0. We denote by
Hyp ` the set of hyperbolic elements inSL p R q .A hyperbolic element g has exactly three distinct fixed points in RP . The at-tracting fixed point is the one corresponding to the largest eigenvalue λ . Therepelling fixed point is the attracting fixed point of g ´ . The remaining fixed pointis called the saddle. A principal line of a hyperbolic g is the line passing throughattracting and repelling fixed points. A principal segment is a geodesic line segmentjoining the attracting fixed point and the repelling fixed point. A geodesic γ in RP is principal if γ is a principal line for some g P Hyp ` .Now we define what we mean by a real projective structure on O . Definition 2.2. A (real) projective structure on a given 2-orbifold O consists of ‚ a homomorphism h : π orb1 p O q Ñ PSL p R q called the holonomy and ‚ a h -equivariant orbifold map dev : r O Ñ RP called the developing map .If B O ‰ H , we further require that the following principal boundary condition holdsfor each boundary component ζ Ă B O : ‚ the holonomy h p ζ q is hyperbolic and ‚ if r ζ is a h p ζ q -invariant component of the lift of ζ , dev p r ζ q is the principalline of h p ζ q .The pair p h , dev q is called the developing pair . Two developing pairs p h , dev q and p h , dev q are equivalent if there is g P PGL p R q such that dev “ g dev and h p¨q “ g h p¨q g ´ .One special property that we are interested in is the convexity of projectivestructures which is defined as follow YMPLECTIC COORDINATES 7
Definition 2.3.
A real projective structure p h , dev q on a 2-orbifold O is convex if dev p r O q is convex domain in RP , i.e., there is an affine patch of RP containing dev p r O q and dev p r O q is a convex subset of it.Hence by a convex projective structure on O we mean a developing pair p h , dev q such that dev p r O q is convex. Lemma 2.4.
Let O be a compact orientable convex projective cone 2-orbifold ofnegative Euler characteristic. Then every closed essential simple closed curve ξ isisotopic to a unique closed principal geodesic in O . Moreover, h p ξ q is a hyperbolicelement. Now we can define the deformation space RP p O q of projective structures on agiven compact 2-orbifold O . Consider the set S p O q of all projective structures on O . We give the C -topology on developing maps to make S p O q a topological space.We say two projective structures p h , dev q and p h , dev q are isotopic if there is adiffeomorphism f : r O Ñ r O commuting with π orb1 p O q such that dev “ dev ˝ f and h “ h . We define RP p O q ˚ to be the quotient space of S p O q by isotopy. The deformation space RP p O q of projective structures on O is the quotient of RP p O q ˚ by equivalence relation given in Definition 2.2.We also define the deformation space C p O q of convex projective structures on O as a subspace of RP p O q that consists of convex projective structures.To relate C p O q with an algebraic object, we define so called the character varietyof Γ : “ π orb1 p O q . The space of representations Hom p Γ , PSL n p R qq can be regarded asan affine algebraic set of PSL n p R q N for some N . In general Hom p Γ , PSL n p R qq mayhave a lot of singularities. In Corollary 4.3, we will see that a sufficient conditionfor ρ P Hom p Γ , G q being smooth is H p Γ; g ρ q “
0. Here g ρ is the Lie algebra ofPSL n p R q regarded as a R Γ-module with the Γ-action given by Ad ˝ ρ .ConsiderHom s p Γ , PSL n p R qq : “ t ρ P Hom p Γ , PSL n p R qq | Z p ρ q “ t u and ρ is irreducible u . This subspace Hom s p Γ , PSL n p R qq Ă Hom p Γ , PSL n p R qq is nonempty and open. Thefollowing lemma suggests that Hom s p Γ , PSL n p R qq is smooth. Lemma 2.5. H p Γ; g ρ q “ for all ρ P Hom s p Γ , PSL n p R qq .Proof. For the sake of convenience, we write simply g in place of g ρ .Take a finite index torsion-free normal subgroup Γ of Γ. Then Γ is the funda-mental group of a surface of negative Euler characteristic, say S .Because Γ is a normal subgroup of finite index, we have the transfer map tr : H ˚ p Γ ; g q Ñ H ˚ p Γ; g q . It is known that H ˚ p Γ; g q res Ñ H ˚ p Γ ; g q tr Ñ H ˚ p Γ; g q is an isomorphism where res : H ˚ p Γ; g q Ñ H ˚ p Γ ; g q is the restriction map inducedfrom the inclusion Γ Ñ Γ. Regarding this fact, we refer the readers to Weibel [30,Lemma 6.7.17]. In particular res is injective and tr is surjective.If S has nonempty boundary, then Γ is a free group and we have H p Γ ; g q “ H p Γ; g q Ñ H p Γ ; g q “ H p Γ; g q “ S is closed. Since g admits an Ad-invariant nondegeneratebilinear form, we have the isomorphism D : H p Γ ; g q Ñ H p Γ ; g q ˚ – H p Γ ; g q arisen from the Poincar´e duality. Recall that the restriction map res : H p Γ; g q Ñ H p Γ ; g q is the same as the edge map H p Γ; g q Ñ H p Γ { Γ ; H p Γ ; g qq in theLyndon-Hochschild-Serre spectral sequence. Therefore we can identify H p Γ; g q SUHYOUNG CHOI AND HONGTAEK JUNG with the image of the restriction map in H p Γ ; g q and this image is an invariantsubspace under the action of the finite group Γ { Γ . It follows that H p Γ; g q res Ñ H p Γ g q D Ñ H p Γ { Γ ; H p Γ ; g qq Ă H p Γ ; g q is injective. The Lyndon-Hochschild-Serre spectral sequence yields H p Γ { Γ ; H p Γ ; g qq – H p Γ; g q . Because H p Γ; g q “ H p Γ; g q “ (cid:3) Now we take the quotient of Hom s p Γ , PSL n p R qq by the conjugation action ofPSL n p R q . This action of PSL n p R q on Hom s p Γ , PSL n p R qq is free as Z p ρ q “ t u .Moreover, due to Kim [22, Lemma 1], the action on Hom s p Γ , PSL n p R qq is proper.This allows us to define the smooth manifold X n p Γ q : “ Hom s p Γ , PSL n p R qq{ PSL n p R q . The space Hom s p π orb1 p O q , PSL p R qq contains special elements. Let O be a closedorientable hyperbolic cone 2-orbifold. A Fuchsian representation is a representa-tion that is of the form π orb1 p O q ρ F Ñ PO p , q ã Ñ PSL p R q where ρ F : π orb1 p O q Ñ PO p , q is a discrete faithful representation that corresponds to an element ofthe Teichm¨uller space. We denote by C T p π orb1 p O qq the connected component ofHom s p π orb1 p O q , PSL p R qq that contains the Fuchsian representations.Now assume that B O ‰ H . We define Fuchsian representations as we did for theclosed orbifolds case. Then C T p π orb1 p O qq is defined to be the set of representations inHom s p π orb1 p O q , PSL p R qq that can be continuously deformed to a Fuchsian througha path ρ t such that ρ t p ζ q is hyperbolic for all t and for all boundary component ζ . Theorem 2.6 (Choi-Goldman [6]) . Let O be a compact orientable cone 2-orbifoldof negative Euler characteristic. Then the holonomy map Hol : C p O q Ñ C T p π orb1 p O qq{ PSL p R q taking rp h , dev qs to r h s is a homeomorphism onto its image.Remark . One can say that C T p π orb1 p O qq{ PSL p R q is the Hitchin component foran orbifold O . Another approach to this object is recently made by Alessandrini-Lee-Schaffhauser [1].In light of Theorem 2.6, we identify C p O q with its image in C T p π orb1 p O qq{ PSL p R q from now on. By pulling back the smooth structure of C T p π orb1 p O qq{ PSL p R q by themap Hol , we can equip C p O q with the smooth structure.Suppose that O has boundary components ζ , ¨ ¨ ¨ , ζ b . Let Γ : “ π orb1 p O q . Let B “ t B , ¨ ¨ ¨ , B b u be a set of conjugacy classes in SL n p R q . For later use, we defineHom B s p Γ , PSL n p R qq : “ t ρ P Hom s p Γ , PSL n p R q | ρ p ζ i q P B i for i “ , , ¨ ¨ ¨ , b u and X B n p Γ q : “ Hom B s p Γ , PSL n p R qq{ PSL n p R q . We also define C B p O q in the same manner. Then C B p O q is a submanifolds of X B p Γ q .We end this section by mentioning the following lemma. Lemma 2.8.
The holonomy ρ of RP p O q has the following properties (a) ρ is irreducible. (b) If, in addition, ρ is a holonomy of C p O q then Z p ρ q “ t u and ρ is stronglyirreducible. YMPLECTIC COORDINATES 9
Proof. (a). It is a result of Choi-Goldman [6] that ρ p π orb1 p O qq does not fix a point.To show that ρ p π orb1 p O qq does not preserve any line in RP , suppose on the contrarythat there is a line l that is invariant under the action of ρ p π orb1 p O qq . Since O doesnot have mirror points, l is a line joining two fixed points p, q of a hyperbolic elementin π orb1 p O q . Then p and q must be fixed by ρ p π orb1 p O qq which is a contradiction.(b). The first assertion follows from (a) and a modified version of Schur’s lemma:If g P GL n p R q is a automorphism on an irreducible π orb1 p O q -module that has atleast one eigenvector then g is a scalar. Recall that if ρ P C p O q then we can lift ρ : π orb1 p O q Ñ PGL p R q to a representation into GL p R q . (See [22, Lemma 5])For the second assertion of (b), we note that every finite index subgroup Γ of π orb1 p O q admits a further finite index subgroup which is isomorphic to a surfacegroup say Γ . Since ρ | Γ is in the PSL p R q -Hitchin component, it is irreducible.Therefore ρ | Γ itself must be irreducible. (cid:3) Splitting, pasting and folding.
Let O be a convex projective compact ori-entable cone 2-orbifold with genus g , b boundary components and c cone points oforders r , ¨ ¨ ¨ , r c . Let r h s be a (conjugacy class) of the holonomy representation forthe convex projective structure on O .Choose a canonical presentationΓ : “ π orb1 p O q “ x x , y , ¨ ¨ ¨ , x g , y g , z , ¨ ¨ ¨ , z b , s , ¨ ¨ ¨ , s c | r “ r “ ¨ ¨ ¨ “ r c “ y where r “ g ź i “ r x i , y i s b ź j “ z j c ź k “ s k and where r i “ s r i i , i “ , , ¨ ¨ ¨ , c .We also choose conjugacy classes B “ t B , ¨ ¨ ¨ , B b u and consider C B p O q the de-formation space of convex projective structures on O with fixed boundary holonomies, h p z i q P B i , i “ , , ¨ ¨ ¨ , b .2.3.1. Splitting along a full 1-suborbifold.
Suppose that O has c b ě s , ¨ ¨ ¨ , s c b be the corresponding order two generators of π orb1 p O q . A full 1-suborbifold is an embedded 1-suborbifold which is a segment withboth endpoints being mirror points. Topologically a full 1-suborbifold is the imageunder the projection r O Ñ O of a segment that joins two fixed points of two order2 elements.Let us say s and s are order two elements of π orb1 p O q . Let p and p be cor-responding order two cone points in O . The convex projective structure on O tellsus that h p s s q is hyperbolic and is conjugate to a matrix of the form ¨˝ λ λ ´ ˛‚ for some λ ą
1. The geodesic segments in dev p r O q joining the isolated fixed pointsof h p s q and h p s q are contained in the principal line r ξ of h p s s q . A principal full1-suborbifold ξ between p and p is, by definition, the image of a principal segmentof h p s s q in O under the covering map r O Ñ O .Suppose that ξ is any full 1-suborbifold between order two cone points p and p . Due to Lemma 4.1 of Choi-Goldman [6] we can isotope ξ to a principal full 1-suborbifold provided O z ξ has negative Euler characteristic. We use the same letter ξ to denote this principal full 1-suborbifold. Then the convex projective structurewith principal boundary condition on O induces the canonical convex projectivestructure with principal boundary condition on the completion of O z ξ . We thus get the splitting map SP : C p O q Ñ C p O z ξ q that equips O z ξ with this canonical convexprojective structure.This map can be best understood in terms of its effect on the holonomy rep-resentation. To describe this, we observe that there is π orb1 p O z ξ q is a subgroup of π orb1 p O q and it admits a presentation x x , y , ¨ ¨ ¨ , x g , y g , z , ¨ ¨ ¨ , z b , z b ` , s , s , ¨ ¨ ¨ , s c | g ź i “ r x i , y i s b ` ź i “ z i c ź i “ s i ,s r “ ¨ ¨ ¨ “ s r c c “ y where the inclusion ι : π orb1 p O z ξ q Ñ π orb1 p O q is induced from the set map on thegenerating set x i ÞÑ x i , y i ÞÑ y i , s i ÞÑ s i z i ÞÑ z i , i “ , , ¨ ¨ ¨ , bz b ` ÞÑ s s . This inclusion induces the smooth map ι ˚ : X p π orb1 p O qq Ñ X p π orb1 p O z ξ qq definedby r ρ s ÞÑ r ρ ˝ ι s . Then the following relation holds Hol ˝ SP “ ι ˚ ˝ Hol . Splitting along a essential simple closed curves.
Let X O be an underlyingspace of O . Let Σ O be the set of singularities and let X O : “ X O z Σ O . By an essential simple closed curve in O we mean a simple closed curve in X O that is nothomotopic to a point, a boundary component or a puncture. Let ξ be an essentialsimple closed curve in O . If each connected component of O z ξ has negative Eulercharacteristic, then by Lemma 4.1 of Choi-Goldman [6], ξ can be isotoped to aprincipal closed geodesic, which is denoted again by ξ . Because h p ξ q is in Hyp ` ,the convex projective structure with principal boundary condition on O induces thecanonical convex projective structure with principal boundary condition on O z ξ .The effect of splitting along ξ varies by connectedness of O z ξ . If O z ξ is dis-connected and has two connected components O and O , we have the splittingmap SP : C p O q Ñ C p O q ˆ C p O q that assigns the induced convex projectivestructure on each O i . At the same time, we can decompose π orb1 p O q into the amal-gamated product of π orb1 p O q “ π orb1 p O q ‹ ξ π orb1 p O q . Let ι i denote the inclusions π orb1 p O i q Ñ π orb1 p O q that induce the map ι ˚ i : X p π orb1 p O qq Ñ X p π orb1 p O i qq asabove. Then we have p Hol ˆ Hol q ˝ SP “ p ι ˚ ˆ ι ˚ q ˝ Hol . If ξ is non-separating so that O z ξ “ : O remains connected, we have the splittingmap SP : C p O q Ñ C p O q and the decomposition of π orb1 p O q into the HNN-extension π orb1 p O q‹ ξ,ξ K . Denote by ι : π orb1 p O q Ñ π orb1 p O q the inclusion with the inducedmap ι ˚ : X p π orb1 p O qq Ñ X p π orb1 p O qq . Then we have the relation Hol ˝ SP “ ι ˚ ˝ Hol .2.3.3.
Splitting along a family.
Now we treat a general situation. Let t ξ , ¨ ¨ ¨ , ξ m u be a set of pairwise disjoint essential simple closed curves or full 1-suborbifolds.Suppose that each connected component of O z Ť mi “ ξ i has also negative Eulercharacteristic. As we did above, we can isotope ξ , ¨ ¨ ¨ , ξ m to pairwise disjoint prin-cipal simple closed geodesics or principal full 1-suborbifolds. The resulting prin-cipal geodesics and principal full 1-suborbifolds will be denoted by the same let-ters ξ , ¨ ¨ ¨ , ξ m . Let O , ¨ ¨ ¨ , O l be the completions of connected components of O z Ť mi “ ξ i . The convex projective structure with principal boundary on O induces YMPLECTIC COORDINATES 11 the convex projective structure with principal boundary on each O i . By the con-struction, we know that for each i , the boundary holonomies of O i are hyperbolic.Therefore this splitting operation induces the splitting map SP : C p O q Ñ C p O q ˆ ¨ ¨ ¨ ˆ C p O l q . As we described above, the holonomy h i of the induced convex projective struc-ture on each O i is the restriction of h to the subgroup π orb1 p O i q Ă π orb1 p O q . Inother words if we denote the inclusion π orb1 p O i q Ñ π orb1 p O q by ι i , we have thecommutative diagram C p O q Hol (cid:15) (cid:15) SP (cid:47) (cid:47) C p O q ˆ ¨ ¨ ¨ ˆ C p O l q Hol ˆ¨¨¨ˆ Hol l (cid:15) (cid:15) X p π orb1 p O qq ι ˚ (cid:47) (cid:47) X p π orb1 p O qq ˆ ¨ ¨ ¨ ˆ X p π orb1 p O l qq where ι ˚ is given by ι ˚ pr h sq “ pr h ˝ ι s , r h ˝ ι s , ¨ ¨ ¨ , r h ˝ ι l sq .We will see that the map SP is a submersion but is not an injection. Pastingand folding are the processes that produce a section of SP .3. Preliminaries on group cohomology
In this section we define group pairs and their parabolic cohomology group tostudy the local structure of character varieties through the cohomology of the orb-ifold fundamental group.We introduce four types of projective resolutions. By using these resolutions, weprove Proposition 3.8 which states that the orbifold fundamental group of com-pact orientable cone orbifolds with negative Euler characteristic has a finite lengthprojective resolution. In Proposition 3.9 we give a characterization of the paraboliccohomology in terms of the bar resolution. They are preliminary results for theconstruction of the symplectic form.Throughout this section we will use G to denote the Lie group PSL n p R q with itsLie algebra g if we do not need to specify n . A representation ρ : Γ Ñ G induces alinear representation Ad ˝ ρ : Γ Ñ GL p g q and this turns g into a (left) R Γ-module.We denote this R Γ-module by g ρ or just g if the representation ρ is clear from thecontext.3.1. Group pairs and the parabolic group cohomology.
The parabolic groupcohomology was first considered by Weil [31]. Here we recall the construction of theparabolic cohomology for the purpose of describing the tangent space of relativecharacter variety X B n p Γ q . We adopt the description for the parabolic group coho-mology from [14].Let Γ be a finitely generated group. By a group pair , we mean a pair p Γ , S q “p Γ , t Γ , ¨ ¨ ¨ , Γ b uq of a group Γ and a collection S of its finitely generated subgroupsΓ , ¨ ¨ ¨ , Γ b . Definition 3.1.
Let p Γ , S q “ p Γ , t Γ , ¨ ¨ ¨ , Γ b uq be a group pair. An auxiliary reso-lution for p Γ , S q consists of the following data: ‚ A projective resolution R ˚ Ñ R of the trivial R Γ-module R ‚ Projective resolutions A i ˚ Ñ R of the trivial R Γ i -module R for each i “ , , ¨ ¨ ¨ , b . These resolutions have the property that A ˚ : “ ‘ bi “ R Γ b Γ i A i ˚ is a direct summand of the projective resolution R ˚ Ñ R .Let p R ˚ , A i ˚ q be an auxiliary resolution for a group pair p Γ , S q . Since A ˚ is adirect summand of R ˚ , we have a chain complex of R Γ-modules R ˚ { A ˚ Ñ R . Forany R Γ-module M , we apply the Hom Γ p´ , M q to R ˚ { A ˚ to get the chain complex p Hom Γ p R ˚ { A ˚ , M q , δ q . We define the relative group cohomology H ˚ p Γ , S ; M q withcoefficients in M to be the cohomology of this chain complex p Hom Γ p R ˚ { A ˚ , M q , δ q .Consider the following long exact sequence associated to the exact sequence0 Ñ A ˚ Ñ R ˚ Ñ R ˚ { A ˚ Ñ ¨ ¨ ¨ Ñ b à i “ H p Γ i ; M q Ñ H p Γ , S ; M q Ñ H p Γ; M q Ñ b à i “ H p Γ i ; M q Ñ ¨ ¨ ¨ . Definition 3.2.
The 1st parabolic cohomology H p Γ , S ; M q of the group pair p Γ , S q with coefficients in M is defined by the image of H p Γ , S ; M q in H p Γ; M q .We mostly use the Γ-module g ρ for some representation ρ : Γ Ñ G as ourcoefficients.We discuss a functorial property of the parabolic group cohomology. Let p Γ , S q be a group pair. Let S be a subset of S . We can construct auxiliary resolutions p R ˚ , A ˚ q for p Γ , S q and p R ˚ , A q for p Γ , S q such that A is a direct summandof A ˚ . Hence we have a chain map R ˚ { A Ñ R ˚ { A ˚ which induces the map H p Γ , S ; M q Ñ H p Γ , S ; M q . Note that this map is not necessarily injective. How-ever, we have injectivity in the parabolic group cohomology. Lemma 3.3.
Let p Γ , S q be a group pair. Let S be a subset of S Then the inducedmap H p Γ , S ; M q Ñ H p Γ , S ; M q is injective.Proof. Let Z p Γ , S ; M q : “ ker p Hom Γ p R { A , M q Ñ Hom Γ p R { A , M qq be thespace of relative cocycles. Define Z p Γ , S ; M q in the same fashion. Then we knowthat Z p Γ , S ; M q and Z p Γ , S ; M q are both embedded in the space of 1-cochainsHom Γ p R , M q . Let B p Γ; M q : “ Image p Hom Γ p R , M q Ñ Hom Γ p R , M qq be thespace of 1-coboundaries. Then the injection Z p Γ , S ; M q Ñ Z p Γ , S ; M q gives riseto the injection Z p Γ , S ; M q B p Γ; M q Ñ Z p Γ , S ; M q B p Γ; M q . The left hand side is, by definition, H p Γ , S ; M q and the right hand side equals H p Γ , S ; M q . This yields the lemma. (cid:3) Models for the group cohomolgy.
The purpose of this section is to con-struct four different models for the group cohomology, each of which has its ownbenefits.Throughout this section, O denotes a compact connected oriented cone orbifoldof negative Euler characteristic with genus g , b boundary components and c conepoints. We choose a base point p in the interior of O and fix a presentation(2) Γ : “ π orb1 p O , p q“ x x , y , ¨ ¨ ¨ , x g , y g , z , ¨ ¨ ¨ , z b , s , ¨ ¨ ¨ , s c | r “ r “ ¨ ¨ ¨ “ r c “ y where r “ g ź i “ r x i , y i s b ź j “ z j c ź k “ s k and where r i “ s r i i , i “ , , ¨ ¨ ¨ , c . YMPLECTIC COORDINATES 13
The bar resolution.
The bar resolution is the standard resolution of the trivial R Γ-module R for any group Γ. For the sake of completeness, we briefly remind thereaders of its construction.We first define the n -chain B n p Γ q , n ą R Γ-module based onthe elements of the form (cid:74) g | g | ¨ ¨ ¨ | g n (cid:75) where g , ¨ ¨ ¨ , g n P Γ zt u . Exceptionally wedefine B p Γ q : “ R Γ r p s . The differential is given by Bp (cid:74) g | g | ¨ ¨ ¨ | g n (cid:75) q “ g (cid:74) g | ¨ ¨ ¨ | g n (cid:75) ` n ´ ÿ i “ p´ q i (cid:74) g | ¨ ¨ ¨ | g i g i ` | ¨ ¨ ¨ | g n (cid:75) ` p´ q n (cid:74) g | ¨ ¨ ¨ | g n ´ (cid:75) for n ą Bp (cid:74) g (cid:75) q “ g (cid:74) p (cid:75) ´ (cid:74) p (cid:75) . It is well-known and is proven in Lemma 3.4 under the general setting that(3) ¨ ¨ ¨ Ñ B p Γ q Ñ B p Γ q Ñ B p Γ q Ñ R Ñ The groupoid bar resolution.
A groupoid is a small category whose mor-phisms are all invertible. For a morphism f in a groupoid, we denote by o p f q and t p f q its domain and range respectively.Given a compact orientable cone orbifold O of negative Euler characteristic, wechoose a point p i at each boundary component ζ i . Define r Γ “ π orb1 p O , t p , p , ¨ ¨ ¨ , p b uq to be the fundamental groupoid of O with base p , ¨ ¨ ¨ , p b . We fix, for each i , a path w i from p to p i . Let ζ i be the boundary loop based at p i so that the complementof x , y , ¨ ¨ ¨ , x g , y g , s , ¨ ¨ ¨ , s c , w , ¨ ¨ ¨ , w b , ζ , ¨ ¨ ¨ , ζ b in the underlying space X O isa union of open disks. We then have the following presentation of r Γ r Γ “ x x , y , ¨ ¨ ¨ , x g , y g , w , ¨ ¨ ¨ , w b , ζ , ¨ ¨ ¨ , ζ b , s , ¨ ¨ ¨ , s c | g ź i “ r x i , y i s b ź i “ w i ζ i w ´ i c ź i “ s i “ s r “ ¨ ¨ ¨ “ s r c c “ y . There is a natural groupoid morphism ext : Γ Ñ r Γ given by x i ÞÑ x i , y i ÞÑ y i , z i ÞÑ w i ζ i w ´ i , s i ÞÑ s i . On the other hand, we have the groupoid morphism ret : r Γ Ñ Γ defined by x i ÞÑ x i , y i ÞÑ y i , w i ÞÑ , ζ i ÞÑ z i , s i ÞÑ s i . Now we define the n -chain r B n p Γ q , n ą R Γ-module over theelements of the form (cid:74) g | g | ¨ ¨ ¨ | g n (cid:75) where g i P r Γ zt u such that t p g i q “ o p g i ` q . The 0-chain r B p Γ q is given by the free R Γ-module on t (cid:74) p (cid:75) , ¨ ¨ ¨ , (cid:74) p b (cid:75) u . The differential B : r B n p Γ q Ñ r B n ´ p Γ q is defined by Bp (cid:74) g | g | ¨ ¨ ¨ | g n (cid:75) q “ ret p g q (cid:74) g | ¨ ¨ ¨ | g n (cid:75) ` n ´ ÿ i “ p´ q i (cid:74) g | ¨ ¨ ¨ | g i g i ` | ¨ ¨ ¨ | g n (cid:75) ` p´ q n (cid:74) g | ¨ ¨ ¨ | g n ´ (cid:75) for n ą Bp (cid:74) g (cid:75) q “ ret p g q (cid:74) t p g q (cid:75) ´ (cid:74) o p g q (cid:75) . Lemma 3.4.
Define the augmentation r ε : r B p Γ q Ñ R by r ε p ř a i (cid:74) p i (cid:75) q “ ř ε p a i q where ε : R Γ Ñ R is the usual augmentation map given by ε p ř n i γ i q “ ř n i . Thenthe complex r B ˚ p Γ q r ε Ñ R Ñ is exact.Proof. It is straightforward to check that
BB “
0. To prove the exactness, we usethe homotopy operator h n : r B n p Γ q Ñ r B n ` p Γ q defined by h n p g (cid:74) g | g | ¨ ¨ ¨ | g n (cid:75) q “ (cid:74) g g | g | g | ¨ ¨ ¨ | g n (cid:75) if g ‰
10 if g “ g g “ ext p g q w i for the generator w i of r Γ that joins p and the origin o p g q of g . Observe that ret p g g q “ g and p g ret p g qq g “ g g g for g P Γ, and g , g P r Γ.Using this observation, one can check that h B ` B h “ Id which proves the exactnessof r B ˚ p Γ q . (cid:3) Therefore we have another free resolution r B i p Γ q (4) ¨ ¨ ¨ Ñ r B i p Γ q Ñ r B i ´ p Γ q Ñ ¨ ¨ ¨ Ñ r B p Γ q Ñ r B p Γ q Ñ R Ñ . The geometric resolution I.
We consider another model which is based on thecanonical cellular decomposition of O . This resolution carries more clear geometricpicture of the group π orb1 p O q . Although this type of resolution may be classical andwell-known, its construction is somehow scattered all over the old literatures, forinstance, [24], [9], [16], and [25]. Here, we give the specific construction that fits forour situation.Our free resolution consists of free R Γ-modules G k “ R Γ r r , ¨ ¨ ¨ , r c s , for k ě G “ R Γ r r , r , ¨ ¨ ¨ , r c s , G “ R Γ r x , y , ¨ ¨ ¨ , x g , y g , z , ¨ ¨ ¨ , z b , s , ¨ ¨ ¨ , s c s , and G “ R Γ r p s . To distinguish the generators of each module from group elements, we enclose thegenerators of modules by the double bracket (cid:74) ¨ (cid:75) .Now we introduce the Fox derivatives [8], the essential ingredients to define thedifferential. Definition 3.5.
Let F be the free group on generators v , ¨ ¨ ¨ , v n . The Fox deriva-tives BB v i : R F Ñ R F , i “ , , ¨ ¨ ¨ , n are R -linear operators with the followingproperties ‚ For every j , B v j B v i “ i “ j i ‰ j . ‚ For every x, y P R F , B xy B v i “ B x B v i ε p y q ` x B y B v i where ε : R F Ñ R is the augmentation map. ‚ (The mean value property) For every x P R F , x “ ε p x q ` n ÿ i “ B x B v i p v i ´ q . YMPLECTIC COORDINATES 15
Using this Fox derivatives, the differential is given by Bp (cid:74) r i (cid:75) q : “ p s i ´ q (cid:74) r i (cid:75) , for (cid:74) r i (cid:75) P G , G , G , ¨ ¨ ¨Bp (cid:74) r i (cid:75) q : “ p ` s i ` ¨ ¨ ¨ ` s r i ´ i q (cid:74) r i (cid:75) , for (cid:74) r i (cid:75) P G , G , G , ¨ ¨ ¨Bp (cid:74) r i (cid:75) q : “ B r i B s i (cid:74) s i (cid:75) “ p ` s i ` ¨ ¨ ¨ ` s r i ´ i q (cid:74) s i (cid:75) , for (cid:74) r i (cid:75) P G , Bp (cid:74) r (cid:75) q : “ g ÿ i “ ˆ B r B x i (cid:74) x i (cid:75) ` B r B y i (cid:74) y i (cid:75) ˙ ` b ÿ i “ B r B z i (cid:74) z i (cid:75) ` c ÿ i “ B r B s i (cid:74) s i (cid:75) Bp (cid:74) x i (cid:75) q : “ p x i ´ q (cid:74) p (cid:75) , Bp (cid:74) y i (cid:75) q : “ p y i ´ q (cid:74) p (cid:75) Bp (cid:74) z i (cid:75) q : “ p z i ´ q (cid:74) p (cid:75) , Bp (cid:74) s i (cid:75) q : “ p s i ´ q (cid:74) p (cid:75) . Together with this differential we get the chain complex of free R Γ-modules(5) ¨ ¨ ¨ Ñ G k Ñ ¨ ¨ ¨ Ñ G Ñ G Ñ G Ñ R Ñ . Lemma 3.6.
The chain complex p G ˚ , Bq is exact.Proof. Using the mean value property of the Fox derivatives, we can show that
BB “ G be the generating set of the presentation (2) and let G “ G Y G ´ . Denoteby } ¨ } the word length with respect to the presentation (2). Observe that G “ R Γ r G s . We also know that R Γ “ R F { N where F is the free group on G and N isthe ideal in R F generated by the elements of the form p x ´ q for x in the normalclosure of r , r , ¨ ¨ ¨ , r c .Suppose that Bp x q “ ε p x q “ x P G . Write x “ ř i n i g i where n i P R zt u and g i P Γ. Since such an expression is unique, the quantity (cid:96) p x q : “ ř i } g i } is well-defined. We show that x “ Bp y q for some y P G by induction on (cid:96) p x q . If (cid:96) p x q “ x is of the form x “ n v ` n v where v , v P G . Since n ` n “
0, we have x “ n p v ´ q ` n p v ´ q . Therefore x “ Bp n (cid:74) v (cid:75) ` n (cid:74) v (cid:75) q . If v i P G ´ , thenwe use the expression ´ n i v i p v ´ i ´ q instead of n i p v i ´ q so that ´ n i v i (cid:74) v ´ i (cid:75) P R Γ r G s “ G . Now suppose that (cid:96) p x q ą
2. Write x “ ř i ě n i g i as before. Observethat at least one, say g , is not the identity element. Let v } g } v } g }´ ¨ ¨ ¨ v be thereduced word representing g and let g “ v } g } v } g }´ ¨ ¨ ¨ v . Then we can write x “ n g p v ´ q ` n g ` ř i ą n i g i . Let x “ n g ` ř i ą n i g i . Since (cid:96) p x q ă (cid:96) p x q and since ε p x q “
0, by the induction hypothesis, we can find y P G such that Bp y q “ x . Then Bp n g (cid:74) v (cid:75) ` y q “ x and this completes the induction. Again if v P G ´ then use ´ n g v p v ´ ´ q instead. This yields the exactness at G .Now we prove the exactness at G . For this let x “ ř i ě d i (cid:74) v i (cid:75) P G be suchthat Bp x q “ d i P R Γ. Being Bp x q “ F , that ř i ě d i p v i ´ q “ ř j ě n j p m j ´ q for n j P R Γ zt u and m i in the normalclosure of r , r , ¨ ¨ ¨ , r c . By applying the Fox derivative BB v i on both sides we get d i “ ÿ j ě n j B m j B v i . Since m j is in the normal closure of the words r , r , ¨ ¨ ¨ , r c , m j is a product ofwords of the forms g r ˘ g ´ and g r ˘ i g ´ . Observe that BB v i p g r ˘ g ´ q “ ˘ g B r B v i . This observation, in particular, implies that B m j B v i can be written as B m j B v i “ f j B r B v i ` c ÿ k “ h j,k B r k B v i for elements f j and h j,k of R Γ which depend only on j . This shows that x is theimage of ÿ j ě p n j f j q (cid:74) r (cid:75) ` c ÿ k “ ÿ j ě n j h j,k (cid:74) r k (cid:75) . This shows the exactness at G .The exactness at G k , k ą Lemma 3.7 (Also proven in [25]) . Let x P R Γ be such that x p s i ´ q “ then x “ x p s r i ´ i ` ¨ ¨ ¨ ` s i ` q for some x P R Γ . If x p s r i ´ i ` ¨ ¨ ¨ ` s i ` q “ then x “ x p s i ´ q for some x P R Γ .Proof. Suppose that x p s i ´ q “
0. Then xs i “ x . Inductively, we have that x “ xs i “ xs i “ ¨ ¨ ¨ “ xs r i ´ i . Therefore, r i x “ x ` xs i ` ¨ ¨ ¨ ` xs r i ´ i . This yields x “ xr i ¨ p s r i ´ i ` ¨ ¨ ¨ ` s i ` q .For the second assertion, assume that x p s r i ´ i ` ¨ ¨ ¨ ` s i ` q “
0. Lifting to thefree group ring R F , we have that x p s r i ´ i `¨ ¨ ¨` s i ` q “ y for some y P N . Observethat, since ε p x p s r i ´ i ` ¨ ¨ ¨ ` s i ` qq “ ε p x q r i “
0, we have ε p x q “
0. Apply the Foxderivatives to get B y B v “ r i B x B v for all v P G zt s i u . Hence by the mean value property, x “ ε p x q ` ÿ v P G B x B v p v ´ q “ B x B s i p s i ´ q ` ÿ v P G zt s i u r i B y B v p v ´ q . Again by the mean value property, we have ÿ v P G zt s i u r i B y B v p v ´ q “ r i ˆ y ´ ε p y q ´ B y B s i p s i ´ q ˙ “ r i ˆ y ´ B y B s i p s i ´ q ˙ . Thus x “ ˆ B x B s i ´ r i B y B s i ˙ p s i ´ q ` yr i . Modulo N , we have that x “ ´ B x B s i ´ r i B y B s i ¯ p s i ´ q . (cid:3) Finally, we show the exactness at G . Let x “ d (cid:74) r (cid:75) ` ř ci “ d i (cid:74) r i (cid:75) be such that Bp x q “ b ą Bp x q of (cid:74) z i (cid:75) is d B r B z i “ d g ź k “ r x k , y k s ź j ă i z j , we know that d “
0. Due to Lemma 3.7, then, we have d i “ d i p s i ´ q for some d i P R Γ as we wanted.If g ą Bp x q of (cid:74) x (cid:75) , which is d B r B x “ d p ´ x y x ´ q . Since y is not a torsion, d p ´ x y x ´ q ‰ d “
0. To conclude we argueas in the previous case.Therefore, it remains the case when b “ g “
0, in which we must have c ě d “
0. Note also that Bp x q “ c ÿ i “ ˆ d B r B s i ` d i B r i B s i ˙ (cid:74) s i (cid:75) “ c ÿ i “ ` d p s s ¨ ¨ ¨ s i ´ q ` d i p ` s i ` ¨ ¨ ¨ s r i ´ i q ˘ (cid:74) s i (cid:75) “ . YMPLECTIC COORDINATES 17
Since r “ s s ¨ ¨ ¨ s c “
1, we get d “ ´ d j p ` s j ` ¨ ¨ ¨ ` s r j ´ j qp s s ¨ ¨ ¨ s j ´ q ´ “ ´ d j p ` s j ` ¨ ¨ ¨ ` s r j ´ j q s j s j ` ¨ ¨ ¨ s c “ ´ d j p ` s j ` ¨ ¨ ¨ ` s r j ´ j q s j ` s j ` ¨ ¨ ¨ s c “ ´ d j B r j B s j s j ` s j ` ¨ ¨ ¨ s c from the coefficient of (cid:74) s j (cid:75) . On the other hand, by looking at the coefficient of (cid:74) s j ` (cid:75) , we have d “ ´ d j ` p ` s j ` ` ¨ ¨ ¨ ` s r j ` ´ j ` qp s s ¨ ¨ ¨ s j q ´ “ ´ d j ` p ` s j ` ` ¨ ¨ ¨ ` s r j ` ´ j ` q s j ` s j ` ¨ ¨ ¨ s c “ ´ d j ` B r j ` B s j ` s j ` s j ` ¨ ¨ ¨ s c . Hence, ´ d “ d B r B s “ d B r B s “ ¨ ¨ ¨ “ d c B r c B s c . Then d p s i ´ q “ i , from which we can conclude that d p x ´ q “ x P Γ. Since c ě
3, Γ is infinite and there is a non-torsion element x P Γ. Therefore d p x ´ q “ x . This leads us to d “ (cid:3) The geometric resolution II.
We present the last projective resolution whichis a slightly refined version of the previous one. This model is particularly usefulfor computing the parabolic cohomology.As we did in the construction of the groupoid bar resolution, we additionallypick a point p i on each ζ i and a path w i from p to p i in such a way that thecomplement of x , y , ¨ ¨ ¨ , x g , y g , s , ¨ ¨ ¨ , s c , w , ¨ ¨ ¨ , w b , ζ , ¨ ¨ ¨ , ζ b in the underlying topological space X O of O is a union of open disks.As before, let r G k “ R Γ r r , ¨ ¨ ¨ , r c s , for k ě r G “ R Γ r r , r , ¨ ¨ ¨ , r c s , r G “ R Γ r x , y , ¨ ¨ ¨ , x g , y g , ζ , ¨ ¨ ¨ , ζ b , w , ¨ ¨ ¨ , w b , s , ¨ ¨ ¨ , s c s , and r G “ R Γ r p , p , ¨ ¨ ¨ , p b s . Again to avoid any potential confusion, we place the generators of each moduleinside the double bracket (cid:74) ¨ (cid:75) . We define the differential by Bp (cid:74) r i (cid:75) q : “ p s i ´ q (cid:74) r i (cid:75) , for (cid:74) r i (cid:75) P r G , r G , r G , ¨ ¨ ¨Bp (cid:74) r i (cid:75) q : “ p ` s i ` ¨ ¨ ¨ ` s r i ´ i q (cid:74) r i (cid:75) , for (cid:74) r i (cid:75) P r G , r G , r G , ¨ ¨ ¨Bp (cid:74) r i (cid:75) q : “ B r i B s i (cid:74) s i (cid:75) “ p ` s i ` ¨ ¨ ¨ ` s r i ´ i q (cid:74) s i (cid:75) , for (cid:74) r i (cid:75) P r G Bp (cid:74) r (cid:75) q : “ g ÿ i “ ˆ B r B x i (cid:74) x i (cid:75) ` B r B y i (cid:74) y i (cid:75) ˙ ` b ÿ i “ B r B z i (cid:74) ζ i (cid:75) ` c ÿ i “ B r B s i (cid:74) s i (cid:75) ` b ÿ i “ B r B z i p ´ z i q (cid:74) w i (cid:75) Bp (cid:74) x i (cid:75) q : “ p x i ´ q (cid:74) p (cid:75) , Bp (cid:74) y i (cid:75) q : “ p y i ´ q (cid:74) p (cid:75) , Bp (cid:74) w i (cid:75) q “ (cid:74) p i (cid:75) ´ (cid:74) p (cid:75) Bp (cid:74) ζ i (cid:75) q : “ p z i ´ q (cid:74) p (cid:75) , Bp (cid:74) s i (cid:75) q : “ p s i ´ q (cid:74) p (cid:75) . By adopting the proof of Lemma 3.6, one can show that(6) ¨ ¨ ¨ Ñ r G Ñ r G Ñ r G Ñ r G Ñ R Ñ Consequences.
We list some consequences of the above construction of var-ious resolutions.Recall that a group Γ is of type FP over R if there is a resolution0 Ñ R Ñ R Ñ R Ñ R Ñ R i is projective and finitely generated R Γ-module.
Proposition 3.8.
Let O be a compact orientable cone orbifold of negative Eulerchatacteristic. Then Γ “ π orb1 p O q is of type FP over R .Proof. We begin with the geometric resolution (5). Let Q be the submodule gener-ated by (cid:74) r (cid:75) , ¨ ¨ ¨ , (cid:74) r c (cid:75) in G . Consider G : “ R Γ r r s , G : “ G {Bp Q q , G “ G , and G j “ j ě . In light of Lemma 3.7, we know that G – R Γ r x , y , ¨ ¨ ¨ , x g , y g , z , ¨ ¨ ¨ , z b , s ‘ c à i “ p R Γ b R Q i K i q where Q i is a finite cyclic subgroup of Γ generated by s i and K i is the kernel of theaugmentation map ε : R Q i Ñ R . Since Q i is finite, we have a section σ : R Ñ R Q i of ε defined by σ p x q “ Q i ř γ P Q i x ¨ γ . Therefore, K i is a projective R Q i -module. Itfollows that R Γ b R Q i K i is a projective R Γ-module. Therefore G is also projective.The new complex(7) ¨ ¨ ¨ Ñ Ñ ¨ ¨ ¨ Ñ Ñ G Ñ G Ñ G Ñ R is clearly exact and hence a projective resolution of length 2. (cid:3) We can also obtain the following description of the parabolic cohomology. Whilethe result itself might be well-known, we are not able to find a good reference forits proof. For the sake of completeness, we include the proof here as well.
Proposition 3.9.
Let S “ tx z y , ¨ ¨ ¨ , x z b yu . In terms of the groupoid bar resolution,the parabolic cohomology is represented by the parabolic cocycles r Z p Γ , S ; g q : “ t u P r Z p Γ; g q | u p (cid:74) ζ i (cid:75) q “ , i “ , , ¨ ¨ ¨ , b u where r Z p Γ; g q “ ker p Hom Γ p r B p Γ q , g q Ñ Hom Γ p r B p Γ q , g qq is the group of usualcocycle.In terms of the bar resolution, the parabolic cohomology is represented by Z p Γ , S ; g q : “ t u P Z p Γ; g q | u p (cid:74) z i (cid:75) q “ z i ¨ X i ´ X i for some X i P g , i “ , , ¨ ¨ ¨ , b u where Z p Γ; g q “ ker p Hom Γ p B p Γ q , g q Ñ Hom Γ p B p Γ q , g qq .Proof. The proof goes as follow. We first characterize the parabolic cohomology interms of the resolution (4) that captures the peripheral structure well. Then wetranslate (4) into the bar resolution (3) and carefully track the parabolic cocycles.For each i , we define A ij , j ą
0, to be the free R rx z i ys -module on the set t (cid:74) x | x | ¨ ¨ ¨ | x j (cid:75) | x , ¨ ¨ ¨ , x j P x z i yzt uu . We let A i “ R rx z i ysr p i s . The differentialis defined in the same way as that of the bar resolution. Then it is clear that A j : “ À bi “ R Γ b A ij is a direct summand of r B j p Γ q and p r B ˚ p Γ q , A ˚ q is an auxiliaryresolution for the pair p Γ , S q .By taking Hom Γ p´ , g q on the complex r B ˚ p Γ q{ A ˚ , we get the chain complex C ˚ p Γ , S ; g q : “ Hom Γ p r B ˚ p Γ q{ A ˚ , g q for H ˚ p Γ , S ; g q . The map j : H p Γ , S ; g q Ñ YMPLECTIC COORDINATES 19 H p Γ; g q in (1) sends the space of 1-cocycles ker p C p Γ , S ; g q Ñ C p Γ , S ; g qq to r Z p Γ , S ; g q . This proves the first part of the lemma.For the second part, we use the homotopy equivalence ext : B ˚ p Γ q Ñ r B ˚ p Γ q .Observe that, by using the cocycle condition, u p ext p (cid:74) z i (cid:75) qq “ u p (cid:74) w i ζ i w ´ i (cid:75) q “ u p (cid:74) w i (cid:75) q ´ z i ¨ u p (cid:74) w i (cid:75) q . Therefore, ext sends r Z p Γ , S ; g q to Z p Γ , S ; g q . (cid:3) The symplectic structure on orbifold Hitchin components
In this section, we construct the Atiyah-Bott-Goldman type symplectic form onthe relative character variety X B n p Γ q of the orbifold group Γ “ π orb1 p O q .In section 4.1, we apply the Poincar´e duality to produce a nondegenerate anti-symmetric bilinear pairing ω P D on T r ρ s X B n p Γ q . Lemma 4.5 and the fact that Γ is oftype FP over R , which is proven in the previous section, guarantee that Γ has thePoincar´e duality map. In Lemma 4.8, we give an explicit form of the fundamentalcycle of Γ which leads to the concrete formula for the pairing ω P D in Theorem4.13. Then we embed X B n p Γ q into X B n p π p X O qq as a open submanifold where X O is some compact surface obtained by removing cone points of O . We show that ω P D is the pull-back of the (well-known) Atiyah-Bott symplectic form on X B n p π p X O qq .In section 4.2, we construct another 2-form ω H defined on a neighborhoodof Hom B p Γ , G q by modifying Huebschmann [17] and Guruprasad-Huebschmann-Jeffrey-Weinstein [14]. From the construction we know that the 2-form ω H is equiv-ariantly closed in a small neighborhood of Hom B p Γ , G q . We show in Theorem 4.23that, after taking the symplectic reduction, ω H “ ´ ω P D on X B n p Γ q .Consequently, we know that ω : “ ω P D “ ´ ω H is a closed non-degenerate 2-formon X B n p Γ q and this form ω will be our Atiyah-Bott-Goldman symplectic form.Throughout this section G denotes the Lie group PSL n p R q and g its Lie algebra.We also adopt the following notation. Notation 4.1.
Let O be a compact orientable cone 2-orbifold of negative Eulercharacteristic. Let ζ , ¨ ¨ ¨ , ζ b be boundary components of O . By choosing a pathfrom a base point of π orb1 p O q to each ζ i , we can view ζ i as an element of π orb1 p O q .Then we denote by B π orb1 p O q a set tx ζ y , ¨ ¨ ¨ , x ζ b yu . Although B π orb1 p O q is definedonly up to conjugation, this ambiguity does not make any difference in our discus-sion.4.1. Construction by the Poincar´e duality.
As in the previous section, denoteby O a compact orientable cone orbifold of negative Euler characteristic and Γ theorbifold fundamental group π orb1 p O q . We continue to use the presentation (2) for Γ.Let F be the free group on t x , y , ¨ ¨ ¨ , x g , y g , s , ¨ ¨ ¨ , s c , z , ¨ ¨ ¨ , z b u and let F : “ x x , y , ¨ ¨ ¨ , x g , y g , s , ¨ ¨ ¨ , s c , z , ¨ ¨ ¨ , z b | r “ ¨ ¨ ¨ “ r c “ y where r i “ s r i i . Let Q i “ x s i y be a cyclic subgroup of F generated by s i . As H p Q i , g ρ q “ ρ P Hom p Q i , G q , Hom p Q i , G q is a smooth embedded sub-space of Hom p Z , G q “ G and so is Hom p F , G q “ G g ` b ˆ Hom p Q , G q ˆ ¨ ¨ ¨ ˆ Hom p Q c , G q .The tangent space of G at g P G can be identified with g as follows. Recallthat the Lie algebra g is the tangent space of G at the identity element. Write R g : G Ñ G the right translation defined by R g p h q “ hg . This map inducesthe tangent map g Ñ T g G . Similarly for ρ “ p g , ¨ ¨ ¨ , g g ` b ` c q P Hom p F , G q ,write R ρ ˚ : g g ` b ` c Ñ T ρ Hom p F , G q the tangent map of the coordinate-wiseright translation R ρ : G g ` b ` c Ñ G g ` b ` c defined by R ρ p h , ¨ ¨ ¨ , h g ` b ` c q “p h g , ¨ ¨ ¨ , h g ` b ` c g g ` b ` c q . We have the conjugation action of G on Hom B p F , G q . For each X P g inducesthe fundamental vector field X defined by the formula X ρ p f q “ dd t f p exp p´ tX q ¨ ρ q ,where f P C p Hom p F , G qq , and where ρ P Hom p F , G q . This gives rise to thelinear map A : X ÞÑ X ρ from g to T ρ Hom B p F , G q .Finally let E : Hom p F , G q Ñ G be the map defined by E p ρ q “ ρ p r q . Then weknow that Hom p Γ , G q “ E ´ p q . Lemma 4.2.
Let G Ñ R Ñ be the resolution (7). The chain complex C ˚ p Γ; g q “ Hom Γ p G , g q that computes H ˚ p Γ; g q fits into the commutative diagram C p Γ; g q (cid:47) (cid:47) “ (cid:15) (cid:15) C p Γ; g q (cid:47) (cid:47) R ρ ˚ (cid:15) (cid:15) C p Γ; g q R E p ρ q˚ (cid:15) (cid:15) T e G A (cid:47) (cid:47) T ρ Hom p F , G q E ˚ (cid:47) (cid:47) T E p ρ q G where R ρ and R E p ρ q are the right translation maps.Proof. For each j “ , , ¨ ¨ ¨ , c , let t j “ t X P g | X ` s j ¨ X ` ¨ ¨ ¨ ` s r j ´ j ¨ X “ u .Observe that C p Γ; g q “ g g ` b ˆ t ˆ ¨ ¨ ¨ ˆ t c . Since Hom p F , G q “ G g ` b ˆ Hom p Q , G q ˆ ¨ ¨ ¨ ˆ Hom p Q c , G q , it is enough tocheck that t j can be identified with T ρ Hom p Q j , G q under the right translation.For this, we show that if ρ t is an analytic curve in Hom p Q j , G q with ρ “ ρ then X i : “ ` ddt ρ t p s i q| t “ ˘ ρ p s i q ´ is in t i . Indeed, since ρ t p s i q r i “
1, we have that0 “ X i ρ p s i q r i ` ρ p s i q X i ρ p s i q r i ´ ` ¨ ¨ ¨ ` ρ p s i q r i ´ X i ρ p s i q“ X i ` s i ¨ X i ` ¨ ¨ ¨ ` s r i ´ i ¨ X i as desired.Commutativity of the diagram can be seen by straightforward computations. (cid:3) The following corollary is known for closed surface groups. Here we record thesame result for orbifold groups.
Corollary 4.3. If H p Γ; g ρ q “ then ρ has a neighborhood that is an analyticallysubmanifold of Hom p Γ , G q .Proof. From Lemma 4.4, the differential E ˚ at ρ is surjective if and only if H p Γ; g ρ q “
0. The result follows from the inverse function theorem. (cid:3)
The following lemma is a relative version of Lemma 4.2.
Lemma 4.4.
There is a chain complex C ˚ par p Γ , B Γ; g q (will be explicitly constructedin the proof (10)) that computes H p Γ , B Γ; g q and that fits into the commutativediagram C p Γ , B Γ; g q (cid:47) (cid:47) “ (cid:15) (cid:15) C p Γ , B Γ; g q (cid:47) (cid:47) R ρ ˚ (cid:15) (cid:15) C p Γ , B Γ; g q R E p ρ q˚ (cid:15) (cid:15) T e G A (cid:47) (cid:47) T ρ Hom B p F , G q E ˚ (cid:47) (cid:47) T E p ρ q G where R ρ and R E p ρ q are the right translation maps.In particular, the right translation identifies H p Γ , B Γ; g ρ q with the (Zariski)tangent space T r ρ s X B n p Γ q at r ρ s . YMPLECTIC COORDINATES 21
Proof.
We begin with a modified version of the resolution (6). Let Q be the sub-module of r G generated by (cid:74) r (cid:75) , ¨ ¨ ¨ , (cid:74) r c (cid:75) . Define r G i “ , i ě r G “ R Γ r r s r G “ r G {Bp Q q r G “ r G . Using the same argument of the proof of Proposition 3.8, we know that r G Ñ R is a projective resolution of R .For each i , we define the R Γ i -module A i ˚ by A i “ R Γ i r p i s , A i “ R Γ i r ζ i s , and A ij “ j ě . Then A ˚ : “ À bi “ R Γ b A i ˚ is a direct summand of the projective resolution r G Ñ R . Therefore we have an auxiliary projective resolution A i ˚ Ñ R of R .Observe that r G { A – R Γ r p s r G { A – R Γ r x , y , ¨ ¨ ¨ , x g , y g , s , ¨ ¨ ¨ , s c , w , ¨ ¨ ¨ , w b s{Bp Q q r G i { A i – r G i , i ě . We apply Hom p´ , g q to the above resolution r G { A ˚ . It follows that chain complexthat computes the relative cohomology H p Γ , B Γ; g q is r C p Γ , B Γ; g q “ g r C p Γ , B Γ; g q “ g g ˆ t ˆ ¨ ¨ ¨ ˆ t c ˆ g b r C p Γ , B Γ; g q “ g where t j “ t X P g | X ` s j ¨ X ` ¨ ¨ ¨ ` s r j ´ j ¨ X “ u .The exact sequence of complexes 0 Ñ A ˚ Ñ r G Ñ r G { A ˚ Ñ r C p Γ , B Γ; g q (cid:47) (cid:47) inclusion (cid:15) (cid:15) r C p Γ , B Γ; g q (cid:47) (cid:47) inclusion (cid:15) (cid:15) r C p Γ , B Γ; g q “ (cid:15) (cid:15) r C p Γ; g q (cid:47) (cid:47) r C p Γ; g q (cid:47) (cid:47) r C p Γ; g q where r C i p Γ; g q “ Hom p r G i , g q . By definition of the parabolic cohomology, we havethat(8) H p Γ , B Γ; g q “ ker p r C p Γ , B Γ; g q Ñ r C p Γ , S ; g qq im p r C p Γ; g q Ñ r C p Γ; g qq X r C p Γ , B Γ; g q . Hence H p Γ , B Γ; g q can be computed from the following subcomplex(9) g ˆ b ź i “ ker p Ad ρ p z i q ´ q Ñ r C p Γ , B Γ; g q Ñ r C p Γ , B Γ; g q where r C p Γ , B Γ; g q and r C p Γ , B Γ; g q are regarded as subspaces of r C p Γ; g q and r C p Γ; g q respectively.We translate the above expression to the bar complex. For this purpose we writedown the explicit chain homotopy equivalence between Hom p r G , g q and Hom p G , g q . The chain homotopy equivalence Hom p r G , g q Ñ Hom p G , g q is induced from thechain map f ˚ : G Ñ r G given by f i “ Id , i ě f : (cid:74) x i (cid:75) ÞÑ (cid:74) x i (cid:75) , (cid:74) y i (cid:75) ÞÑ (cid:74) y i (cid:75) , (cid:74) z i (cid:75) ÞÑ (cid:74) w i (cid:75) ` (cid:74) ζ i (cid:75) ´ z i (cid:74) w i (cid:75) , (cid:74) s i (cid:75) ÞÑ (cid:74) s i (cid:75) f : (cid:74) p (cid:75) ÞÑ (cid:74) p (cid:75) It follows that the subcomplex (9) can be translated into(10) C ˚ par p Γ , B Γ; g q : g Ñ g g ˆ t ˆ ¨ ¨ ¨ ˆ t c ˆ h ˆ ¨ ¨ ¨ ˆ h b Ñ g where h i “ im p ´ Ad ρ p z i q q .In light of Lemma 4.2, we know that the tangent space of Hom p F , G q canbe identified with a subspace g g ˆ t ˆ ¨ ¨ ¨ ˆ t c ˆ g ˆ ¨ ¨ ¨ ˆ g b of C p Γ , B Γ; g q by the right translation R ρ . Here g i “ g for all i “ , , ¨ ¨ ¨ , b . We now showthat each h i Ă g i is the tangent space to the orbit of the conjugation action B i “ t gρ p z i q g ´ | g P G u through ρ p z i q . For if g t is an analytic curve in B i passing through g “ ρ p z i q P B i , then we can find an analytic curve h t in G with h “ g t “ h t g h ´ t . Taking the derivative at t “ p dg t dt | t “ q g ´ “ X i ´ g X i g ´ where X i “ dh t dt | t “ P g i . It follows that T ρ p z i q B i Ă h i .The reverse inclusion is achieved by setting h t “ exp tX i . We thus have the identi-fication R ρ ˚ : C p Γ , B Γ; g q Ñ T ρ Hom B p F , G q .Finally, it is a straightforward computation to show that the diagram in thestatement is commutative. (cid:3) Lemma 4.5.
Let O be a compact orientable cone orbifold of negative Euler char-acteristic. Let Γ “ π orb1 p O q . Then H k p Γ; R Γ q “ for k ‰ and H p Γ; R Γ q – ∆ where ∆ is the kernel of the augmentation À x z yPB Γ R Γ b x z y R Ñ R .Proof. Let C i Ă B O be the boundary component corresponding to the generator z i P Γ. Take a finite index torsionfree normal subgroup N in Γ. N is necessarily thefundamental group of a compact surface with boundary, say S and we have a finitecovering map p : S Ñ O with the deck group D : “ Γ { N “ t γ , ¨ ¨ ¨ , γ d u . Since thereis no singular points on the boundary of O , we know that D acts simply transitivelyon the set of connected components of p ´ p C i q . Hence N admits a presentation N “ x x , y , ¨ ¨ ¨ , x g , y g ,γ z γ ´ , ¨ ¨ ¨ , γ z b γ ´ , γ z γ ´ , ¨ ¨ ¨ , γ z b γ ´ , ¨ ¨ ¨ , γ d z γ ´ d , ¨ ¨ ¨ , γ d z b γ ´ d | ź i r x i , y i s ź i,j γ j z i γ ´ j “ y . Let B N “ tx γ j z i γ ´ j yu j “ , , ¨¨¨ ,di “ , , ¨¨¨ ,b . Then we know that p N, B N q is a P D Z -pair inthe sense of Bieri-Eckmann [2]. Let ∆ N be the kernel of the augmentation map À x z yPB N R N b x z y R Ñ R .We have the Lyndon-Hochschild-Serre spectral sequence whose 2nd page is E p,q “ H p p Γ { N ; H q p N ; R Γ qq that converges to H p ` q p Γ; R Γ q . Since N is the fundamental group of a compactsurface S with boundary, we know that H q p N ; R Γ q “ q ‰ H p N ; R Γ q “ À di “ γ i ∆ N . Hence we have that H p Γ; R Γ q “
0. Because the action of Γ { N on À di “ γ i ∆ N is the left multiplication, the space of the fixed points is isomorphic (asabelian groups) to ∆ N . There is a further isomorphism ∆ N Ñ ∆ given by sending x b r P R N b x γ i z j γ ´ i y R to xγ i b r P R Γ b x z j y R . This proves that H p Γ; R Γ q – ∆.To show H p Γ; R Γ q “ H ˚ p N ; R Γ q Ñ H ˚ p Γ; R Γ q .As in the proof of Lemma 2.5, this transfer map precomposed with the restriction YMPLECTIC COORDINATES 23 res : H ˚ p Γ; R Γ q Ñ H ˚ p N ; R Γ q becomes an isomorphism. In particular tr is surjec-tive. Since H p N ; R Γ q “
0, so is H p Γ; R Γ q .Finally from the resolution (7), we know that H q p Γ; R Γ q “ q ą (cid:3) Proposition 4.6.
Let O be a compact orientable cone orbifold of negative Eulercharacteristic and Γ “ π orb1 p O q . Then p Γ , B Γ q is a P D R -pair in the sense of Bieri-Eckmann [2] . Namely, there is a relative fundamental class r O , B O s P H p Γ , B Γ; R q such that Xr O , B O s : H ˚ p Γ , B Γ; R q Ñ H ´˚ p Γ; R q and Xr O , B O s : H ˚ p Γ; R q Ñ H ´˚ p Γ , B Γ; R q are isomorphisms.Proof. Due to Lemma 3.8, Lemma 4.5 and Theorem 6.2 of [2], p Γ , B Γ q is a P D R -pair. (cid:3) Recall that we also have the cup product Y : H p Γ , B Γ; g q b H p Γ; g q Ñ H p Γ , B Γ; R q . Composing this cup product with the cap product Xr O , B O s , we get the pairing ω P D : H p Γ , B Γ; g q b H p Γ; g q Y Ñ H p Γ , B Γ; R q Xr O , B O s Ñ H p Γ; R q “ R . We now present the explicit form of ω P D . The groupoid bar resolution is thebest choice for this purpose.
Notation 4.7.
This makes many expressions much simpler, we adopt the followingconvention: (cid:74) a ˘ b | c (cid:75) “ (cid:74) a | c (cid:75) ˘ (cid:74) b | c (cid:75) for all a, b, c P Γ zt u . For instance, (cid:115) Br x, y sB y ˇˇˇˇ y (cid:123) “ (cid:74) x ´ xyx ´ y ´ ˇˇ y (cid:75) “ (cid:74) x | y (cid:75) ´ (cid:74) xyx ´ y ´ | y (cid:75) . Lemma 4.8.
Let c “ g ÿ i “ ˆ (cid:115) B r B x i ˇˇˇˇ x i (cid:123) ` (cid:115) B r B y i ˇˇˇˇ y i (cid:123) ˙ ` b ÿ i “ (cid:115) B r B z i ˇˇˇˇ z i (cid:123) ` c ÿ i “ (cid:115) B r B s i ˇˇˇˇ s i (cid:123) ´ c ÿ i “ r i (cid:115) B r i B s i ˇˇˇˇ s i (cid:123) be an element of B p Γ q b R . Then r c : “ ext p c q ´ b ÿ i “ p (cid:74) w ´ i | w i ζ i (cid:75) ´ (cid:74) w i ζ i | w ´ i (cid:75) q P r B p Γ q b R represents the relative fundamental class r O , B O s P H p Γ , B Γ; R q .Proof. Observe that the element(11) (cid:74) r (cid:75) ´ c ÿ i “ r i (cid:74) r i (cid:75) P r G b R is the relative fundamental class. Now we explicitly write down the chain homotopyequivalence f ˚ from r G ˚ to r B ˚ p Γ q .The chain map f ˚ sends the generator (cid:74) p i (cid:75) P r G to (cid:74) p i (cid:75) P r B p Γ q and actssimilarly in degree 1. In degree 2, we define f ˚ by (cid:74) r (cid:75) ÞÑ g ÿ i “ ˆ (cid:115) B r B x i ˇˇˇˇ x i (cid:123) ` (cid:115) B r B y i ˇˇˇˇ y i (cid:123) ˙ ` b ÿ i “ ˆ (cid:115) B r B z i w i ˇˇˇˇ ζ i (cid:123) ` (cid:115) B r B z i ˇˇˇˇ w i (cid:123) ´ (cid:115) B r B z i w i ζ i w ´ i ˇˇˇˇ w i (cid:123) ˙ ` c ÿ i “ (cid:115) B r B s i ˇˇˇˇ s i (cid:123)(cid:74) r i (cid:75) ÞÑ (cid:115) B r i B s i ˇˇˇˇ s i (cid:123) and by the zero map for all higher degrees. Our chain map f ˚ takes the element(11) to c : “ g ÿ i “ ˆ (cid:115) B r B x i ˇˇˇˇ x i (cid:123) ` (cid:115) B r B y i ˇˇˇˇ y i (cid:123) ˙ ` b ÿ i “ ˆ (cid:115) B r B z i w i ˇˇˇˇ ζ i (cid:123) ` (cid:115) B r B z i ˇˇˇˇ w i (cid:123) ´ (cid:115) B r B z i w i ζ i w ´ i ˇˇˇˇ w i (cid:123) ˙ ` c ÿ i “ (cid:115) B r B s i ˇˇˇˇ s i (cid:123) ´ c ÿ i “ r i (cid:115) B r i B s i ˇˇˇˇ s i (cid:123) . By direct computation, we know that c and r c are homologues. Indeed c ´ r c is theboundary of the 3-chain b ÿ i “ ˆ ´ (cid:115) B r B z i ˇˇˇˇ w i ζ i w ´ i ˇˇˇˇ w i (cid:123) ´ (cid:115) B r B z i ˇˇˇˇ w i ζ i ˇˇˇˇ ζ ´ i (cid:123) ´ (cid:74) w i ζ i | w ´ i | w i (cid:75) ` (cid:74) w ´ i | w i ζ i | ζ ´ i (cid:75) ` (cid:115) B r B z i w i ˇˇˇˇ ζ i ˇˇˇˇ ζ ´ i (cid:123) ˙ . Hence the lemma follows. (cid:3)
Recall that we have an auxiliary resolution p r B ˚ p Γ q , A ˚ q for the group pair p Γ , B Γ q . For the description of the peripheral part A ˚ , see the proof of Proposi-tion 3.9. Lemma 4.9.
Let O be a compact orientable cone orbifold of negative Euler char-acteristic, and Γ “ π orb1 p O q . Let p r B ˚ p Γ q , A ˚ q be an auxiliary resolution for a grouppair p π orb1 p O q , B π orb1 p O qq . Let u P Hom Γ p r B p Γ q{ A , g q and v P Hom Γ p r B p Γ q , g q .Then for any (cid:74) x | y (cid:75) P r B p Γ q b R , we have x u Y v, (cid:74) x | x (cid:75) y “ Tr p u p (cid:74) x (cid:75) q ret p x q ¨ v p (cid:74) x (cid:75) qq . Proof.
This is the Alexander-Whitney diagonal approximation theorem. (cid:3)
Lemma 4.10.
Let u be in Z p Γ , B Γ; g q Ă Hom Γ p r B p Γ q{ A , g q or in Z p Γ , B Γ; g q Ă Hom Γ p B p Γ q , g q . Then for each s i , there exists T i P g such that u p (cid:74) s i (cid:75) q “ s i ¨ T i ´ T i .Moreover if T i P g are other choices with the same property u p (cid:74) s i (cid:75) q “ s i ¨ T i ´ T i then we have Tr p T i v p (cid:74) s i (cid:75) qq “ Tr p T i v p (cid:74) s i (cid:75) qq for any v P Z p Γ , B Γ; g q .Proof. The proof is along the same lines as that of Lemma 3.7. More precisely, let T i “ ´ r i ` u p (cid:74) s i (cid:75) q ` u p (cid:74) s i (cid:75) q ` ¨ ¨ ¨ ` u p (cid:74) s r i ´ i (cid:75) q ˘ . Then using the cocycle condition, we have that r i p s i ¨ T i ´ T i q “ ´ s i ¨ u p (cid:74) s i (cid:75) q ´ s i ¨ u p (cid:74) s i (cid:75) q ´ ¨ ¨ ¨ ´ s i ¨ u p (cid:74) s r i ´ i (cid:75) q` u p s i q ` p u p (cid:74) s i (cid:75) q ` s i ¨ u p (cid:74) s i (cid:75) qq ` p u p (cid:74) s i (cid:75) q ` s i ¨ u p (cid:74) s i (cid:75) qq ` ¨ ¨ ¨¨ ¨ ¨ ` p u p (cid:74) s i (cid:75) q ` s i ¨ u p (cid:74) s r i ´ i (cid:75) qq“ p r i ´ q u p (cid:74) s i (cid:75) q ´ s i ¨ u p (cid:74) s r i ´ i (cid:75) q“ p r i ´ q u p (cid:74) s i (cid:75) q ` u p (cid:74) s i (cid:75) q ´ u p (cid:74) s r i i (cid:75) q“ r i u p (cid:74) s i (cid:75) q . Therefore the result follows.
YMPLECTIC COORDINATES 25
To prove the second statement, we observe that s i ¨ p T i ´ T i q “ T i ´ T i foreach i . Let D i “ T i ´ T i . Since v P Z p Γ , B Γ; g q we can also choose V i P g so that v p (cid:74) s i (cid:75) q “ s i ¨ V i ´ V i for each i . Then we haveTr p T i v p (cid:74) s i (cid:75) qq ´ Tr p T i v p (cid:74) s i (cid:75) qq “ Tr p D i v p (cid:74) s i (cid:75) qq“ Tr p D i p s i ¨ V i ´ V i qq“ Tr p D i s i ¨ V i q ´ Tr p D i V i q“ Tr p D i V i q ´ Tr p D i V i q“ (cid:3) The following proposition and its proof is a slight variation of that of Guruprasad-Huebschmann-Jeffrey-Weinstein [14].To avoid cumbersome notation, we sometimes write u p x q rather than u p (cid:74) x (cid:75) q . Wealso use the map p¨q : B p Γ q Ñ B p Γ q defined by ř n i (cid:74) g i (cid:75) “ ř n i (cid:74) g ´ i (cid:75) . Proposition 4.11.
Let O be a compact oriented cone orbifold of negative Eu-ler characteristic and Γ “ π orb1 p O q . Let r ρ s P X B n p Γ q . Let u P Z p Γ , B Γ; g ρ q Ă Hom Γ p r B p Γ q{ A , g ρ q and v P r Z p Γ , B Γ; g ρ q Ă Hom Γ p r B p Γ q , g ρ q . Let T i be theelements in g defined in Lemma 4.10. Then ω P D pr u s , r v sq “ ´ g ÿ i “ Tr ˆ u ˆ B r B x i ˙ v p x i q ˙ ´ g ÿ i “ Tr ˆ u ˆ B r B y i ˙ v p y i q ˙ ´ b ÿ i “ Tr ˆ u ˆ B r B z i ˙ v p z i q ˙ ´ c ÿ i “ Tr ˆ u ˆ B r B s i ˙ v p s i q ˙ ´ c ÿ i “ Tr p T i v p s i qq ´ b ÿ i “ Tr p X i v p z i qq . where u “ u ˝ ext , v “ v ˝ ext , and X i “ ´ u p (cid:74) w i (cid:75) q .Proof. By the definition of ω P D we have that ω P D pr u s , r v sq “ x u Y v, r c y“ x u Y v, c y ´ b ÿ i “ x u Y v, (cid:74) w ´ i | w i ζ i (cid:75) ´ (cid:74) w i ζ i | w ´ i (cid:75) y . where c “ ext p c q . Then by making use of Lemma 4.9 we can evaluate x u Y v, (cid:74) w ´ i | w i ζ i (cid:75) y “ Tr p u p (cid:74) w ´ i (cid:75) q ret p w ´ i q ¨ v p (cid:74) w i ζ i (cid:75) qq“ Tr p X i v p (cid:74) w i ζ i (cid:75) qq“ Tr p X i v p (cid:74) w i (cid:75) qq . For the last equality, we use the cocycle condition to get v p (cid:74) w i ζ i (cid:75) q “ v p (cid:74) w i (cid:75) q .Likewise, we have that x u Y v, (cid:74) w i ζ i | w ´ i (cid:75) y “ ´ Tr p X i p ret p w i ζ i q ¨ v p (cid:74) w ´ i (cid:75) qqq“ Tr p X i p z i ¨ v p (cid:74) w i (cid:75) qqq . Observe that v p (cid:74) z i (cid:75) q “ v p (cid:74) ext p z i q (cid:75) q “ v p (cid:74) w i ζ i w ´ i (cid:75) q “ v p (cid:74) w i (cid:75) q ´ z i ¨ v p (cid:74) w i (cid:75) q . Com-bining all, we get b ÿ i “ x u Y v, (cid:74) w ´ i | w i ζ i (cid:75) ´ (cid:74) w i ζ i | w ´ i (cid:75) y “ b ÿ i “ Tr p X i v p (cid:74) z i (cid:75) qq . By using Lemma 4.9 and Lemma 4.10 and the identityTr p u p (cid:74) g (cid:75) q g ¨ v p (cid:74) h (cid:75) qq “ ´ Tr p u p (cid:74) g ´ (cid:75) q v p (cid:74) h (cid:75) qq , we can expand x u Y v, c y “ x u Y v , c yx u Y v , c y “ ´ g ÿ i “ Tr ˆ u ˆ B r B x i ˙ v p x i q ˙ ´ g ÿ i “ Tr ˆ u ˆ B r B y i ˙ v p y i q ˙ ´ b ÿ i “ Tr ˆ u ˆ B r B z i ˙ v p z i q ˙ ´ c ÿ i “ Tr ˆ u ˆ B r B s i ˙ v p s i q ˙ ´ c ÿ i “ Tr p T i v p s i qq . This yields the theorem. (cid:3)
Corollary 4.12.
Let j : H p Γ , B Γ; g q Ñ H p Γ , g q be the map in (1). For r u s P ker j Ă H p Γ , B Γ; g q and r v s P im j “ H p Γ , B Γ; g q Ă H p Γ; g q , we have ω P D pr u s , r v sq “ . Proof.
Being r u s P ker j means that u “ δX for some X P Hom Γ p r B p Γ q , g q . Define X P Hom Γ p r B p Γ q{ A , g q to be X p (cid:74) p (cid:75) q “ X p (cid:74) p (cid:75) q and X p (cid:74) p i (cid:75) q “ i “ , , ¨ ¨ ¨ , b so that X P Hom Γ p r B p Γ q{ A , g q . Then r u ´ δX s “ r u s in H p Γ , B Γ; g q .Now we apply Proposition 4.11 to conclude. Namely, we show that each term inthe formula for ω P D vanishes.Since r u s “ r u ´ δX s we can replace u with u ´ δX . Observe that p u ´ δX qp (cid:74) γ (cid:75) q “ o p γ q “ t p γ q “ p . This means that p u ´ δX q ˝ ext ˆ B r B x i ˙ “ p u ´ δX q ˝ ext ˆ B r B y i ˙ “ , for i “ , , ¨ ¨ ¨ , g , p u ´ δX q ˝ ext ˆ B r B s i ˙ “ i “ , , ¨ ¨ ¨ , c , and p u ´ δX q ˝ ext ˆ B r B z i ˙ “ i “ , , ¨ ¨ ¨ , b . Moreover since p u ´ δX qp (cid:74) s i (cid:75) q “
0, we can set T i “ i .This shows that the first five terms in the formula for ω P D are zero.Recall that in the statement of Proposition 4.11 we found X i as ´ X i “ p u ´ δX qp (cid:74) w i (cid:75) q “ δX p (cid:74) w i (cid:75) q ´ δX p (cid:74) w i (cid:75) q “ X p (cid:74) p i (cid:75) q . Since u “ δX P Hom Γ p r B p Γ q{ A , g q we have0 “ u p (cid:74) ζ i (cid:75) q “ δX p (cid:74) ζ i (cid:75) q “ z i ¨ X p (cid:74) p i (cid:75) q ´ X p (cid:74) p i (cid:75) q “ X i ´ z i ¨ X i . This implies that, for each i ,Tr p X i v p z i qq “ Tr p X i p v p (cid:74) w i (cid:75) q ´ z i ¨ v p (cid:74) w i (cid:75) qqq “ Tr pp X i ´ z ´ i ¨ X i q v p (cid:74) w i (cid:75) qq “ . It follows that ω P D pr u s , r v sq “ xp u ´ δX q Y v, r c y “ (cid:3) Recall that H p Γ , B Γ; g q “ H p Γ , B Γ; g q{ ker j . In particular, due to Corollary4.12, ω P D gives rise to the well-defined paring ω P D : H p Γ , B Γ; g q b H p Γ , B Γ; g q Ñ R . We denote by the same letter ω P D to denote the induced pairing. By Lemma 4.4,and the properties of cup and cap products, it is immediate that ω P D is a 2-formon X B n p Γ q via the right translation. This 2-form on X B n p Γ q will also be denoted by ω P D . YMPLECTIC COORDINATES 27
Theorem 4.13.
Let O be a compact oriented cone orbifold of negative Eulercharacteristic and Γ “ π orb1 p O q . Let r ρ s P X B n p Γ q . Let u, v P Z p Γ , B Γ; g q Ă Hom Γ p B p Γ q , g q be parabolic cocycles in terms of the bar resolution. We choose X i , T j P g such that u p z i q “ z i ¨ X i ´ X i and u p s j q “ s j ¨ T j ´ T j for each i “ , , ¨ ¨ ¨ , b and j “ , , ¨ ¨ ¨ , c respectively. Then ω P D can be written explicitly as ω P D pr u s , r v sq “ x u Y v, c y ´ b ÿ i “ Tr p X i v p z i qq“ ´ g ÿ i “ Tr ˆ u ˆ B r B x i ˙ v p x i q ˙ ´ g ÿ i “ Tr ˆ u ˆ B r B y i ˙ v p y i q ˙ ´ b ÿ i “ Tr ˆ u ˆ B r B z i ˙ v p z i q ˙ ´ c ÿ i “ Tr ˆ u ˆ B r B s i ˙ v p s i q ˙ ´ c ÿ i “ Tr p T i v p s i qq ´ b ÿ i “ Tr p X i v p z i qq . Proof.
Let
X, Y be elements of r B p Γ q such that X p (cid:74) p i (cid:75) q “ X i i “ , , ¨ ¨ ¨ , b i “ , Y p (cid:74) p i (cid:75) q “ Y i i “ , , ¨ ¨ ¨ , b i “ . Let r u “ u ˝ ret ´ δX and r v “ v ˝ ret ´ δY . Then r u can be seen as a member of Z p Γ , B Γ; g q Ă Hom Γ p r B p Γ q{ A , g q . Observe that r u ˝ ext “ u and r v ˝ ext “ v . Nowapply Proposition 4.11 to get the conclusion. (cid:3) Let X O be an underlying space of O . Let Σ O be the set of singularities. Recallthat we have defined X O : “ X O z Σ O . X O is a manifold and each puncture of X O corresponds to a singularity of O . Hence the fundamental group is given by π p X O q “ x x , y , ¨ ¨ ¨ , x g , y g , z , ¨ ¨ ¨ , z b , s , ¨ ¨ ¨ , s c | g ź i “ r x i , y i s b ź i “ z i c ź i “ s i y . Lemma 4.14.
Let O be a compact oriented cone 2-orbifold of negative Euler char-acteristic and Γ “ π orb1 p O q . We have a smooth embedding I : X B n p Γ q Ñ X B n p π p X O qq for some choice B of conjugacy classes of boundary holonomies.Proof. The map I is defined by precomposing ρ P X B n p π orb1 p O qq with the projectionmap π p X O q Ñ π orb1 p O q .We know that X n p π p X O qq is foliated by leaves of the form Ť Z X Z n p π p X O qq .Lemma 4.10 tells us that the derivative d I of I at each point of X B n p Γ q mapsisomorphically the tangent space H p Γ , B Γ; g ρ q of X B n p Γ q to the tangent space H p π p X O q , B π p X O q ; g I p ρ q q of a leaf. Therefore, the image of I must be containedin a single leaf, say X B n p π p X O qq . (cid:3) Remark . The image I p C B p O qq is not in the Hitchin component of X O . In fact,it is not even an Anosov representation because every element of I p C B p O qq is notfaithful. Theorem 4.16.
Let O be a compact oriented cone 2-orbifold of negative Eulercharacteristic and Γ “ π orb1 p O q . Then the pairing ω P D is a symplectic form on X B n p Γ q via the right translation. Proof.
By works of [14] and [21], each leaf X B n p π p X O qq of X n p π p X O qq carries thesymplectic form ω X O K . By comparing the explicit formula for ω O (Theorem 4.13) andthat of ω X O K [21, Theorem 5.6], we know that I ˚ ω X O K “ ω O . This proves that ω O is closed. Moreover we know that d I p H p Γ , B Γ; g qq “ H p π p X O q , B π p X O q ; g q .Since ω X O K is nondegenerate, so is ω O . (cid:3) In particular we have the following result for n “ Proposition 4.17.
Let O be a compact oriented cone 2-orbifold of negative Eu-ler characteristic. The image of the embedding I : C B p O q Ñ X B p π p X O qq is asymplectic submanifold. Definition 4.18.
We will call ω P D on X B n p π orb1 p O qq or on C B p O q the Atiyah-Bott-Goldman symplectic form and denote it simply by ω O .4.2. Construction by equivariant de Rham bicomplex.
Let G be a Lie groupacting on a smooth manifold M . Recall that the equivariant de Rham complex A p,qG p M q is defined as follows: A j,pG p M q is the set of G -equivariant homogeneouspolynomials of degree j on the Lie algebra g of G with values in p -forms on M . Wehave two differentials d and δ G defined byd : A j,pG p M q Ñ A j,p ` G p M q the usual de Rham differential δ G : A j,pG p M q Ñ A j ` ,p ´ G p M q δ G p α qp X q “ ´ ι X p α p X qq where X is the fundamental vector field generated by X .An element α P A j,pG p M q is said to be equivariantly closed if p d ` δ G qp α q “
0. If M is a symplectic manifold with a symplectic form ω and the G -action issymplectomorphic, then ω becomes a d-closed element of A , G p M q . In general ω may not be equivariantly closed. An element (if exists) µ P A , G p M q making ω ` µ equivariantly closed is called a moment map .We consider M “ G q where G acts on M as conjugations on each component.There is another differential δ : A j,pG p G q q Ñ A j,pG p G q ` q defined by δf “ p´ q q ` q ÿ i “ p´ q i p δ i f q where δ i f is induced by p x , ¨ ¨ ¨ , x q q ÞÑ p x , ¨ ¨ ¨ , x q q for i “ p x , ¨ ¨ ¨ , x q q ÞÑp x , ¨ ¨ ¨ , x q ´ q for i “ q and p x , ¨ ¨ ¨ , x q q ÞÑ p x , ¨ ¨ ¨ , x i ´ x i , ¨ ¨ ¨ , x q q for 1 ď i ď q .Let ω be the Cartan-Maurer form on G . That is, ω g p X q is the unique left invariantvector field such that ω g p X q “ X g . Let ι i : G ˆ G Ñ G be the projection onto i thfactor, i “ ,
2. Define Ω : “ ι ˚ ω ¨ ι ˚ ω where ¨ is the trace form. Ω is an element of A , G p G q . Remark . Our convention of operations of Lie algebra valued forms is p ω ¨ ω qp X, Y q “ ω p X q ¨ ω p Y q ´ ω p Y q ¨ ω p X qr ω, ω sp X, Y q “ r ω p X q , ω p Y qs ´ r ω p Y q , ω p X qs . This coincides with the one used in Huebschmann [17], [18] and Weinstein [32].Under this convention the structure equation takes of the formd ω “ ´ r ω, ω s . YMPLECTIC COORDINATES 29
We also define λ : “ r ω, ω s ¨ ω P A , G p G q θ : “ X ¨ p ω ` ω q P A , G p G q . Lemma 4.20 (Weinstein [32]) . We have the following identities dΩ “ δλδ Ω “ δ G Ω “ ´ δθ d λ “ δ G λ “ d θδ G θ “ . Recall that the group F was defined by the presentation F : “ x x , y , ¨ ¨ ¨ , x g , y g , s , ¨ ¨ ¨ , s c , z , ¨ ¨ ¨ , z b | r “ ¨ ¨ ¨ “ r c “ y . Let E : Hom p F , G q ˆ p F q q Ñ G q be the evaluation map E p ρ, x , x , ¨ ¨ ¨ , x q q “ p ρ p x q , ρ p x q , ¨ ¨ ¨ ρ p x q qq . Here G actstrivially on p F q q . Then E is G -equivariant so that it induces E ˚ : A j,pG p G q q Ñ A j,pG p Hom p F , G q ˆ p F q q q . Observe that A j,pG p Hom p F , G q ˆ p F q q q “ A j,pG p Hom p F , G qq b A j, G pp F q q q . Following Huebschmann [17], define ω c “ x E ˚ Ω , ´ c y P A , G p Hom p F , G qq where c is the one define in Lemma 4.8. Here we regard c as an element of B p F qb R Ă R r F ˆ F s . Notice thatd ω c “ x E ˚ dΩ , ´ c y“ x E ˚ δλ, ´ c y“ x E ˚ λ, ´B c y“ x E ˚ λ, r ´ ÿ z i y“ r ˚ λ ´ ÿ z ˚ i λ. Here we view r and z i as the maps from Hom p F , G q to G defined by ρ ÞÑ E p ρ, r q and ρ ÞÑ E p ρ, z i q respectively. Now we choose any regular neighborhood O Ă g forthe exponential map at the identity and consider the pull-back diagram H B η (cid:15) (cid:15) p p r , x z , ¨¨¨ , x z b q (cid:47) (cid:47) O ˆ C ˆ ¨ ¨ ¨ ˆ C b exp ˆ Id (cid:15) (cid:15) Hom B p F , G q p r ,z , ¨¨¨ ,z b q (cid:47) (cid:47) G ˆ C ˆ ¨ ¨ ¨ ˆ C b . Since O Ă g is contractible, we can construct an adjoint invariant homotopy op-erator h : A j,pG p g q Ñ A j,p ´ G p g q such that d h ` h d “ Id. Using this homotopyoperator, we define ω “ η ˚ ω c ´ p r ˚ h exp ˚ λ. Then we have(12) d ω “ ´ ÿ p z i ˚ λ. In view of the proof of Theorem 2 of [17],(13) δ G ω “ ´ d µ ` ÿ p z i ˚ θ where µ : H B Ñ g ˚ is given by µ “ ψ ˝ p r and ψ : g Ñ g ˚ is the dualization withrespect to the trace form Tr.Let C be a conjugacy class in G . To find a closed 2-form on H B , we need τ P A , G p C q such that d τ “ λ and δ G τ “ ´ θ . At each p P C , the right translationby p identifies the tangent space T p C with the subspace t Ad p X ´ X | X P g u of g “ T e G . The 2-form τ on C is then defined by τ p Ad p X ´ X, Ad p Y ´ Y q “ p Tr p X Ad p Y q ´ Tr p Y Ad p X qq . Lemma 4.21 (Guruprasad-Huebschmann-Jeffrey-Weinstein [14]) . We have d τ “ λ, δ G τ “ ´ θ. Now define the 2-form ω H on H B by ω H “ η ˚ x E ˚ Ω , ´ c y ´ p r ˚ h exp ˚ λ ` b ÿ i “ p z i ˚ τ i . Proposition 4.22. ω H is closed on H B and a moment map is given by µ “ ψ ˝ p r . Proof.
We first observe that ω H is d-closed. Indeed, from (12) and Lemma 4.21, wehave d ω H “ ´ b ÿ i “ p z ˚ i λ ` b ÿ i “ p z ˚ i d τ i “ . For the second assertion regarding the moment map, we use (13) and Lemma4.21 to compute δ G ω H “ ´ d µ ` b ÿ i “ p z ˚ i θ ` b ÿ i “ p z ˚ i δ G τ i “ ´ d µ. Since µ P A , G p H B q , δ G µ “ ω H ` µ is equivariantlyclosed meaning that µ “ ψ ˝ p r is a moment map for the G -action on H B . (cid:3) Let ρ P µ ´ p q . We know that η p ρ q P Hom p F , G q induces the representation ρ : Γ Ñ G . In view of Lemma 4.4 the tangent space at ρ P µ ´ p q can be identifiedwith Z p Γ , B Γ; g ρ q . The fundamental vector fields X , X P g span the subspace B p Γ , g q . Theorem 4.23.
Let ρ P µ ´ p q . Assume that the induced representation ρ : Γ Ñ G is in Hom B s p Γ , G q . Then ω H,ρ | Z p Γ , B Γ; g ρ q descends to H p Γ , B Γ; g ρ q and de-fines the closed 2-form, denoted by the same symbol ω H , on X B n p Γ q . Moreover theinduced closed 2-form ω H on X B n p Γ q coincides with the right translation of ´ ω P D .Proof.
By the construction, we have δ G ω H “ µ ´ p q . This means that ω H vanishes along B p Γ; g q . Therefore ω H descends to H p Γ , B Γ; g q . By d-closednessof ω H , we know that the induced 2-form is also closed on X B n p Γ q . Since the G -action on Hom B s p Γ , G q Ă µ ´ p q is proper, we can understand this process as theMarsden-Weinstein quotient. YMPLECTIC COORDINATES 31
Recall Theorem 4.13. Since ω P D is anti-symmectric, we have that ω P D pr u s , r v sq “ px u Y v, c y ´ x v Y u, c yq ´ b ÿ i “ Tr p X i v p z i q ´ Y i u p z i qq“ px u Y v, c y ´ x v Y u, c yq ´ b ÿ i “ Tr p X i z i ¨ Y i ´ Y i z i ¨ X i q . (14)The last term coincides with ř bi “ τ p z i ¨ X i ´ X i , z i ¨ Y i ´ Y i q “ ř bi “ z ˚ i τ i . We claimthat the first term px u Y v, c y ´ x v Y u, c yq equals x E ˚ Ω , ´ c yp u, v q . For this, itsuffices to show that x E ˚ Ω , (cid:74) x | y (cid:75) yp u, v q “ px u Y v, (cid:74) x | y (cid:75) y ´ x v Y u, (cid:74) x | y (cid:75) yq for (cid:74) x | y (cid:75) P B p F q b R .Recall that (Lemma 4.2) the right translation identifies Z p Γ; g q with T ρ Hom p Γ , G q .Thus given u P Z p Γ; g q and x P Γ, we have that E x ˚ u “ R ρ p x q˚ p u p x qq where themap E x : Hom p Γ , G q Ñ G is defined by ρ ÞÑ ρ p x q . Then we compute x E ˚ Ω , (cid:74) x | y (cid:75) yp u, v q “ p ι ω ¨ ι ˚ ω q p ρ p x q ,ρ p y qq p E ˚ u, E ˚ v q“ ´ ω ρ p x q p E x ˚ u q ¨ ω ρ p y q p E y ˚ v q ´ ω ρ p x q p E x ˚ v q ¨ ω ρ p y q p E y ˚ u q ¯ “ ` Tr p Ad ρ p x q ´ p u p x qq v p y qq ´ Tr p Ad ρ p x q ´ p v p x qq u p y qq ˘ “ p Tr p u p x q x ¨ v p y qq ´ Tr p v p x q x ¨ u p y qqq . By comparing this to the Alexander-Whitney approximation theorem, Lemma4.9, we conclude that x E ˚ Ω , (cid:74) x | y (cid:75) yp u, v q “ px u Y v, (cid:74) x | y (cid:75) y ´ x v Y u, (cid:74) x | y (cid:75) yq as wewanted.Finally observe that, on the space Hom B p Γ , G q , the second term p r ˚ h exp ˚ λ of ω H vanishes. Therefore ´ ω H equals (14) on X B n p Γ q . (cid:3) Global Darboux coordinates
In this section, we prove our main theorem. Recall that we have the splittingoperation that decomposes a given convex projective orbifold into simpler peaces ofconvex projective orbifolds. We show the decomposition theorem for C B p O q whichis analogous to a previous result of the authors [7, Theorem 4.5.7]. Finally by usingour version of the action-angle principle [7, Theorem 3.4.5], we obtain the globalDarboux coordinates for the symplectic manifold p C p O q , ω O q .Throughout this section, we continue to use B Γ to denote a set of subgroupsgenerated by boundary components. See Notation 4.1.5.1.
Local decomposition.
We will prove that the tangent space can be decom-posed into the direct sum of symplectic subspaces under the splitting operations.Indeed, we did show the same result for compact surfaces. What is new here is thatthe splittings can be done along full 1-suborbifolds. We first deal with this newcase.Let O be a compact oriented cone 2-orbifold of negative Euler characteristic,Γ : “ π orb1 p O q . Let ξ be a principal full 1-suborbifold joining two order 2 cone pointsof O . Note that O “ O z ξ has also negative Euler characteristic. The orbifoldfundamental group Γ : “ π orb1 p O q of O is a subgroup of Γ and we denote by ι : Γ Ñ Γ the inclusion. We can find presentationsΓ “ x x , y , ¨ ¨ ¨ , x g , y g , z , ¨ ¨ ¨ , z b , s , ¨ ¨ ¨ , s c | g ź i “ r x i , y i s b ź i “ z i c ź i “ s i “ s r “ ¨ ¨ ¨ “ s r c c “ y andΓ “ x x , y , ¨ ¨ ¨ , x g , y g , z , ¨ ¨ ¨ , z b ` , s , ¨ ¨ ¨ , s c | g ź i “ r x i , y i s b ` ź i “ z i c ź i “ s i “ s r “ ¨ ¨ ¨ “ s r c c “ y such that r “ r “ ι p z b ` q “ s s . Recall that z , ¨ ¨ ¨ , z b ` correspond tothe boundary components of O .Let S “ B Γ Y tx s s yu . We consider the group pair p Γ , S q . In view of Lemma 3.3, H p Γ , S ; g q is a subspace of H p Γ , B Γ; g q . Hence one can compute ω O pr u s , r v sq by regarding r u s and r v s as members of H p Γ , B Γ; g q . Proposition 5.1.
Keep the notations as above. Let r ρ s P X B n p Γ q be such that r ρ ˝ ι s P X B n p Γ q for some B . Then for any r u s , r v s P H p Γ , S ; g ρ q we have ω O pr u s , r v sq “ ω O p ι ˚ r u s , ι ˚ r v sq . where ι ˚ : H p Γ , S ; g q Ñ H p Γ , B Γ ; g q is the map induced from ι : Γ Ñ Γ .Proof. Recall that the map ι : Γ Ñ Γ induces the equivariant chain map ι ˚ : B ˚ p Γ q Ñ B ˚ p Γ q defined by ι ˚ p (cid:74) x | y (cid:75) q “ (cid:74) ι p x q| ι p y q (cid:75) .Let r “ ś gi “ r x i , y i s ś bi “ z i . The relative fundamental cycle c of O can bewritten as c “ ι ˚ c ´ (cid:74) r | s s (cid:75) ` (cid:74) r | s (cid:75) ` (cid:74) r s | s (cid:75) ´ (cid:74) s | s (cid:75) ´ (cid:74) s | s (cid:75) where c is the relative fundamental cycle of O .As r u s , r v s P H p Γ , S ; g q , one can choose representatives u, v of r u s , r v s so that u p s s q “ v p s s q “
0. Then by the cocycle condition v p s s q “ v p s q ` s ¨ v p s q ,we have Tr p u p r q r ¨ v p s s qq “ Tr p u p r q r ¨ v p s qq ` Tr p u p r qp r s q ¨ v p s qq and Tr p u p r s qp r s q ¨ v p s qq “ Tr pp u p r q ` r ¨ u p s qqp r s q ¨ v p s qq . Therefore we get x u Y v, c y “ x u Y v, ι ˚ c y` Tr p u p s q s ¨ v p s qq ´
12 Tr p u p s q s ¨ v p s qq ´
12 Tr p u p s q s ¨ v p s qq . Since u p s s q “
0, we know that 0 “ u p s q ` s ¨ u p s q . Since s “
1, we can deduce u p s q “ ´ s ¨ u p s q “ u p s q . Similarly, we see v p s q “ v p s q . Hence it follows thatTr p u p s q s ¨ v p s qq ´
12 Tr p u p s q s ¨ v p s qq ´
12 Tr p u p s q s ¨ v p s qq“ ´ Tr p u p s q v p s qq `
12 Tr p u p s q v p s qq `
12 Tr p u p s q v p s qq“ ´ Tr p u p s q v p s qq `
12 Tr p u p s q v p s qq `
12 Tr p u p s q v p s qq“ . YMPLECTIC COORDINATES 33
Finally, our choice of cocycles u, v tells us that X b ` “
0. Combining all, we get ω O pr u s , r v sq “ x u Y v, ι ˚ c y ` Tr p X b ` v p z b ` qq ´ b ` ÿ i “ Tr p X i v p z i qq“ ω O p ι ˚ r u s , ι ˚ r v sq . as desired. (cid:3) We unify the local decomposition theorem in the full generality. Let O be acompact oriented cone 2-orbifold of negative Euler characteristic. Suppose thatwe are given a collection t ξ , ¨ ¨ ¨ , ξ m u of pairwise disjoint essential simple closedcurves or full 1-suborbifolds such that the completions of connected components O , ¨ ¨ ¨ , O l of O z Ť mi “ ξ i have negative Euler characteristic. As before, we denoteΓ : “ π orb1 p O q and Γ i : “ π orb1 p O i q .Let S “ B Γ Y tx ξ y , ¨ ¨ ¨ , x ξ m yu . Here we should clarify what we mean by x ξ i y when ξ i is a full 1-suborbifold. Suppose that the lift r ξ i of ξ i in the universal coveris joining the fixed points of the corresponding generators in π orb1 p O q , say, s and s . Then x ξ i y is, by convention, x s s y .For each i , there is the inclusion ι i : Γ i Ñ Γ which induces ι ˚ i : H p Γ , S ; g q Ñ H p Γ i , B Γ i ; g q . Theorem 5.2.
Let O be a compact oriented cone 2-orbifold of negative Euler char-acteristic and Γ “ π orb1 p O q . Let t ξ , ¨ ¨ ¨ , ξ m u be a set of pairwise disjoint essen-tial simple closed curves or full 1-suborbifolds such that the completions of con-nected components O , ¨ ¨ ¨ , O l of O z Ť mi “ ξ i have negative Euler characteristic. Let r ρ s P X B n p Γ q be such that, for each i , r ρ ˝ ι i s P X B i n p Γ i q for some choice of conjugacyclasses B i . Then for any r u s , r v s P H p Γ , S ; g q we have ω O pr u s , r v sq “ l ÿ i “ ω O i p ι ˚ i r u s , ι ˚ i r v sq . Remark . Again, r u s and r v s should be understood (by Lemma 3.3) as membersof H p Γ , B Γ; g q so that the left hand side ω O pr u s , r v sq makes sense. Proof.
This is a combination of Corollary 4.4.3 of [7] and Proposition 5.1.Renumbering if necessary, we may assume that ξ , ¨ ¨ ¨ , ξ k represent essential sim-ple closed curves and remaining ξ k ` , ¨ ¨ ¨ , ξ m are full 1-suborbifolds. Apply Corol-lary 4.4.3 of [7]. Although Corollary 4.4.3 of [7] is stated for compact surfaces, thesame assertion and the proof work for compact orbifolds. Then we use Proposition5.1 inductively for each full 1-suborbifold ξ i , i ě k ` (cid:3) Hamiltonian action and global decomposition.
Now we deal with theglobal decomposition theorem. Throughout this subsection we focus on n “ G denotes PSL p R q .We briefly summarize Goldman’s work on Hamiltonian flows. By an invariantfunction , we mean a smooth function f : U Ñ R defined on an nonempty con-jugation invariant subset U of G such that f p ghg ´ q “ f p h q for all h P U and g P G . Given an invariant function f we can associate so call the Goldman function f : U Ñ g whose defining property is that dd t | t “ f p h exp tX q “ Tr p f p h q X q forall X P g .We fix invariant functions (cid:96) and m defined on the set Hyp ` of hyperbolic ele-ments of G . (cid:96) p g q : “ log ˇˇˇˇ λ p g q λ p g q ˇˇˇˇ , m p g q : “ log | λ p g q| where λ i p g q is the i th largest eigenvalue of g P Hyp ` . We can observe that p (cid:96), m q is a set of complete invariants for elements in Hyp ` . Namely, two elements g, h P Hyp ` are conjugate if and only if p (cid:96) p g q , m p g qq “ p (cid:96) p h q , m p h qq .Let ξ , ¨ ¨ ¨ , ξ m be essential simple closed curves and let ξ m ` , ¨ ¨ ¨ , ξ m ` m e befull 1-suborbifolds. Set m “ m ` m e and M “ m ` m e . Suppose that ξ , ¨ ¨ ¨ , ξ m are mutually disjoint, non-isotopic and each component of O z Ť mi “ ξ i has negativeEuler characteristic.We apply the argument of Choi-Jung-Kim [7] with respect to the closed curves ξ , ¨ ¨ ¨ , ξ m . This process identifies π orb1 p O q with the fundamental group of a graphof groups p Γ , G q where Γ “ π orb1 p O q . The graph G has the vertex set V p G q : “t O , ¨ ¨ ¨ , O l u and we join O i and O j , possibly i “ j , by an edge if and only if theyare glued along some ξ k . We choose a base vertex say O and a maximal spanningtree T of G . Then V p G q admits the natural partial ordering ď with the minimalelement O . The vertex group Γ O i of O i is isomorphic to π orb1 p O i q and the edgegroup Γ ξ i of ξ i is the infinite cyclic generated by the abstract generator e i . If ξ i , i ď m , is a loop in G , then we a need extra generator e K i . For each edge ξ i , we havetwo homomorphisms p¨q ˘ : Γ ξ i Ñ Γ p ξ i q ˘ where p ξ i q ` and p ξ i q ´ are the terminal andthe initial vertex of ξ i respectively. All these generators are subject to the followingrelations(15) e ` i “ e ´ i and(16) e K i e ` i p e K i q ´ “ e ´ i . Since Γ O j is isomorphic to π orb1 p O j q , we can choose a presentation of each Γ O i Γ O j “ x x j, , y j, , ¨ ¨ ¨ , x j,g j , y j,g j , z j, , ¨ ¨ ¨ , z j,b j , s j, , ¨ ¨ ¨ , s j,c j | g j ź i “ r x j,i , y j,i s b j ź i “ z j,i c j ź i “ s j,i “ s r j, j, “ ¨ ¨ ¨ s r j,cj j,c j “ y such that ‚ the homomorphisms Γ ξ a Ñ Γ ξ ˘ a at an edge ξ a connecting O p ă O q aregiven by e ` a “ z q,i and e ´ a “ z ´ p,j for some i, j . ‚ if ξ i is in O k then the lift of ξ i in the universal cover joins the fixed pointsof s k,i and s k,i ` for some i .To make our notation consistent, we formally define e ` i , i ą m , to be z ´ k,b k ` “ s k, s k, if ξ i is a full 1-suborbifold in O k whose lift in the universal cover joins thefixed points of s k, and s k, .Invariant functions (cid:96) and m induce smooth functions on C B p O q defined by (cid:96) i pr ρ sq : “ (cid:96) p ρ p ξ i qq , i “ , , ¨ ¨ ¨ , m and m i pr ρ sq : “ m p ρ p ξ i qq for each i “ , , ¨ ¨ ¨ , m .Now following Goldman [11, Theorem 4.3] and Choi-Jung-Kim [7, Theorem 4.5.6],we define the flows L it and M it along ξ i for i “ , , ¨ ¨ ¨ , m . The flow along theprincipal full 1-suborbifold ξ i , i ą m is something new. For readers’ convenience,we present their definitions. – Case I: ξ i is a simple closed curve and in T . Suppose that ξ i joins O p and O q , O p ă O q . Then for each t P R , we define L it by YMPLECTIC COORDINATES 35 L it pr ρ sqp x q “ $’’’’’’&’’’’’’% ρ p x q if x P Γ O r , O r ě O q exp p t(cid:96) p ρ p e ` i qqq ρ p x q exp p´ t(cid:96) p ρ p e ` i qqq if x P Γ O r , O r ě O q exp p t(cid:96) p ρ p e ` i qqq ρ p x q exp p´ t(cid:96) p ρ p e ` i qqq if x “ e K r and ξ ` r , ξ ´ r ě O q ρ p x q exp p´ t(cid:96) p ρ p e ` i qqq if x “ e K r and ξ ` r ě O q exp p t(cid:96) p ρ p e ` i qqq ρ p x q if x “ e K r and ξ ´ r ě O q and similarly M it with (cid:96) replaced by m . Note that L it pr ρ sq respects the relations(15) and (16). – Case II: ξ i is a simple closed curve and not in T . This case we have simplerexpressions L it pr ρ sqp x q “ ρ p x q exp p t(cid:96) p ρ p e ` i qqq if x “ e K i ρ p x q otherwiseand again similarly M it . – Case III: ξ i is a full 1-suborbifold. In this case, we only have flows of the type L it . Suppose that the lift of ξ i in the universal cover of O j joins fixed points of ordertwo generators s j, and s j, . Then we understand the word z j,b j ` as s j, s j, . Foreach t P R , we define L it pr ρ sqp x q “ exp p t(cid:96) p ρ p e ` j qqq ρ p x q exp p´ t(cid:96) p ρ p e ` j qqq if x “ s j, , s j, ρ p x q otherwise . Lemma 5.4.
The flows L it , M it defined above are Hamiltonian, complete and com-mutative each other. Consequently, they give rise to the Hamiltonian action of theabelian Lie group R M .Proof. The embedding I is symplectic and preserves the flows. Thus the flows L it , M it , i “ , , ¨ ¨ ¨ , m being commutative and Hamiltonian are consequences of [7,Theorem 4.5.6]. Completeness of L t , ¨ ¨ ¨ , L mt , M t , ¨ ¨ ¨ , M m t follows from section3.4.1 and section 3.4.2 of Choi-Goldman [6]. (cid:3) The moment map µ : C B p O q Ñ p R M q ˚ of this action is given by invariants of ρ p ξ i q , that is, r ρ s ÞÑ p (cid:96) pr ρ sq , ¨ ¨ ¨ , (cid:96) m pr ρ sq , m pr ρ sq , ¨ ¨ ¨ , m m pr ρ sqq . Here we identify R M with its dual by the canonical inner product. See section 4.5of [7] for details. Lemma 5.5.
The R M -action is free.Proof. Suppose that m “ m “ m e “ R M -action is free. The lemma follows byinduction on m . (cid:3) Suppose that we are given a value y in the image of the moment map µ . Sincethe invariant functions (cid:96) and m form a set of complete invariants, the value of µ determines the conjugacy classes of ρ p e ` i q say C i . That is, µ ´ p y q is the set of classesof representations r ρ s in C B p O q such that ρ p e ` i q P C i for each i “ , , ¨ ¨ ¨ , m . Thisalso defines a set of conjugacy class B i for each i “ , , ¨ ¨ ¨ , l such that the imageof the projection C B p O q Ñ C p O i q lies in C B i p O i q .For each y in the image of the moment map µ , the splitting map SP induces themap SP y : µ ´ p y q Ñ C B p O q ˆ ¨ ¨ ¨ ˆ C B l p O l q . Recall that, in terms of holonomy, SP y is given by r ρ s ÞÑ pr ρ ˝ ι s , r ρ ˝ ι s , ¨ ¨ ¨ , r ρ ˝ ι l sq . We observe that SP y descends to the map SP y : µ ´ p y q{ R M Ñ C B p O q ˆ ¨ ¨ ¨ ˆ C B l p O l q . Lemma 5.6.
For every y P Image p µ q , the R M -action on µ ´ p y q is proper. This lemma is implicitly shown in Choi-Goldman [6]. To be more precise weinclude two proofs for this lemma. The first proof is indirect; we prove that µ ´ p y q is a fiber bundle over the image. The second proof is a variant of the authors’previous paper [7] which is more intrinsic but slightly complicated. Proof 1.
Let S “ B Γ Ytx ξ y , ¨ ¨ ¨ , x ξ m yu . One can identify T r ρ s µ ´ p y q with H p Γ , S ; g ρ q .There is a Mayer-Vietoris type short exact sequence0 Ñ m à i “ H px ξ i y ; g q δ Ñ H p Γ , S ; g q d SP y Ñ l à i “ H p Γ i , B Γ i ; g q Ñ δ in H p Γ , S ; g q is the tangent directions of the R M -action.This shows that SP y is a local diffeomorphism and that SP y : µ ´ p y q Ñ C B p O qˆ¨ ¨ ¨ ˆ C B l p O l q is a submersion. Since SP y is known to be one-to-one and onto weknow that SP y is a diffeomorphism. In particular, µ ´ p y q Ñ µ ´ p y q{ R M is also asubmersion whose fiber coincides with that of SP y .Since µ ´ p y q Ñ µ ´ p y q{ R M is a submersion, we know that each fiber of µ ´ p y q Ñ µ ´ p y q{ R M is an embedded submanifold and since Lemma 5.5 shows that R M actssimply transitively on each fiber, we know that each fiber is homeomorphic to R M .By Corollary 31 of Meigniez [26], we know that µ ´ p y q Ñ µ ´ p y q{ R M is a fiberbundle. Hence the action is proper. (cid:3) Proof 2.
Let H p y q “ t ρ P Hom p π p X O q , G q | r ρ s P µ ´ p y qu . A slight modification of the proof of Lemma 4.5.4 of [7] shows that the R M -actionon H p y q is proper.Now we construct an equivariant section of H p y q Ñ µ ´ p y q . For each ρ P H p y q ,we have a unique limit map ξ ρ : B r O Ñ RP . Choose four points x , ¨ ¨ ¨ , x P B r O ,no two of which are in the same orbit. We also choose four generic points v , ¨ ¨ ¨ , v in RP . We define the section s : µ ´ p y q Ñ H by declaring that s pr ρ sq is the unique ρ P H p y q such that ξ ρ p x i q “ v i . Then s is clearly the desired equivariant section.For a given compact set K of µ ´ p y q , we know that t t P R M | t ¨ s p K qX s p K q ‰ Hu is compact. Since s is equivariant, we have t t P R M | t ¨ s p K q X s p K q ‰ Hu “ t t P R M | t ¨ K X K ‰ Hu . Therefore, the R M -action on µ ´ p y q is also proper. (cid:3) Since the R M -action on µ ´ p y q is proper, we can take the Marsden-Weinsteinquotient µ ´ p y q{ R M . On the other hand, the right hand side admits the symplecticstructure given by the orthogonal sum ω O ‘ ¨ ¨ ¨ ‘ ω O l .By appealing to Theorem 5.2 and Proposition 4.17, we obtain the followingdecomposition theorem Theorem 5.7.
Let O be a compact oriented cone 2-orbifold of negative Euler char-acteristic. Let ξ , ¨ ¨ ¨ , ξ m be pairwise disjoint essential simple closed curves or full 1-suborbifolds such that the completion of each connected component O i of O z Ť mi “ ξ i has also negative Euler characteristic. Then the map SP y : µ ´ p y q{ R M Ñ C B p O q ˆ ¨ ¨ ¨ ˆ C B l p O l q defined above is a symplectomorphism.Proof. From the proof of Lemma 5.6, we can conclude that SP y is a diffeomorphism.Since SP y is induced from ι i : Γ i Ñ Γ, Theorem 5.2 tells us that SP y is asymplectomorphism. (cid:3) YMPLECTIC COORDINATES 37
Construction of global coordinates.
To make upcoming indices neat, weintroduce the following notation.
Notation 5.8.
Let O be a closed oriented cone 2-orbifold of negative Euler char-acteristic, with genus g , c cone points and c b order two cone points. Let I “ I p g, c, c b q : “ g ´ ` c ´ t c b { u if c b is even2 g ´ ` c ´ t c b { u ´ c b is odd ,J “ J p g, c, c b q : “ g ´ ` c ´ t c b { u and K “ K p g, c q : “ g ´ ` c. We prove that O can be decomposed into so called elementary orbifolds of types(P1),(P2),(P3) and (P4). This is analogues to the pair-of-pants decomposition of asurface but we require more building blocks.For the definition of elementary orbifolds, we refer readers to Choi-Goldman [6]. Lemma 5.9.
There is a decomposition of O by essential simple closed curves ξ , ¨ ¨ ¨ , ξ J and full 1-suborbifolds ξ J ` , ¨ ¨ ¨ , ξ K such that each component of O z Ť Ki “ ξ i is an elementary orbifold of the type (P1), (P2), (P3), or (P4).Proof. If O has order 2 cone points, we pair them by full 1-suborbifolds ξ J ` , ¨ ¨ ¨ , ξ K .By splitting O along ξ J ` , ¨ ¨ ¨ , ξ K , we get another orbifold O of the same genus g with t c b { u boundary components, c ´ t c b { u cone points and at most one order twocone point. Let C “ t ξ , ¨ ¨ ¨ , ξ J u be a pair-of-pants decomposition of X O . Thenit is clear that C decomposes O into 2 g ´ ` c ´ t c b { u connected components E , ¨ ¨ ¨ , E g ´ ` c ´ t c b { u each of which is an elementary orbifold of the type (P1),(P2), (P3), or (P4). (cid:3) If an elementary orbifold E of type (P1), (P2), (P3) or (P4) has an order 2cone point, then the deformation space C B p E q becomes singleton. Call such an ele-mentary orbifold exceptional . Every non-exceptional elementary orbifold E of type(P1), (P2), (P3) or (P4) has two dimensional deformation space C B p E q . Observethat among the pieces E i of Lemma 5.9, there appears at most one exceptionalelementary orbifold depending on the parity of c b .There is the smooth map SP : C p O q Ñ (cid:52) given by the restriction on eachcomponent or ι where (cid:52) is the subset of C p E q ˆ ¨ ¨ ¨ ˆ C p E g ´ ` c ´ t c b { u q whichsatisfies the condition that if γ i and γ j are gluing boundaries, then they have thesame invariants. Note that SP restricts on each leaf µ ´ p y q to SP y .Our first coordinates are called length coordinates µ “ p (cid:96) , ¨ ¨ ¨ , (cid:96) K , m , ¨ ¨ ¨ , m J q . Lemma 5.10.
There is a section of the principal R M -fiber bundle map SP : C p O q Ñ (cid:52) admits a global section say s where M “ J ` t c b { u “ g ´ ` c ´ t c b { u .Proof. We know, from Choi-Goldman [6], that (cid:52) “ p R ` q I ˆ L J ˆ p R ` q K ´ J where L “ tp l, m q P R | l ą , m ` l ą u . Indeed, there are parameters p s i , t i q defined in Choi-Goldman [6] that parametrize C B i p E i q when E i is non-exceptional.This p s i , t i q parameters take all values in R ` “ tp x, y q P R | x, y ą u . Theycorrespond to the first factor p R ` q I . The second factor comes from the range ofinvariants p (cid:96), m q associated to simple closed geodesics. The last factor is the rangeof (cid:96) associated to principal full 1-suborbifolds.Since, (cid:52) is contractible, SP is the trivial R M -bundle. (cid:3) Remark . In Choi-Goldman [6], the parameter s is defined merely as a param-eter for the solution space of the system of linear equations [6, eqns. (23), (24)]and is not specified explicitly. In this paper we use γ (eq. (20) in [6]) as a freeparameter for this system and declare this variable s .The twisting coordinates θ , ¨ ¨ ¨ , θ K , ϕ , ¨ ¨ ¨ , ϕ J are defined by integration fromthe image of s along the Hamiltonian flow. We finally define internal coordinates s i : “ log s i ˝ SP and t i : “ log t i ˝ SP . Altogether, we get the global coordinates for C p O q . Observe that there are 16 g ´ ` c ´ c b coordinates, which is equal to the dimension of C p O q . These coordinates,however, may not be Darboux.5.4. Existence of global Darboux coordinates.
We now show that C p O q ad-mits global Darboux coordinates. We begin with the global coordinates constructedin the previous section. Theorem 5.12.
Let E i be a non-exceptional elementary orbifold of the type (P1),(P2), (P3), or (P4). Let p s i , t i q be the coordinates for C B i p E i q . Then ω E i “ d t i ^ d s i . Proof.
We use Mathematica to evaluate the formula for ω E i (Theorem 4.13). Math-ematica files are available at [19]. (cid:3) Remark . For (P1), a pair-of-pants case, the same result is already shown byH. Kim [21, Theorem 5.8].By following the arguments of Choi-Jung-Kim [7] directly, one can obtain theparallel results. We list here some of key facts without proofs.
Lemma 5.14.
For each i , the Hamiltonian vector field X s i is of the form X s i “ BB t i ` K ÿ j “ a j BB θ j ` J ÿ j “ b j BB ϕ j for some smooth functions a j and b j . Lemma 5.15.
For each i , the Hamiltonian vector field X s i is complete and X s , ¨ ¨ ¨ , X s I , X (cid:96) , ¨ ¨ ¨ , X (cid:96) K , X m , ¨ ¨ ¨ , X m J span a Lagrangian subspace of T r ρ s C p O q at each r ρ s . As a consequence, the above Hamiltonian vector fields induces the Hamiltonian R I ` J ` K -action on C p O q . Let us define the function F : C p O q Ñ R I ` J ` K by(17) F pr ρ sq “ p s pr ρ sq , ¨ ¨ ¨ , s I pr ρ sq , (cid:96) pr ρ sq , ¨ ¨ ¨ , (cid:96) K pr ρ sq , m pr ρ sq , ¨ ¨ ¨ , m J pr ρ sqq . Then the R I ` J ` K -action preserves each fiber of F . To make use of Theorem 3.4.5of [7], we should have the following result Lemma 5.16.
The R I ` J ` K -action on C p O q is free, proper. In addition, the actionis transitive on each fiber. Therefore F is a fiber bundle such that each fiber isLagrangian and diffeomorphic to R I ` J ` K .Proof. By Lemma 5.14, the R I ` J ` K -action is free and fiberwise transitive. More-over, F is a submersion since F is a projection with respect to the Goldman co-ordinates. Therefore, each fiber of F is homeomorphic to R I ` J ` K . Now we applyCorollary 31 of [26]. (cid:3) YMPLECTIC COORDINATES 39
Now we can deduce our main theorem:
Theorem 5.17.
Let O be a closed oriented cone 2-orbifold of negative Euler char-acteristic, with genus g and c cone points. Then C p O q admits a global Darbouxcoordinates system.Proof. Apply Theorem 3.4.5 of [7] to the function F defined in (17). Then thereare complementary coordinates p s , ¨ ¨ ¨ , s I , (cid:96) , ¨ ¨ ¨ , (cid:96) K , m , ¨ ¨ ¨ , m J q such that ω O “ I p g,c,c b q ÿ i “ d s i ^ d s i ` K p g,c q ÿ i “ d (cid:96) i ^ d (cid:96) i ` J p g,c,c b q ÿ i “ d m i ^ d m i . This completes the proof. (cid:3)
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Department of Mathematical Sciences, KAIST, Daejeon 34141, Republic of Korea
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