THE SYMPLECTIC GLOBAL

COORDINATES ON THE MODULI SPACE

OF REAL PROJECTIVE STRUCTURES

HONG CHAN KIM

A convex real projective structure on a smooth surface M is a representation of M as a quotient of a convex domain Ω ⊂ ℝℙ^{2} by a discrete group Γ ⊂ PGL(3, ℝ) acting properly and freely on Ω. If χ(M) < 0, then the equivalence classes of convex real projective structures form a moduli space 𝔓(M) which is an extension of the Teichmüller space 𝔗(M).

Wolpert [17] proved that the Weil-Petersson Kähler form of the Teichmüller space 𝔗(M) of a closed surface Σ(g, 0) with χ(M) < 0 is expressed by

ω = ∑_{i=1}^{g} dℓ_{i} ∧ dθ_{i}

where ℓ_{i}, θ_{i} are Fenchel-Nielsen coordinates on 𝔗(M). In this paper, I will prove 𝔓(M) has analogous properties.

1991 Mathematics Subject Classification. 53D30, 53A20.

Key words and phrases. symplectic structure, moduli space, real projective structure, group cohomology, fundamental cycle, symplectic global coordinates.

The author was partially supported by Korea Institute for Advanced Study.

2 HONG CHAN KIM

In Section 1, we study the set of positive hyperbolic elements Hyp_{+} of PGL(3, ℝ) since the holonomy group Γ of a convex real projective structure lies in Hyp_{+}. In Section 2, we show the parameters (ℓ, m) on 𝔓(M) extend Fenchel-Nielsen’s length parameter ℓ. Let π be the fundamental group of M and G a connected algebraic Lie group. In Section 3, we study local properties of Hom(π, G)/G since 𝔓(M) embeds onto an open subset of Hom(π, PGL(3, ℝ))/PGL(3, ℝ). In Section 4, we induce a symplectic form ω on 𝔓(M) for a closed surface M by Fox’s calculus and the fundamental cycle. In Section 5, we generalize 𝔓(M) for a compact oriented surface M with boundary. Restriction to boundary components defines the symplectic leaves of the parabolic foliation of a Poisson structure on 𝔓(M). We show the modified form ω̃ is a symplectic form on each parabolic leaf. We actually calculate the symplectic forms ω̃ for the pair of pants with using Mathematica. In last Section 6, we induce generalized twist parameters θ, β and prove that 𝔓(M) is symplectomorphic to ℝ^{16g−16}.

Theorem 0.1. Let M = Σ(g, 0) a closed smooth surface. Then the symplectic form on the moduli space 𝔓(M) of convex real projective structures is

ω = ∑_{i=1}^{3g−3} dℓ_{i} ∧ dθ_{i} + ∑_{i=1}^{3g−3} dm_{i} ∧ dβ_{i} + ∑_{j=1}^{2g−2} dt_{j} ∧ ds_{j}

where ℓ_{i}, m_{i} are length parameters, θ_{i}, β_{i} are twisting parameters, and s_{j}, t_{j} are internal parameters on 𝔓(M).

Acknowledgements

This work has benefited from discussions and correspondence with J. Millson, R. Schwartz, and S. Wolpert. Especially I wish to thank my advisor W. Goldman for introducing me this subject and his sincere guidance. I also grateful to the referee for reading this paper.

1. Real projective structures on a compact oriented surface

1.1. The real projective plane ℝℙ^{2} is the space of all lines through the origin in ℝ^{3}. Let A be an element of GL(3, ℝ), the group of linear transformations of ℝ^{3}. Then A preserves lines through the origin and induces a projective transformation of ℝℙ^{2}. The group of projective transformations of ℝℙ^{2} is denoted by PGL(3, ℝ). Since the scalar matrices ℝ^{∗} in GL(3, ℝ) act trivially on ℝℙ^{2}, we have an exact sequence

{1} → ℝ^{∗} → GL(3, ℝ) → PGL(3, ℝ) → {1}.

The homomorphism GL(3, ℝ) → SL(3, ℝ) defined by

A ↦ (detA)^{−1/3}A

induces an isomorphism PGL(3, ℝ) → SL(3, ℝ) as analytic groups. Thus from now on we shall identify the groups PGL(3, ℝ) and SL(3, ℝ).

1.2. An element A ∈ SL(3, ℝ) is called positive hyperbolic if it has three distinct positive real eigenvalues. The set of positive hyperbolic elements of SL(3, ℝ) is denoted by Hyp_{+}. If A is positive hyperbolic, then it can be represented by the diagonal matrix

(1)  [ \begin{matrix}λ & 0 & 0 \\ 0 & μ & 0 \\ 0 & 0 & ν \end{matrix} ]

with

(2)  λμν = 1, 0 < λ < μ < ν

4 HONG CHAN KIM

via an SL(3, ℝ)-conjugation. We define

(λ, τ) : Hyp_{+} → ℝ^{2}

by λ(A) to be the smallest eigenvalue of A and τ(A) the sum of the other two eigenvalues. For the A above,

λ(A) = λ, τ(A) = μ + ν.

As you see the pair (λ(A), τ(A)) is invariant under SL(3, ℝ)-conjugation.

The following two Propositions 1.1 and 1.2 are works from Goldman [8].

Proposition 1.1. Consider the action of SL(3, ℝ) on Hyp_{+} by conjugation. Then (λ, τ) : Hyp_{+} → ℝ^{2} is an SL(3, ℝ)-invariant fibration with the image

I = \{(λ, τ) ∈ ℝ^{2} | 0 < λ < 1, 2/\sqrt{λ} < τ < λ + λ^{−2}\}.
If (λ, τ) ∈ I, then A ∈ Hyp_{+} is determined by
λ = λ(A)
μ = \frac{1}{2} \{τ(A) − \sqrt{τ(A)^{2} − 4/λ(A)}\}
ν = \frac{1}{2} \{τ(A) + \sqrt{τ(A)^{2} − 4/λ(A)}\}

up to SL(3, ℝ)-conjugation.

Another pair of invariants of a positive hyperbolic element is more geometric. They are related to the translation lengths of A. I will explain this in Section 2. We define

ℓ(A) = log (\frac{ν}{λ}) , m(A) = \frac{3}{2} log(μ)

where A is a positive hyperbolic projective transformation represented by the diagonal matrix (1) with λμν = 1 and 0 < λ < μ < ν.

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 5

Proposition 1.2. Consider the action of SL(3, ℝ) on Hyp_{+} by conjugation. Then (ℓ, m) : Hyp_{+} → ℝ^{2} is an SL(3, ℝ)-invariant fibration with the image

I′ = \{(ℓ, m) ∈ ℝ^{2} | ℓ > 0, −\frac{1}{2} ℓ < m < \frac{1}{2} ℓ \}.

If (ℓ, m) ∈ I′, then A ∈ Hyp_{+} is determined by
λ = exp (−\frac{ℓ(A)}{2} − \frac{m(A)}{3})
μ = exp (\frac{2 m(A)}{3})
ν = exp ( \frac{ℓ(A)}{2} − \frac{m(A)}{3})

up to SL(3, ℝ)-conjugation. Therefore (λ, τ) and (ℓ, m) are two sets of coordinates on the conjugacy classes in Hyp_{+}.

1.3. Let Ω be an open subset of ℝℙ^{2}. A map ϕ : Ω → ℝℙ^{2} is called locally projective if for each component W ⊂ Ω, there exists a projective transformation g ∈ PGL(3, ℝ) such that ϕ|_{W} = g|_{W} . Clearly a locally projective map is a local diffeomorphism. Let M be a connected smooth surface. A real projective structure or ℝℙ^{2}-structure on M is a maximal collection {(U_{α}, ψ_{α})} such that

(1) {U_{α}} is an open covering of M.

(2) For each α, ψ_{α} : U_{α} → ℝℙ^{2} is a diffeomorphism onto its image.

(3) The change of coordinates is locally projective ; If (U_{α}, ψ_{α}) and (U_{β}, ψ_{β}) are two coordinate charts with U_{α} ∩ U_{β} ≠ ∅, then ψ_{β} ∘ ψ_{α}^{−1} : ψ_{α}(U_{α} ∩ U_{β}) → ψ_{β}(U_{α} ∩ U_{β}) is locally projective. A smooth surface with an ℝℙ^{2}-structure is called an ℝℙ^{2}-manifold.

Let M, N be smooth surfaces. If f : M → N is a smooth map which is a local diffeomorphism and N is an ℝℙ^{2}-manifold, then we can give

6 HONG CHAN KIM

an ℝℙ^{2}-structure on M via f. In particular every covering space of an ℝℙ^{2}-manifold has the canonically induced ℝℙ^{2}-structure.

The following Development Theorem is the fundamental fact about ℝℙ^{2}-structures.

Theorem 1.3 (Ch. Ehresmann [5]). Let p : M̃ → M denote a universal covering map of an ℝℙ^{2}-manifold M and π the corresponding group of covering transformations.

(1) There exist a projective map dev : M̃ → ℝℙ^{2} and homomorphism h : π → PGL(3, ℝ) such that for each γ ∈ π the following diagram commutes.

M̃ ⟶^{dev} ℝℙ^{2}
γ↓ ↓^{h(γ)}
M̃ ⟶_{dev} ℝℙ^{2}

(2) Suppose (dev′, h′) is another pair satisfying above conditions, then there exists a projective transformation g ∈ PGL(3, ℝ) such that dev′ = g∘dev and h′ = ι_{g}∘h where ι_{g} : PGL(3, ℝ) → PGL(3, ℝ) denotes the inner automorphism defined by g ; that is h′(γ) = (ι_{g} ∘ h)(γ) = g ∘ h(γ) ∘ g^{−1}.

M̃ ⟶^{dev} ℝℙ^{2} ⟶^{g} ℝℙ^{2}
γ↓ ↓^{h(γ)} ↓^{h′(γ)}
M̃ ⟶_{dev} ℝℙ^{2} ⟶_{g} ℝℙ^{2}

The projective map dev : M̃ → ℝℙ^{2} is called a developing map. The homomorphism h : π → PGL(3, ℝ) is called the holonomy homomorphism. The image Γ = h(π) ⊂ PGL(3, ℝ) is called the holonomy group.

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 7

The developing image dev(M̃) ⊂ ℝℙ^{2} is a Γ-invariant open set. In many cases the developing map is a diffeomorphism from M̃ onto dev(M̃). In this case the holonomy homomorphism is an isomorphism from π onto a discrete subgroup of PGL(3, ℝ) which acts properly and freely on dev(M̃). In general it is not. We can find examples in Sullivan and Thurston’s paper [15].

A domain Ω ⊂ ℝℙ^{2} is called convex if there exist a projective line l ⊂ ℝℙ^{2} such that Ω ⊂ (ℝℙ^{2} − l) and Ω is a convex subset of the affine plane ℝℙ^{2} − l ; that is if x, y ∈ Ω, then the line segment x̄y lies in Ω. By definition, ℝℙ^{2} itself is not convex. A convex domain Ω is called strictly convex if Ω is a proper subset of ℝℙ^{2} − l. A geodesic on M is a curve g ⊂ M such that for each component g̃_{0} ⊂ p^{−1}(g) ⊂ M̃, the developing map takes g̃_{0} into a line in ℝℙ^{2}.

An ℝℙ^{2}-structure on M is convex if dev is a diffeomorphism onto a convex domain in ℝℙ^{2}. If M is a convex ℝℙ^{2}-manifold with boundary, then we always assume each boundary component is geodesic. The following fundamental facts are from Goldman’s paper [8].

Proposition 1.4. Let M be an ℝℙ^{2}-manifold. Then the following statements are equivalent.

(1) M has a convex ℝℙ^{2}-structure.

(2) M is projectively isomorphic to a quotient Ω/Γ where Ω ⊂ ℝℙ^{2} is a convex domain and Γ ⊂ PGL(3, ℝ) is a discrete group acting properly and freely on Ω.

(3) Every path in M is homotopic (relative end points) to a unique geodesic path.

If M is a convex ℝℙ^{2}-manifold, then we can identify M = Ω/Γ where Ω is the developing image and Γ the holonomy group.

8 HONG CHAN KIM

Theorem 1.5 (N. Kuiper [12]). Let M = Ω/Γ be a compact oriented surface with a convex ℝℙ^{2}-structure. Suppose that χ(M) < 0. Then the following statements hold.

(1) Ω ⊂ ℝℙ^{2} is a strictly convex domain with C^{1} boundary and therefore contains no affine line.

(2) Either ∂Ω is a conic in ℝℙ^{2} or is not C^{1+ε} for any 0 < ε < 1.

(3) If A ∈ Γ is nontrivial, then A ∈ Hyp_{+}. Furthermore every homotopically nontrivial closed curve on M is freely homotopic to a unique closed geodesic.

2. The geometric meaning of invariants (ℓ, m)

In this section we will show the invariants (ℓ, m) extend Fenchel-Nielsen’s length parameter. See Chapter 2 of Abikoff’s book [1] about Fenchel-Nielsen coordinates on the Teichmüller space.

2.1. For any A ∈ Hyp_{+} there exist three non-collinear fixed points and an A-invariant line in ℝℙ^{2}. We define

• Fix_{−}(A) ⇔ the repelling fixed point of A ⇔ the fixed point corresponding to an eigenvector in ℝ^{3} for the smallest eigenvalue λ,

• Fix_{0}(A) ⇔ the saddle fixed point of A ⇔ the fixed point corresponding to an eigenvector in ℝ^{3} for the middle eigenvalue μ,

• Fix_{+}(A) ⇔ the attracting fixed point of A ⇔ the fixed point corresponding to an eigenvector in ℝ^{3} for the largest eigenvalue ν,

• σ(A) ⇔ the principal line of A ⇔ the line joining the repelling and attracting fixed points of A

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 9

Before we discuss the geometric meaning of the invariants (ℓ, m), we need some knowledge about the cross-ratio and Hilbert distance.

Let ℂ̄ denote ℂ ∪ {∞} the extended complex numbers and D_{4}(ℂ̄) the subset of ℂ̄ × ℂ̄ × ℂ̄ × ℂ̄ consisting of distinct four points. The cross-ratio is the mapping X : D_{4}(ℂ̄) → ℂ̄ defined by

X\{w_{1}, w_{2}, w_{3}, w_{4}\} = \frac{(w_{1} − w_{3})(w_{2} − w_{4})}{(w_{1} − w_{2})(w_{3} − w_{4})},

and is invariant under the GL(2, ℂ)-action on D_{4}(ℂ̄). Let a_{i}, 1 ≤ i ≤ 4, be collinear distinct four points of ℝℙ^{2}. Then there exists B ∈ SL(3, ℝ) such that the second homogeneous coordinate of each B(a_{i}) is zero. Through the identification

(3)  [ \begin{matrix}x \\ 0 \\ z \end{matrix} ] = \{ \begin{matrix}x/z & if z ≠ 0 \\ ∞ & if z = 0 , \end{matrix}

we can think of B(a_{i}) ∈ ℝ ∪ {∞} the extended real line. Furthermore they are distinct. Let CD_{4}(ℝℙ^{2}) denote the subset of ℝℙ^{2} × ℝℙ^{2} × ℝℙ^{2} × ℝℙ^{2} consisting of all collinear distinct four points. The crossratio CR : CD_{4}(ℝℙ^{2}) → ℝ is defined by

CR{a_{1}, a_{2}, a_{3}, a_{4}} = X{B(a_{1}), B(a_{2}), B(a_{3}), B(a_{4})}

where B ∈ SL(3, ℝ) satisfying B(a_{i}) lies in the extended real line ℝ ∪ {∞} through the identification (3) for each i. Suppose B′ is another such element in SL(3, ℝ); then we can show B^{−1}B′ ∈ SL(2, ℝ) via the identification

(4)  [ \begin{matrix}a & b \\ c & d \end{matrix} ] ↔ [ \begin{matrix}a & 0 & b \\ 0 & 1 & 0 \\ c & 0 & d \end{matrix} ] .

\overline{}x  a  b  y

Figure 1. The Hilbert distance

Therefore the cross-ratio CR on CD_{4}(ℝℙ^{2}) is independent of the choice of B ∈ SL(3, ℝ).

The Hilbert distance h : ℝℙ^{2} × ℝℙ^{2} → ℝ^{+} is defined by

h(a, b) = inf_{x⃗y} (log CR{x, a, b, y})

where the points a, b lie on the oriented segment x⃗y with a is the first, b the second point and inf runs over all such x⃗y. The Hilbert distance defines a metric on a strictly convex subset Ω of ℝℙ^{2} and it is called the Hilbert metric. From Theorem 1.5, if ∂Ω is a conic in the Klein model of unit disc hyperbolic space, then the Hilbert metric is the hyperbolic metric.

2.2. Convex real projective structures are an extension of hyperbolic structures. We are going to adapt basic properties of hyperbolic structures to convex real projective structures. For a hyperbolic manifold M, let Ω be the developing image in ℍ^{2} and A be an element of the holonomy group Γ ⊂ PSL(2, ℝ). Then A is a hyperbolic element. The translation length t_{A} of A is defined by

t_{A} = inf_{z∈Ω} ρ(z, A(z))

where ρ is the hyperbolic metric on Ω. Then the translation length t_{A} of A is achieved if and only if z lies on the principal line of A. We can extend the concept of translation length to real projective structures.

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 11

Proposition 2.1. Let a ∈ ℝℙ^{2} and A ∈ Hyp_{+}, then the Hilbert distance

h(a, A(a)) = log CR{Fix_{−}(A), a, A(a), Fix_{+}(A)}.

Proof. Since A^{−n}(a) → Fix_{−}(A) and A^{m}(a) → Fix_{+}(A) as n, m → ∞, it is enough to show that

CR{A^{−n}(a), a, A(a), A^{m}(a)} > CR{A^{−n−1}(a), a, A(a), A^{m+1}(a)}.

Without loss of generality we may let

a = [ \begin{matrix}x \\ 0 \\ z \end{matrix} ] , A = [ \begin{matrix}λ & 0 & 0 \\ 0 & μ & 0 \\ 0 & 0 & ν \end{matrix} ]

since there exists B ∈ SL(3, ℝ) such that B(a) = [x′, 0, z′]^{t} and CR is invariant under SL(3, ℝ)-conjugation.

X{A^{−n}(a), a, A(a), A^{m}(a)}
= X {[λ^{−n}x, 0, ν^{−n}z]^{t}, [x, 0, z]^{t}, [λx, 0, νz]^{t}, [λ^{m}x, 0, ν^{m}z]^{t}}
= \frac{(α^{−n}w − αw)(w − α^{m}w)}{(α^{−n}w − w)(αw − α^{m}w)} = \frac{(α^{−n} − α)(1 − α^{m})}{(α^{−n} − 1)(α − α^{m})}

where α = λ/ν and w = x/z. By an easy computation we can show for 0 < α < 1

\frac{(α^{−n} − α)(1 − α^{m})}{(α^{−n} − 1)(α − α^{m})} > \frac{(α^{−n−1} − α)(1 − α^{m+1})}{(α^{−n−1} − 1)(α − α^{m+1})},

and the proposition is proved. q.e.d.

For any A ∈ Hyp_{+}, A can be uniquely decomposed as HV up to SL(3, ℝ)-conjugation where

(5) H = [ \begin{matrix}λ\sqrt{μ} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ν\sqrt{μ} \end{matrix} ] and V = [ \begin{matrix}1/\sqrt{μ} & 0 & 0 \\ 0 & μ & 0 \\ 0 & 0 & 1/\sqrt{μ} \end{matrix} ] .

We call H the horizontal factor and V the vertical factor of the decomposition of a positive hyperbolic transformation. Sometimes H will be called the pure hyperbolic factor.

Consider H. By Proposition 2.1, the Hilbert distance between a and H(a) for any a = [1 − s, 0, s]^{t} ∈ σ(A) = σ(H) is

h(a, H(a))
= log CR \{[ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] , [ \begin{matrix}1 − s \\ 0 \\ s \end{matrix} ] , [ \begin{matrix}λ\sqrt{μ} (1 − s) \\ 0 \\ ν\sqrt{μ} s \end{matrix} ] , [ \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ]\}

= log \begin{matrix}(∞ − \frac{λ}{ν} \frac{1−s}{s}) & (\frac{1−s}{s} − 0) \\ (∞ − \frac{1−s}{s}) & (\frac{λ}{ν} \frac{1−s}{s} − 0) \end{matrix} = log (\frac{ν}{λ}) = ℓ(A).

We call ℓ(A) = h(a, H(a)) the horizontal translation length and it is the length of the boundary component represented by A.

Consider V. Then the stationary set is the line joining [1, 0, 0]^{t} and [0, 0, 1]^{t} and the fixed point [0, 1, 0]^{t}. Without loss of generality we assume μ > 1. Then for any a = [1 − s, y, s]^{t} in the segment joining [1−s, 0, s]^{t} and [0, 1, 0]^{t}, the point V (a) goes toward [0, 1, 0]^{t} since μ > 1.

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 13

[figure: Fix_{−}(A) = [1, 0, 0]^{t}; Fix_{0}(A); Fix_{+}(A) = [0, 0, 1]^{t}; a; H(a)]

Figure 2. The horizontal translation length

Then the Hilbert distance between a and V (a) is h(a, V (a))

= log CR \{[ \begin{matrix}1 − s \\ 0 \\ s \end{matrix} ] , [ \begin{matrix}1 − s \\ y \\ s \end{matrix} ] , [ yμ_{\frac{1−s}{\sqrt{μ}}}^{\frac{1−s}{\sqrt{μ}}} ] , [ \begin{matrix}0 \\ 1 \\ 0 \end{matrix} ]\}
= log X \{B [ \begin{matrix}1 − s \\ 0 \\ s \end{matrix} ] , B [ \begin{matrix}1 − s \\ y \\ s \end{matrix} ] , B [ yμ_{\frac{1−s}{\sqrt{μ}}}^{\frac{1−s}{\sqrt{μ}}} ] , B [ \begin{matrix}0 \\ 1 \\ 0 \end{matrix} ]\}

where B = [ \begin{matrix}(1 − s)^{−1} & 0 & 0 \\ s & μ & s − 1 \\ 0 & 1 & 0 \end{matrix} ]

= log X \{[ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] , [ \begin{matrix}1 \\ 0 \\ y \end{matrix} ] , [ \begin{matrix}1/\sqrt{μ} \\ 0 \\ yμ \end{matrix} ] , [ \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ]\}
= log \begin{matrix}(∞ − y^{−1}μ^{− \frac{3}{2}} ) & (y^{−1} − 0) \\ (∞ − y^{−1}) & (y^{−1}μ^{− \frac{3}{2}} − 0) \end{matrix} = log (μ^{\frac{3}{2}} ) = \frac{3}{2} log (μ) = m(A).

[figure: Fix_{−}(A); Fix_{0}(A) = [0, 1, 0]^{t}; Fix_{+}(A); a; V (a); [1 − s, 0, s]^{t}]

Figure 3. The vertical translation length

We call m(A) = h(a, V (a)) the vertical translation length. We can easily compute the relations among H, V and A.

ℓ(H) = log (\frac{ν\sqrt{μ}}{λ\sqrt{μ}}) = log (\frac{ν}{λ}) = ℓ(A)
m(H) = \frac{3}{2} log(1) = 0
ℓ(V) = log (\frac{1/\sqrt{μ}}{1/\sqrt{μ}}) = log(1) = 0
m(V) = \frac{3}{2} log(μ) = m(A).

The above equations imply the horizontal translation lengths of A and H are the same and the vertical translation lengths of A and V are the same. Therefore H is the pure hyperbolic structure of A. If V = I, then A is derived from the hyperbolic structures.

3. Local properties of Hom(π, SL(3, ℝ))/SL(3, ℝ)

3.1. Let M be a compact oriented smooth surface, M̃ a fixed universal covering space and π = π_{1}(M) the corresponding group of covering transformations. Consider (f, N) where N is an ℝℙ^{2}-manifold and f : M → N a diffeomorphism. Such a pair is equivalent to a developing

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 15

pair (dev, h) where dev : M̃ → ℝℙ^{2} and h : π → SL(3, ℝ). We say two pairs (f, N) and (f′, N′) are equivalent if there exists a projective isomorphism g : N → N′ such that g ∘ f is isotopic to f′ ; that is there exists a diffeomorphism g′ : M → M such that g′ is isotopic to the identity map I_{M} and the following diagram commutes :

M ⟶^{f} N
g′↓ ↓^{g}
M ⟶_{f′} N′

The set of equivalence classes is denoted by ℝℙ^{2}(M) and is called the deformation space or moduli space of ℝℙ^{2}-structures on M. The deformation space ℝℙ^{2}(M) has the natural topology that is locally equivalent to the quotient space Hom(π, SL(3, ℝ))/SL(3, ℝ).

Let 𝔓(M) denote the subset of ℝℙ^{2}(M) corresponding to the convex ℝℙ^{2}-structures. Goldman showed 𝔓(M) is homeomorphic to ℝ^{−8·χ(M)}.

Theorem 3.1 (W. Goldman [8]). Let M be a compact oriented surface with χ(M) < 0. Then 𝔓(M) is an open subset of ℝℙ^{2}(M) and the restriction of hol : ℝℙ^{2}(M) → Hom(π, SL(3, ℝ))/SL(3, ℝ) to 𝔓(M) is an embedding of 𝔓(M) onto a Hausdorff real analytic manifold of dimension −8 · χ(M).

3.2. Since 𝔓(M) embeds into Hom(π, SL(3, ℝ))/SL(3, ℝ), we need to know the basic properties of Hom(π, G)/G where π denote the fundamental group of a compact oriented smooth surface M, and G is a connected Lie group. Suppose M = Σ(g, n) a smooth surface with genus g, n boundary components and χ(M) = 2 − 2g − n < 0. Then π admits 2g + n generators A_{1}, B_{1}, …, A_{g}, B_{g}, C_{1}, …, C_{n} with a single relation

R = A_{1}B_{1}A_{1}^{−1}B_{1}^{−1} ⋯ A_{g}B_{g}A_{g}^{−1}B_{g}^{−1}C_{1} ⋯ C_{n}.

16 HONG CHAN KIM

Then Hom(π, G) may be identified with the collection of all (2g + n)- tuples

(A_{1}, B_{1}, …, A_{g}, B_{g}, C_{1}, …, C_{n}) ⊂ G^{2g+n}

elements of G satisfying

R(A_{1}, B_{1}, …, A_{g}, B_{g}, C_{1}, …, C_{n}) = I.

Since R : G^{2g+n} → G is a polynomial equation and

Hom(π, G) = R^{−1}(I) ⊂ G^{2g+n},

if G is an algebraic Lie group, then Hom(π, G) is an algebraic variety.

In general Hom(π, G) is not smooth. Suppose ϕ ∈ Hom(π, G) and Z(ϕ) is the centralizer of ϕ(π) in G. Goldman [7] showed ϕ is a nonsingular point of Hom(π, G) if and only if dim Z(ϕ)/Z(G) = 0 where Z(G) denotes the center of G. Let Hom(π, G)^{−} be the set of nonsingular points of Hom(π, G). Then G acts freely on the smooth Zariski open subset Hom(π, G)^{−}. But unfortunately Hom(π, G)^{−}/G is generally not Hausdorff. Let Hom(π, G)^{−−} be the subset of Hom(π, G)^{−} consisting of homomorphisms whose image does not lie in a parabolic subgroup of G. Then Hom(π, G)^{−−} is an Zariski open subset of Hom(π, G)^{−} and Hom(π, G)^{−−}/G is a Hausdorff smooth manifold of dimension −dim G· χ(M). For more detail see Goldman’s paper [7].

Suppose ϕ ∈ Hom(π, G) is an irreducible representation. Then ϕ ∈ Hom(π, G)^{−−}. Since the holonomy representation of a real projective structure from the fundamental group π of a compact oriented surface to an algebraic Lie group G is irreducible, we restrict our interest to Hom(π, G)^{−−}.

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 17

3.3. Hom(π, G)^{−−}/G is the quotient space of Hom(π, G)^{−−} with ϕ′ ∼ ϕ if and only if there exists g ∈ G such that

ϕ′(γ) = g ∘ ϕ(γ) ∘ g^{−1}

for all γ ∈ π. Let ϕ ∈ Hom(π, G)^{−−} and [ϕ] ∈ Hom(π, G)^{−−}/G its equivalence class. We want to show the tangent space to Hom(π, G)^{−−}/G at [ϕ] is isomorphic to the first group cohomology H^{1}(π; 𝔤_{Adϕ}). To do this we need some knowledge about group cohomology [14].

Suppose π is a group and V is a π-module. Thus there exists a linear representation ρ : π → End(V). We define C^{0}(π; V_{ρ}) = V and C^{p}(π; V_{ρ}) consists of all maps π^{p} → V for p > 0. The coboundary operator d : C^{p}(π; V_{ρ}) → C^{p+1}(π; V_{ρ}) is defined by

df(g_{1}, …, g_{p+1}) = ρ(g_{1})f(g_{2}, …, g_{p+1})

+ ∑_{i=1}^{p}(−1)^{i}f(g_{1}, …, g_{i}g_{i+1}, …, g_{p+1}) + (−1)^{p+1}f(g_{1}, …, g_{p})

where f ∈ C^{p}(π; V_{ρ}), g_{1}, …, g_{p+1} ∈ G. Since d∘d = 0, we can construct the cohomology space called the group cohomology.

H^{p}(π; V_{ρ}) = \frac{Z^{p}(π; V_{ρ})}{B^{p}(π; V_{ρ})} = \frac{kernel (d : C^{p}(π; V_{ρ}) → C^{p+1}(π; V_{ρ}))}{image (d : C^{p−1}(π; V_{ρ}) → C^{p}(π; V_{ρ}))}.

Consider coboundary operators for p = 0 and 1. For any v ∈ C^{0}(π; V_{ρ}) = V, f ∈ C^{1}(π; V_{ρ}) and x, y ∈ π,

(6)  dv(x) = ρ(x)v − v
(7)  df(x, y) = ρ(x)f(y) − f(xy) + f(x).

Therefore f ∈ Z^{1}(π; V_{ρ}) if and only if f(xy) = f(x) + ρ(x)f(y) for all x, y ∈ π and f ∈ B^{1}(π; V_{ρ}) if and only if there exists v ∈ V such that f(x) = dv(x) = ρ(x)v − v for all x ∈ π.

18 HONG CHAN KIM

Let 𝔤 be the Lie algebra of G, then 𝔤 is a π-module 𝔤_{Adϕ} by the composition

π →^{ϕ} G →^{Ad} Aut(𝔤).

Recall Ad(g)(X) = (r_{g^{−1}})_{∗}X where g ∈ G, X ∈ 𝔤, and r_{g^{−1}} is the right translation.

Proposition 3.2 (A. Weil [16]). Let π be the fundamental group of a compact oriented surface, G a connected algebraic Lie group, and ϕ ∈ Hom(π, G)^{−−}. Then the tangent space to Hom(π, G)^{−−}/G at [ϕ] is isomorphic to the first group cohomology H^{1}(π; 𝔤_{Adϕ}).

4. The symplectic structure on Hom(π, G)^{−−}/G

Atiyah and Bott [2] initiated a new approach to the study of moduli spaces of homomorphisms from fundamental groups of surfaces to compact connected Lie groups by methods of the gauge theory. In particular they found natural symplectic structures on certain smooth open subsets of these moduli spaces.

In this paper I will extend the symplectic form to the compact oriented surface with boundary. Before doing this, we first study the symplectic form on the moduli space of a closed surface.

4.1. Fox [6] developed a noncommutative differential calculus for words in a free group. It turns out Fox’s calculus is a very useful tool for calculations in spaces of representations Hom(π, G). The goal of this section is to determine the explicit formula for the symplectic 2-form on Hom(π, G)^{−−}/G using Fox’s calculus. We start with the summary of Fox’s calculus.

Let Π be a free group with generators x_{1}, …, x_{n}, and ℤΠ its integral group ring. For every element ∑ n_{i}σ_{i} ∈ ℤΠ, we assume n_{i} = 0 for all

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 19

but finitely many. The augmentation homomorphism ε : ℤΠ → ℤ is a ring homomorphism defined by

ε(∑ n_{i}σ_{i}) = ∑ n_{i}.

The Fox derivation of ℤΠ is a ℤ-linear map D : ℤΠ → ℤΠ satisfying

D(m_{1}m_{2}) = D(m_{1})ε(m_{2}) + m_{1}D(m_{2})

where m_{1}, m_{2} ∈ ℤΠ. The integral group ring ℤΠ is a Π-bimodule where Π acts on the left by ordinary left-multiplication and on the right by trivial. Since for any x, y ∈ Π

D(xy) = D(x)ε(y) + xD(y) and ε(y) = 1,

the Fox derivation is a 1-cocycle on Π with coefficients in ℤΠ.

Proposition 4.1 (R. Fox [6]). Suppose x_{1}, …, x_{n} are the generators for group Π and Der(Π) is the set of all Fox derivations. We define (D ∘ m)(x) = D(x)ε(m) for D ∈ Der(Π), m ∈ ℤΠ and x ∈ Π. Then Der(Π) is freely generated as a right ℤΠ-module by n element ∂_{i} = ∂/∂x_{i}, i = 1, …, n such that (∂/∂x_{i})(x_{j}) = δ_{ij}I where I is the identity element of Π.

For any word w ∈ Π, computing the Fox derivation ∂w is quite mechanical. The following are examples which we need later. For any generators A, B ∈ Π, we have

• ∂_{A}(AB) = ∂_{A}(A)ε(B) + A∂_{A}(B) = I ;

• ∂_{B}(AB) = ∂_{B}(A)ε(B) + A∂_{B}(B) = A ;
• ∂_{A}(A^{−1}) = −A^{−1} ;
• ∂_{A}(ABA^{−1}) = I − ABA^{−1} ;
• ∂_{B}(ABA^{−1}) = A ;
• ∂_{A}(ABA^{−1}B^{−1}) = I − ABA^{−1} ;

20 HONG CHAN KIM

• ∂_{B}(ABA^{−1}B^{−1}) = A − ABA^{−1}B^{−1}. From the above examples we can see for any word w ∈ Π, ε(∂_{i}w) equals the total exponent sum of the letter x_{i} in the word w. Fox proves that these free derivations satisfy a very useful rule of differential calculus ; the Mean Value Theorem.

Proposition 4.2 (Fox[6]). Suppose x_{1}, …, x_{n} are the generators for a group Π. Then for any m ∈ ℤΠ,

∑_{i=1}^{n}(∂_{i}m)(x_{i} − I) = m − ε(m)I.

4.2. To write the explicit formula for the symplectic structure ω on Hom(π, G)^{−−}/G, we need more knowledge about the group homology theory [4]. Let Π be a group and ℤΠ the integral group ring. Let C_{n}(Π) be the ℤ-module freely generated by Π^{n} = Π × ⋯ × Π and C_{0}(Π) = ℤ. We define the boundary operator ∂_{n} : C_{n}(Π) → C_{n−1}(Π) by for n ≥ 2

∂_{n}(u_{1}, …, u_{n}) = ε(u_{1})(u_{2}, …, u_{n})

+ ∑_{i=1}^{n−1}(−1)^{i}(u_{1}, …, u_{i}u_{i+1}, …, u_{n}) + (−1)^{n}(u_{1}, …, u_{n−1})ε(u_{n}).

and ∂_{1}(u) = 0. Since ∂_{n} ∘ ∂_{n+1} = 0, we can construct the homology space called the group homology.

H_{n}(Π) = \frac{Z_{n}(Π)}{B_{n}(Π)} = \frac{kernel (∂_{n} : C_{n}(Π) → C_{n−1}(Π))}{image (∂_{n+1} : C_{n+1}(Π) → C_{n}(Π))}.

For n = 2, ∂_{2} : C_{2}(Π) → C_{1}(Π) is defined by

(8)  ∂_{2}(u, v) = ε(u)v − uv + uε(v).

Our interesting case is a closed surface group π = Π/R where π is a group generated by 2g generators A_{1}, B_{1}, …, A_{g}, B_{g} and with one

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 21

relation

(9)  R = A_{1}B_{1}A_{1}^{−1}B_{1}^{−1} ⋯ A_{g}B_{g}A_{g}^{−1}B_{g}^{−1}.

Consider a 2-chain on π

(10) Z_{R} = ∑_{i=1}^{g} ((\frac{∂R}{∂A_{i}} , A_{i}) + (\frac{∂R}{∂B_{i}} , B_{i})) = ∑_{i} n_{i}(x_{i}, y_{i}) ∈ ℤ(π × π).

Then its boundary is

∂Z_{R} = ∑_{i=1}^{g} \{(ε(\frac{∂R}{∂A_{i}})A_{i} − \frac{∂R}{∂A_{i}} A_{i} + (\frac{∂R}{∂A_{i}})ε(A_{i}))

+ (ε(\frac{∂R}{∂B_{i}})B_{i} − \frac{∂R}{∂B_{i}} B_{i} + (\frac{∂R}{∂B_{i}})ε(B_{i}))\}

= ∑_{i=1}^{g} \{(0 − \frac{∂R}{∂A_{i}} (A_{i} − I)) + (0 − \frac{∂R}{∂B_{i}} (B_{i} − I))\}
= − ∑_{i=1}^{g} \{\frac{∂R}{∂A_{i}} (A_{i} − I) + \frac{∂R}{∂B_{i}} (B_{i} − I)\}
= −(R − I) = 0

by the Mean Value Theorem 4.2 and the fact ε(∂R/∂x) is the total exponent sum of x in R. The following Proposition 4.3 is due to Goldman [7] and Lyndon [13].

Proposition 4.3. Let R be the relation (9) for the fundamental group π of a closed surface of genus g. Let Z_{R} be the 2-cycle (10) on π. Then its homology class [Z_{R}] generates H_{2}(π).

We call Z_{R} the fundamental cycle of the fundamental group π. In next section I will extend the concept of the fundamental cycle Z_{R} to compact oriented surfaces with boundary.

4.3. Finally we can formulate the explicit formula of the symplectic form on Hom(π, G)^{−−}/G where π is the fundamental group of a closed surface and G is a connected algebraic Lie group. Let 𝔤 be the

22 HONG CHAN KIM

Lie algebra of G with an Ad-invariant nondegenerate symmetric bilinear form β : 𝔤 × 𝔤 → ℝ. If 𝔤 = 𝔰𝔩(3, ℝ), then we could take β is just the trace form. Let u, v ∈ Z^{1}(π; 𝔤_{Adϕ}). We define a ℤ-linear map

β_{∗}(u, v) : ℤ(π × π) → ℝ

by

(11)  β_{∗}(u, v) (∑_{i=1}^{k} n_{i}(x_{i}, y_{i})) = ∑_{i=1}^{k} n_{i}{β(u(x_{i}), x_{i}.v(y_{i}))}

where x_{i}.v(y_{i}) means Adϕ(x_{i})v(y_{i}) for (x_{i}, y_{i}) ∈ π × π ⊂ C_{2}(π). Then β_{∗}(u, v) ∈ Z^{2}(π; ℝ).

Theorem 4.4. Let π be the fundamental group of a closed surface. Define

ω : H^{1}(π; 𝔤_{Adϕ}) × H^{1}(π; 𝔤_{Adϕ}) → ℝ by
(12)  ω([u], [v]) = β_{∗}(u, v)Z_{R}

where Z_{R} is the fundamental cycle (10) of π. Then ω is a symplectic form on Hom(π, G)^{−−}/G.

5. A symplectic leaf on the moduli space of a compact oriented surface with boundary

In this section I give an explicit formula for the symplectic structure on a symplectic leaf on the moduli space of real projective structures of a compact oriented surface with boundary.

5.1. First I recall some basic definitions and properties. Let M be a smooth m-dimensional manifold and C^{∞}(M) the set of all smooth functions on M. A Poisson manifold M is a smooth manifold endowed with a Poisson bracket { , } : C^{∞}(M) × C^{∞}(M) → C^{∞}(M) such that

(1) {f, g} = −{g, f} (skew-symmetry)

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 23

(2) {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0 (the Jacobi identity)

(3) {h, fg} = {h, f}g + f{h, g} (the Leibniz rule). If M is a Poisson manifold, then C^{∞}(M) has the Lie algebra structure and {f, ·} is a vector field, called the Hamiltonian vector field of f and denoted by X_{f}. The Poisson bracket is determined by a skew-symmetric bilinear form on T^{∗}M ; that is there exists a 2-vector (or bivector) field P ∈ ∧^{2}T M such that

{f, g} = P (df ∧ dg).

P will be called the Poisson bivector of M and locally expressed by

P = ∑_{1≤i<j≤m} a^{ij} \frac{∂}{∂x^{i}} ∧ \frac{∂}{∂x^{j}}

where a^{ij}’s are locally defined smooth functions on M.

A symplectic manifold is a smooth manifold M endowed with a nondegenerate closed 2-form ω. For any f ∈ C^{∞}(M), ω allows us to associate a vector field X_{f} dual to the differential df ; that is

ω(X_{f}, ·) = df(·).

This allows us to define {, } on C^{∞}(M) by

{f, g} = X_{f} · g = dg(X_{f}) = ω(X_{g}, X_{f}).

Then {, } is skew-symmetric since ω is alternating and satisfies the Jacobi identity and the Leibniz rule since ω is closed and X_{f} is dual to the differential df.

It turns out that symplectic manifolds are Poisson manifolds but there are many Poisson manifolds that are not symplectic. The moduli space of real projective structures on a compact oriented surface with boundary is an example. A function f ∈ C^{∞}(M) is a Casimir function if {f, g} = 0 for all g ∈ C^{∞}(M). If the only Casimir functions are

24 HONG CHAN KIM

constant, then {, } comes from a symplectic structure. But the above moduli space has nonconstant Casimir functions. Let f : G → ℝ be an Ad-invariant function (e.g. trace function). For any γ ∈ π we can define an associated function f_{γ} : Hom(π, G)^{−−}/G → ℝ by

f_{γ}([ϕ]) = f ∘ ϕ(γ).

Then f_{γ} does not depend on the choice of ϕ since f is Ad-invariant. We call f_{γ} the Goldman function. From Audin [3], f_{C_{1}}, …, f_{C_{n}} are Casimir functions where C_{1}, …, C_{n} are the boundary generators of π.

Let N be a submanifold of a Poisson manifold M. Generally {f, g}|_{N} is not related to f|_{N} and g|_{N}. If {f, g}|_{N} depends only on f|_{N} and g|_{N} for all f, g ∈ C^{∞}(M), then N is called a Poisson submanifold. Moreover if {, }|_{N} can be defined from a symplectic form, then N is called a symplectic submanifold. Since any Poisson manifold has a symplectic foliation, a foliation whose leaves are the maximal symplectic submanifolds, we will find a symplectic foliation of the moduli space 𝔓(M) and formulate the explicit symplectic form on its leaf.

5.2. Let M be a compact oriented surface with genus g, n boundary components and χ(M) = 2 − 2g − n < 0. Let π be the fundamental group of M, then π has 2g + n generators

A_{1}, …, A_{g}, B_{1}, …, B_{g}, C_{1}, …, C_{n}

with a single relation

R = ∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{n} C_{j} = I

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 25

where [A_{i}, B_{i}] = A_{i}B_{i}A_{i}^{−1}B_{i}^{−1}. We extend the concept of fundamental cycle Z_{R} of the group π with boundary generators C_{1}, …, C_{n}. Let

(13) Z_{R} = ∑_{k=1}^{g} (\frac{∂R}{∂A_{k}} , A_{k}) + ∑_{k=1}^{g} (\frac{∂R}{∂B_{k}} , B_{k}) + ∑_{h=1}^{n} (\frac{∂R}{∂C_{h}} , C_{h}) .

Then the boundary of Z_{R} is

∂Z_{R} = ∑_{i=1}^{g} \{(ε(\frac{∂R}{∂A_{i}})A_{i} − \frac{∂R}{∂A_{i}} A_{i} + (\frac{∂R}{∂A_{i}})ε(A_{i}))\}
+ ∑_{i=1}^{g} \{(ε(\frac{∂R}{∂B_{i}})B_{i} − \frac{∂R}{∂B_{i}} B_{i} + (\frac{∂R}{∂B_{i}})ε(B_{i}))\}
+ ∑_{j=1}^{n} \{(ε(\frac{∂R}{∂C_{j}})C_{j} − \frac{∂R}{∂C_{j}} C_{j} + (\frac{∂R}{∂C_{j}})ε(C_{j}))\}
= ∑_{i=1}^{g} \{(0 − \frac{∂R}{∂A_{i}} (A_{i} − I)) + (0 − \frac{∂R}{∂B_{i}} (B_{i} − I))\}
+ ∑_{j=1}^{n} \{(C_{j} − \frac{∂R}{∂C_{j}} (C_{j} − I))\} = ∑_{j=1}^{n} C_{j}
− ∑_{i=1}^{g} \{\frac{∂R}{∂A_{i}} (A_{i} − I) + \frac{∂R}{∂B_{i}} (B_{i} − I)\} − ∑_{j=1}^{n} \{\frac{∂R}{∂C_{j}} (C_{j} − I)\}
= ∑_{j=1}^{n} C_{j} − (R − I) = ∑_{j=1}^{n} C_{j}

by the Mean Value Theorem 4.2 and the fact ε(∂R/∂x) is the total exponent sum of x in R. So the boundary of Z_{R} is the sum of boundary generators. Hence we will call Z_{R} the fundamental relative cycle of π.

We define a bilinear form ω : Z^{1}(π; 𝔤_{Adϕ}) × Z^{1}(π; 𝔤_{Adϕ}) → ℝ by

(14)  ω(u, v) = β_{∗}(u, v)Z_{R}

where Z_{R} is the fundamental relative cycle (13) of π and β_{∗} is defined as before in (11).

26 HONG CHAN KIM

In the following proposition 5.2, we first meet an obstruction for ω to be symplectic. To do this we need a lemma.

Lemma 5.1. For any u, v ∈ Z^{1}(π; 𝔤_{Adϕ}) and x, y ∈ π,

β(u(x), x.v(y)) + β(v(x), x.u(y))
= β(u(xy), v(xy)) − β(u(x), v(x)) − β(u(y), v(y)).

Proof. Since u is a 1-cocycle, u(xy) = u(x) + x.u(y). So we have

β(u(xy), v(xy)) = β(u(x) + x.u(y), v(x) + x.v(y)) = β(u(x), v(x)) + β(u(x), x.v(y)) + β(v(y), x.u(y)) + β(u(y), v(y))

since β is symmetric and β(x.u(y), x.v(y)) = β(u(y), v(y)). It proves the lemma. q.e.d.

In the following proposition we know ω is skew-symmetric unless π has boundary generators C_{1}, …, C_{n}.

Proposition 5.2. For any u, v ∈ Z^{1}(π; 𝔤_{Adϕ}),

(15)  ω(u, v) + ω(v, u) + ∑_{j=1}^{n} β(u(C_{j}), v(C_{j})) = 0

where C_{1}, …, C_{n} are the boundary generators of π.

Proof. From the examples of Fox’s calculus, the fundamental relative cycle has the following expression.

Z_{R} = ∑_{k=1}^{g} ( ∏_{i=1}^{k−1}[A_{i}, B_{i}](1 − A_{k}B_{k}A_{k}^{−1}), A_{k})
+ ∑_{k=1}^{g} ( ∏_{i=1}^{k−1}[A_{i}, B_{i}](A_{k} − A_{k}B_{k}A_{k}^{−1}B_{k}^{−1}), B_{k})
+ ∑_{h=1}^{n} ( ∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{h−1} C_{j}, C_{h}) .

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 27

For the convenience of notation, ∏_{i=1}^{0} F (i) means the identity element of π where F (i) ∈ π. Then

ω(u, v) = β_{∗}(u, v)Z_{R}

= ∑_{k=1}^{g} β (u(∏_{i=1}^{k−1}[A_{i}, B_{i}]), ∏_{i=1}^{k−1}[A_{i}, B_{i}].v(A_{k}))

− ∑_{k=1}^{g} β (u(∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}B_{k}A_{k}^{−1}), ∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}B_{k}A_{k}^{−1}.v(A_{k}))

+ ∑_{k=1}^{g} β (u(∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}), ∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}.v(B_{k}))

− ∑_{k=1}^{g} β (u(∏_{i=1}^{k−1}[A_{i}, B_{i}][A_{k}, B_{k}]), ∏_{i=1}^{k−1}[A_{i}, B_{i}][A_{k}, B_{k}].v(B_{k}))

+ ∑_{h=1}^{n} β (u(∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{h−1} C_{h}), ∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{h−1} C_{j}.v(C_{h})) .

Let β̄(x) = β(u(x), v(x)). Then by Lemma 5.1, ω(u, v) + ω(v, u)

= ∑_{k=1}^{g} β̄ (∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}) − ∑_{k=1}^{g} β̄ (∏_{i=1}^{k−1}[A_{i}, B_{i}]) − ∑_{k=1}^{g} β̄ (A_{k})

− ∑_{k=1}^{g} β̄ (∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}B_{k}) + ∑_{k=1}^{g} β̄ (∏_{i=1}^{k}[A_{i}, B_{i}]B_{k}) + ∑_{k=1}^{g} β̄ (A_{k})

+ ∑_{k=1}^{g} β̄ (∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}B_{k}) − ∑_{k=1}^{g} β̄ (∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}) − ∑_{k=1}^{g} β̄ (B_{k})

− ∑_{k=1}^{g} β̄ (∏_{i=1}^{k}[A_{i}, B_{i}]B_{k}) + ∑_{k=1}^{g} β̄ (∏_{i=1}^{k}[A_{i}, B_{i}]) + ∑_{k=1}^{g} β̄ (B_{k})

+ ∑_{h=1}^{n} β̄ (∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{h} C_{j}) − ∑_{h=1}^{n} β̄ (∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{h−1} C_{j})

− ∑_{h=1}^{n} β̄ (C_{h}) = −β̄(I) + β̄(R) − ∑_{h=1}^{n} β̄ (C_{h})

= − ∑_{h=1}^{n} β (u(C_{h}), v(C_{h})) . q.e.d.

5.3. The boundary ∂M of M is a disjoint union of (∂M)_{j}, j = 1, …, n such that each boundary component is diffeomorphic to the unit circle S^{1}. Let π_{j} be the fundamental group of (∂M)_{j}. Then for each j, π_{j} is the cyclic group < C_{j} > generated by C_{j}.

Let ϕ ∈ Hom(π, G)^{−−} and ψ_{j} : π_{j} → π the inclusion, then ϕ_{j} ∈ Hom(π_{j}, G)^{−−} where ϕ_{j} = ϕ ∘ ψ_{j}. Define

C^{p}({π_{j}}; 𝔤_{Adϕ}) = ⨁_{j=1}^{n} C^{p}(π_{j}; 𝔤_{Adϕ_{j}})

= {(f_{1}, …, f_{n}) : π_{1}^{p} × ⋯ × π_{n}^{p} → 𝔤_{Adϕ_{1}} × ⋯ × 𝔤_{Adϕ_{n}}}

and the boundary map d : C^{p}({π_{j}}; 𝔤_{Adϕ}) → C^{p+1}({π_{j}}; 𝔤_{Adϕ}) by

d(f_{1}, …, f_{n}) = (df_{1}, …, df_{n}).

Then by definition

B^{p}({π_{j}}; 𝔤_{Adϕ}) = ⨁_{j=1}^{n} B^{p}(π_{j}; 𝔤_{Adϕ_{j}}).

We confuse 𝔤_{Adϕ_{j}} with 𝔤_{Adϕ} since ϕ_{j} is the restriction ϕ to π_{j}. Because d ∘ d = 0, {C^{p}({π_{j}}; 𝔤_{Adϕ}), d} forms a cochain complex.

We define α : C^{p}(π; 𝔤_{Adϕ}) → C^{p}({π_{j}}; 𝔤_{Adϕ}) by α(f) = (f_{1}, …, f_{n}) where f_{j} = f|_{π_{j}} = f ∘ ψ_{j} : π_{j}^{p} → 𝔤_{Adϕ}. Then the following diagram commutes.

C^{p−1}(π; 𝔤_{Adϕ}) ⟶^{α} C^{p−1}({π_{j}}; 𝔤_{Adϕ})
d↓ ↓_{d}
C^{p}(π; 𝔤_{Adϕ}) ⟶_{α} C^{p}({π_{j}}; 𝔤_{Adϕ})

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 29

We define Z_{par}^{p}(π; 𝔤_{Adϕ}) = {f ∈ Z^{p}(π; 𝔤_{Adϕ}) | α(f) ∈ B^{p}({π_{j}}; 𝔤_{Adϕ})}. Then we have the following relation.

Proposition 5.3.

(16)  B^{p}(π; 𝔤_{Adϕ}) ⊆ Z_{par}^{p}(π; 𝔤_{Adϕ}) ⊆ Z^{p}(π; 𝔤_{Adϕ}).

Proof. By definition Z_{par}^{p}(π; 𝔤_{Adϕ}) ⊆ Z^{p}(π; 𝔤_{Adϕ}). Suppose g ∈ B^{p}(π; 𝔤_{Adϕ}), then there exists f ∈ C^{p−1}(π; 𝔤_{Adϕ}) such that df = g. Since the above diagram commutes,

α(g) = α(d(f)) = (α ∘ d)(f) = (d ∘ α)(f) = d(α(f)).

Therefore α(g) ∈ B^{p}({π_{j}}; 𝔤_{Adϕ}). Hence g ∈ Z_{par}^{p}(π; 𝔤_{Adϕ}). q.e.d.

We define H_{par}^{p}(π; 𝔤_{Adϕ}) = Z_{par}^{p}(π; 𝔤_{Adϕ})/B^{p}(π; 𝔤_{Adϕ}) and call the p-th parabolic cohomology with coefficients 𝔤_{Adϕ}. Then we have a long exact sequence

⋯ → H_{par}^{p−1}(π; 𝔤_{Adϕ}) → H^{p−1}(π; 𝔤_{Adϕ}) → H^{p−1}({π_{j}}; 𝔤_{Adϕ})
→ H_{par}^{p}(π; 𝔤_{Adϕ}) → H^{p}(π; 𝔤_{Adϕ}) → H^{p}({π_{j}}; 𝔤_{Adϕ}) → · · · .

We get a relation H_{par}^{1}(π; 𝔤_{Adϕ}) ⊆ H^{1}(π; 𝔤_{Adϕ}) since H^{0}({π_{j}}; 𝔤_{Adϕ}) = 0. Suppose M is closed. Then H^{p}({π_{j}}; 𝔤_{Adϕ}) = 0 for all p. That means H_{par}^{p}(π; 𝔤_{Adϕ}) is the same as H^{p}(π; 𝔤_{Adϕ}) if M is a closed surface. For more detail about the parabolic cohomology, see Guruprasad, Huebschmann, Jeffrey, and Weinstein’s joint paper [10].

Consider p = 1 since we are interested in H^{1}(π; 𝔤_{Adϕ}) the tangent space of Hom(π, G)^{−−}/G. Let u ∈ Z_{par}^{1}(π; 𝔤_{Adϕ}), then u ∈ Z^{1}(π; 𝔤_{Adϕ}) and α(u) ∈ B^{1}({π_{j}}; 𝔤_{Adϕ}). That means u(xy) = u(x)+x.u(y) for x, y ∈ π and there exist D_{1}, …, D_{n} ∈ C^{0}(π_{j}; 𝔤_{Adϕ}) = 𝔤 such that u_{j} = dD_{j} where u_{j} = u|_{π_{j}}. Since π_{j} =< C_{j} >,

(17)  u(C_{j}) = u_{j}(C_{j}) = dD_{j}(C_{j}) = C_{j}.D_{j} − D_{j}.

30 HONG CHAN KIM

Weil [16] called the 1-cocycle satisfying the condition (17) a parabolic element. We will use his terminology.

5.4. Let D be an element of the Lie algebra 𝔤 = C^{0}(π; 𝔤_{Adϕ}), then dD ∈ B^{1}(π; 𝔤_{Adϕ}) such that dD(x) = x.D − D for any x ∈ π.

Proposition 5.4. For any u, v ∈ Z^{1}(π; 𝔤_{Adϕ}) and D ∈ 𝔤,

(18)  ω(dD, v) = ∑_{j=1}^{n} β(D, v(C_{j}))
(19)  ω(u, dD) = ∑_{j=1}^{n} β(u(C_{j}^{−1}), D).

Proof. Since for any x, y ∈ π, β_{∗}(dD, v)(x, y) = β(dD(x), x.v(y)) = β(x.D − D, x.v(y)) = β(x.D, x.v(y)) − β(D, x.v(y)) = β(D, v(y)) − β(D, x.v(y)) and u(x^{−1}) = −x^{−1}.u(x), we can compute the following.

ω(dD, v) = β_{∗}(dD, v)Z_{R}

= ∑_{k=1}^{g} (β(D, v(A_{k})) − β(D, ∏_{i=1}^{k−1}[A_{i}, B_{i}].v(A_{k})))
− ∑_{k=1}^{g} (β(D, v(A_{k})) − β(D, ∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}B_{k}A_{k}^{−1}.v(A_{k})))
+ ∑_{k=1}^{g} (β(D, v(B_{k})) − β(D, ∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}.v(B_{k})))
− ∑_{k=1}^{g} (β(D, v(B_{k})) − β(D, ∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}B_{k}A_{k}^{−1}B_{k}^{−1}.v(B_{k})))
+ ∑_{h=1}^{n} (β(D, v(C_{h})) − β(D, ∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{h−1} C_{j}.v(C_{h})))
= − ∑_{k=1}^{g} β(D, ∏_{i=1}^{k−1}[A_{i}, B_{i}].v(A_{k})) − ∑_{k=1}^{g} β(D, ∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}.v(B_{k}))

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 31

− ∑_{k=1}^{g} β(D, ∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}B_{k}.v(A_{k}^{−1}))
− ∑_{k=1}^{g} β(D, ∏_{i=1}^{k−1}[A_{i}, B_{i}]A_{k}B_{k}A_{k}^{−1}.v(B_{k}^{−1}))
− ∑_{h=1}^{n} β(D, ∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{h−1} C_{j}.v(C_{h})) + ∑_{h=1}^{n} β(D, v(C_{h}))
= −β(D, v(R)) + ∑_{h=1}^{n} β(D, v(C_{h})) = ∑_{h=1}^{n} β(D, v(C_{h})).

From Equation (18) and Proposition 5.2,

ω(u, dD) = −ω(dD, u) − ∑_{j=1}^{n} β(dD(C_{j}), u(C_{j}))

= − ∑_{j=1}^{n} β(D, u(C_{j})) − ∑_{j=1}^{n} β(C_{j}.D − D, u(C_{j}))

= − ∑_{j=1}^{n} β(D, u(C_{j})) − ∑_{j=1}^{n} β(C_{j}.D, u(C_{j})) + ∑_{j=1}^{n} β(D, u(C_{j}))

= ∑_{j=1}^{n} β(D, −C_{j}^{−1}.u(C_{j})) = ∑_{j=1}^{n} β(D, u(C_{j}^{−1})). q.e.d.

5.5. We know ω is not skew-symmetric on Z^{1}(π; 𝔤_{Adϕ}) if M has nonempty boundary. To make ω skew-symmetric, we restrict our domain to the subspace Z_{par}^{1}(π; 𝔤_{Adϕ}) ⊆ Z^{1}(π; 𝔤_{Adϕ}) and define a bilinear form ω̃ : Z_{par}^{1}(π; 𝔤_{Adϕ}) × Z_{par}^{1}(π; 𝔤_{Adϕ}) → ℝ by

(20) ω̃(u, v) = ω(u, v) − ∑_{j=1}^{n} β(D_{j}, v(C_{j}))

where u|_{π_{j}} = u_{j} = dD_{j}.

Suppose we have another D̄_{j} ∈ 𝔤 such that u|_{π_{j}} = dD_{j} = dD̄_{j}. Then

C_{j}.D̄_{j} − D̄_{j} = dD̄_{j}(C_{j}) = dD_{j}(C_{j}) = C_{j}.D_{j} − D_{j}.

Hence C_{j}.(D̄_{j} − D_{j}) = (D̄_{j} − D_{j}). Therefore there exists F_{j} ∈ 𝔤 such that D̄_{j} = D_{j} + F_{j} and C_{j}.F_{j} = F_{j} for each j = 1, …, n. Then

C_{j}^{−1}.F_{j} = C_{j}^{−1}.(C_{j}.F_{j}) = (C_{j}^{−1}C_{j}).F_{j} = F_{j}.

Let v|_{π_{j}} = dD′_{j}. Then

β(D̄_{j}, v(C_{j})) = β(D_{j} + F_{j}, v(C_{j}))
= β(D_{j}, v(C_{j})) + β(F_{j}, v(C_{j}))
= β(D_{j}, v(C_{j})) + β(F_{j}, dD′_{j}(C_{j}))
= β(D_{j}, v(C_{j})) + β(F_{j}, C_{j}.D′_{j} − D′_{j})
= β(D_{j}, v(C_{j})) + β(F_{j}, C_{j}.D′_{j}) − β(F_{j}, D′_{j}))
= β(D_{j}, v(C_{j})) + β(C_{j}^{−1}.F_{j}, D′_{j}) − β(F_{j}, D′_{j}) = β(D_{j}, v(C_{j})).

So ∑_{j=1}^{n} β(D_{j}, v(C_{j})) does not depend on the choices of D_{j}’s. Therefore ω̃ is well-defined on Z_{par}^{1}(π; 𝔤_{Adϕ}). If M is closed, then ω̃ = ω.

Proposition 5.5. ω̃ is skew-symmetric on Z_{par}^{1}(π; 𝔤_{Adϕ}). Proof. Recall Proposition 5.2. Then ω̃(u, v) + ω̃(v, u)

= ω(u, v) − ∑_{j=1}^{n} β(D_{j}, v(C_{j})) + ω(v, u) − ∑_{j=1}^{n} β(D′_{j}, u(C_{j})) = − ∑_{j=1}^{n}{β(u(C_{j}), v(C_{j})) + β(D_{j}, v(C_{j})) + β(D′_{j}, u(C_{j}))} = − ∑_{j=1}^{n}{β(C_{j}.D_{j} − D_{j}, v(C_{j})) + β(D_{j}, v(C_{j})) + β(D′_{j}, u(C_{j}))} = − ∑_{j=1}^{n}{β(C_{j}.D_{j}, v(C_{j})) + β(D′_{j}, u(C_{j}))} = − ∑_{j=1}^{n}{β(C_{j}.D_{j}, C_{j}.D′_{j} − D′_{j}) + β(D′_{j}, C_{j}.D_{j} − D_{j})} = − ∑_{j=1}^{n}{β(C_{j}.D_{j}, C_{j}.D′_{j})−β(C_{j}.D_{j}, D′_{j})+β(D′_{j}, C_{j}.D_{j})−β(D′_{j}, D_{j})} = 0. Therefore ω̃ is skew-symmetric. q.e.d.

Consider dD ∈ B^{1}(π; 𝔤_{Adϕ}) ⊆ Z_{par}^{1}(π; 𝔤_{Adϕ}). Then dD|_{π_{j}} = dD for all j. Therefore by Equation (18) and skew-symmetry, for any v ∈

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 33

Z_{par}^{1}(π; 𝔤_{Adϕ})

(21)  ω̃(dD, v) = ω(dD, v) − ∑_{j=1}^{n} β(D, v(C_{j})) = 0
(22)  ω̃(v, dD) = − ω̃(dD, v) = 0

Therefore we can induce ω̃ on H_{par}^{1}(π; 𝔤_{Adϕ}).

Theorem 5.6. Define ω̃ : H_{par}^{1}(π; 𝔤_{Adϕ}) × H_{par}^{1}(π; 𝔤_{Adϕ}) → ℝ by

(23)  ω̃([u], [v]) = ω̃(u, v) = ω(u, v) − ∑_{j=1}^{n} β(D_{j}, v(C_{j}))

where u, v ∈ Z_{par}^{1}(π; 𝔤_{Adϕ}) representing [u], [v] ∈ H_{par}^{1}(π; 𝔤_{Adϕ}) and u|_{π_{j}} = dD_{j}. Then ω̃ is a symplectic form on H_{par}^{1}(π; 𝔤_{Adϕ}).

Proof. (Well-definedness on H_{par}^{1}(π; 𝔤_{Adϕ})) Suppose u, ū are two representatives of [u]. Then there exists dD ∈ B^{1}(π; 𝔤_{Adϕ}) such that ū−u = dD. Similarly there exist v, v̄ ∈ Z_{par}^{1}(π; 𝔤_{Adϕ}) and dD′ ∈ B^{1}(π; 𝔤_{Adϕ}) such that v̄ − v = dD′. Then

ω̃(ū, v̄) = ω̃(u + dD, v + dD′)
= ω̃(u, v) + ω̃(dD, v) + ω̃(u, dD′) + ω̃(dD, dD′) = ω̃(u, v).

Therefore ω̃ is well-defined on H_{par}^{1}(π; 𝔤_{Adϕ}).

(Bilinearity and skew-symmetry) ω̃ is clearly bilinear since ω and β are bilinear. By Proposition 5.5 and well-definedness, ω̃ is skewsymmetric on H_{par}^{1}(π; 𝔤_{Adϕ}).

(Nondegeneracy) Let [u] ∈ H_{par}^{1}(π; 𝔤_{Adϕ}) such that ω̃([u], ·) = 0. Let u ∈ Z_{par}^{1}(π; 𝔤_{Adϕ}) be a representative of [u]. For any integer 1 ≤ k ≤ n − 1, we define v_{C_{k,k+1}} ∈ Z_{par}^{1}(π; 𝔤_{Adϕ}) by v_{C_{k,k+1}}(A_{i}) = v_{C_{k,k+1}}(B_{i}) = v_{C_{k,k+1}}(C_{j}) = 0 for all i, j but v_{C_{k,k+1}}(C_{k}) ≠ 0 and v_{C_{k,k+1}}(C_{k+1}) ≠ 0. Recall that R = ∏_{i=1}^{g}[A_{i}, B_{i}] ∏_{j=1}^{n} C_{j} = I. By the cocycle condition 0 = v_{C_{k,k+1}}(R) = v_{C_{k,k+1}}(C_{k}) + C_{k}.v_{C_{k,k+1}}(C_{k+1}). Hence we have the relation v_{C_{k,k+1}}(C_{k+1}) = −C_{k}^{−1}.v_{C_{k,k+1}}(C_{k}). Since ω̃(u, ·) vanishes,

0 = ω̃(u, v_{C_{k,k+1}})
= ω(u, v_{C_{k,k+1}}) − β(D_{k}, v_{C_{k,k+1}}(C_{k})) − β(D_{k+1}, v_{C_{k,k+1}}(C_{k+1}))
= ω(u, v_{C_{k,k+1}}) − β(D_{k}, v_{C_{k,k+1}}(C_{k})) − β(D_{k+1}, −C_{k}^{−1}.v_{C_{k,k+1}}(C_{k}))
= ω(u, v_{C_{k,k+1}}) − β(D_{k} − C_{k}.D_{k+1}, v_{C_{k,k+1}}(C_{k})).

Therefore

β(D_{k} − C_{k}.D_{k+1}, v_{C_{k,k+1}}(C_{k})) = ω(u, v_{C_{k,k+1}})
= β (u(∏_{i=1}^{g}[A_{i}, B_{i}]C_{1} ⋯ C_{k−1}), (∏_{i=1}^{g}[A_{i}, B_{i}]C_{1} ⋯ C_{k−1}).v_{C_{k,k+1}}(C_{k}))
+ β (u(∏_{i=1}^{g}[A_{i}, B_{i}]C_{1} ⋯ C_{k}), (∏_{i=1}^{g}[A_{i}, B_{i}]C_{1} ⋯ C_{k}).v_{C_{k,k+1}}(C_{k+1}))
= β (−u((∏_{i=1}^{g}[A_{i}, B_{i}]C_{1} ⋯ C_{k−1})^{−1}), v_{C_{k,k+1}}(C_{k}))
+ β (−u((∏_{i=1}^{g}[A_{i}, B_{i}]C_{1} ⋯ C_{k})^{−1}), v_{C_{k,k+1}}(C_{k+1}))
= β (−u(C_{k} ⋯ C_{n}), v_{C_{k,k+1}}(C_{k}))
+ β (−u(C_{k+1} ⋯ C_{n}), −C_{k}^{−1}.v_{C_{k,k+1}}(C_{k})))
= β (−u(C_{k} ⋯ C_{n}) + C_{k}.u(C_{k+1} ⋯ C_{n}), v_{C_{k,k+1}}(C_{k}))
= β (−u(C_{k}), v_{C_{k,k+1}}(C_{k})) .

Since β is nondegenerate and v_{C_{k,k+1}}(C_{k}) ≠ 0, for each k, we have a relation

(24)  D_{k} − C_{k}.D_{k+1} = −u(C_{k}).

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 35

Since u(C_{k}) = dD(C_{k}) = C_{k}.D_{k} − D_{k}, by Equation (24) we have D_{k} = D_{k+1} for all k.

Let D = D_{1} = ⋯ = D_{n} and ū = u − dD. Then ū is a parabolic element. I claim that ū = 0. By the definition of ū, ū(C_{j}) = u(C_{j})−dD(C_{j}) = 0 for all j. We define v_{A_{l}} ∈ Z_{par}^{1}(π; 𝔤_{Adϕ}) by v_{A_{l}}(A_{i}) = v_{A_{l}}(B_{i}) = v_{A_{l}}(C_{j}) = 0 for all i, j but v_{A_{l}}(A_{l}) ≠ 0 ∈ 𝔤. Similarly we define v_{B_{l}} for l = 1, …, g. Since ω̃(ū, v_{B_{g}}) = ω̃(u, v_{B_{g}}) − ω̃(dD, v_{B_{g}}) = 0, we have ω(ū, v_{B_{g}}) = ∑_{j=1}^{n} β(D̄_{j}, v_{B_{g}}(C_{j})) = 0. Hence

0 = ω(ū, v_{B_{g}}) = − β (ū((∏_{i=1}^{g−1}[A_{i}, B_{i}]A_{g})^{−1}), v_{B_{g}}(B_{g}))

+ β (ū((∏_{i=1}^{g−1}[A_{i}, B_{i}][A_{g}, B_{g}])^{−1}), v_{B_{g}}(B_{g}))

= − β (ū(B_{g}A_{g}^{−1}B_{g}^{−1}C_{1} ⋯ C_{n}), v_{B_{g}}(B_{g}))

+ β (ū(C_{1} ⋯ C_{n}), v_{B_{g}}(B_{g})) .

Since ū(C_{1} ⋯ C_{n}) = 0 and v_{B_{g}}(B_{g}) ≠ 0,

(25)  ū(B_{g}A_{g}^{−1}B_{g}^{−1}) = 0.

Similarly ω̃(ū, v_{A_{g}}) = 0 induces

0 = ω(ū, v_{A_{g}}) = − β (ū(A_{g}B_{g}A_{g}^{−1}B_{g}^{−1}C_{1} ⋯ C_{n}), v_{A_{g}}(A_{g}))
+ β (ū(B_{g}^{−1}C_{1} ⋯ C_{n}), v_{A_{g}}(A_{g})) .

Since ū(C_{1} ⋯ C_{n}) = 0 and v_{A_{g}}(A_{g}) ≠ 0,

(26)  ū(B_{g}^{−1}) = ū(A_{g}B_{g}A_{g}^{−1}B_{g}^{−1}).

From Equations (25) and (26),

ū(B_{g}^{−1}) = ū(A_{g}) + A_{g}.ū(B_{g}A_{g}^{−1}B_{g}^{−1}) = ū(A_{g}).

36 HONG CHAN KIM

From Equation (25),

0 = ū(B_{g}A_{g}^{−1}B_{g}^{−1})
= ū(B_{g}) + B_{g}.ū(A_{g}^{−1}) + B_{g}A_{g}^{−1}.ū(B_{g}^{−1})
= ū(B_{g}) − B_{g}A_{g}^{−1}.ū(A_{g}) + B_{g}A_{g}^{−1}.ū(B_{g}^{−1})
= ū(B_{g}) − B_{g}A_{g}^{−1}.ū(B_{g}^{−1}) + B_{g}A_{g}^{−1}.ū(B_{g}^{−1})
= ū(B_{g}).

Therefore ū(B_{g}) = 0 and ū(A_{g}) = ū(B_{g}^{−1}) = −B_{g}^{−1}.ū(B_{g}) = 0. By induction ū(A_{i}) = ū(B_{i}) = 0 for all i = 1, …, g. Therefore ū = u−dD = 0. That is [u] = 0 ∈ H_{par}^{1}(π; 𝔤_{Adϕ}).

(Closedness) To do this we represent H_{par}^{1}(π; 𝔤_{Adϕ}) as the cohomology H_{par}^{1}(M; adP_{ϕ}) with coefficients in the flat vector bundle. Then H^{1}(M; adP) is a tangent space of F(P)/G(P) flat connections modulo gauge transformations that is identified with Hom(π, G)/G, and isomorphic to the de Rham cohomology H_{DR}^{1}(M; 𝔤). For more detail see Goldman [7] and Millson [9]. Let ∂M be the disjoint union ∪_{j=1}^{n}∂M_{j}. Define [σ ⊗ ξ] ∈ H_{par}^{1}(M; adP_{ϕ}) if σ ⊗ ξ ∈ Z^{1}(M; adP_{ϕ}) and for each j = 1, …, n there exists f_{j} ∈ C^{∞}(∂M_{j}) such that

(σ ⊗ ξ)|_{∂M_{j}} = df_{j} ⊗ ξ

where σ is a 1-cocycle and ξ is a 𝔤-valued section on M. Define

ω̃ : H_{par}^{1}(M; adP) × H_{par}^{1}(M; adP) → ℝ by

ω̃([σ ⊗ ξ], [σ′ ⊗ ξ′]) = ∫_{M}(σ ∧ σ′) B(ξ, ξ′) − ∑_{j=1}^{n} ∫_{∂M_{j}}(f_{j} ∧ σ′) B(ξ, ξ′).

Then the above 2-form ω̃ does not involve the connection A explicitly, so it is invariant under the translations of F(P). Therefore ω̃ is parallel and hence closed. q.e.d.

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 37

5.6. Consider the moduli space 𝔓(M) of convex real projective structures on M = Σ(g, n). Since 𝔓(M) embeds onto an open subset of Hom(π, SL(3, ℝ))/SL(3, ℝ), dim 𝔓(M) = −dim SL(3, ℝ) · χ(M) = 16g − 16 + 8n.

Let {z_{1}, …, z_{N}} be a coordinates of 𝔓(M) where N = 16g−16+8n. Since the holonomy homomorphism h : π → G = SL(3, ℝ) is isomorphic to its image holonomy group Γ, we identify the elements of π and those image in G up to conjugation. Therefore each element of π can be expressed by {z_{1}, …, z_{N}} and A.X = AXA^{−1} for A ∈ π and X ∈ 𝔤 = 𝔰𝔩(3, ℝ). For each k, define u_{z_{k}} : π → 𝔤 by

(27)  u_{z_{k}}(A) = (∂_{z_{k}}A)A^{−1}

since tr{(∂_{z_{k}}A)A^{−1}} = ∂_{z_{k}}{log detA} = 0. Then for any A, B ∈ π,

u_{z_{k}}(AB) = (∂_{z_{k}}AB)(AB)^{−1} = {(∂_{z_{k}}A)B + A(∂_{z_{k}}B)}B^{−1}A^{−1}
= (∂_{z_{k}}A)A^{−1} + A(∂_{z_{k}}B)B^{−1}A^{−1} = u_{z_{k}}(A) + A.u_{z_{k}}(B).

Therefore u_{z_{k}} ∈ Z^{1}(π; 𝔤) for each k.

We find a necessary and sufficient condition for u_{z_{k}} to be parabolic. Since 𝔓(M) corresponds convex real projective structures, every element of π has three distinct positive real eigenvalues. For each boundary generator C_{j} there exists an invertible matrix P_{j} ∈ G such that P_{j}^{−1}C_{j}P_{j} = E_{j} the diagonal matrix (1). Even though C_{j} has repeated eigenvalues there exists P_{j} ∈ G such that P_{j}^{−1}C_{j}P_{j} = J_{j} the Jordan canonical form. Then we still have a sufficient condition for u_{z_{k}} to be parabolic.

Proposition 5.7. Suppose z_{k} is a coordinate in {z_{1}, …, z_{N}}. Then u_{z_{k}} is a parabolic element if and only if ∂_{z_{k}}(P_{j}^{−1}C_{j}P_{j}) = ∂_{z_{k}}(E_{j}) = 0 for each boundary generator C_{j} where j = 1, …, n.

38 HONG CHAN KIM

Proof. Suppose u_{z_{k}}(C) = dD(C). Then

(∂_{z_{k}}C)C^{−1} = u_{z_{k}}(C) = dD(C) = C.D − D = CDC^{−1} − D.

Therefore (∂_{z_{k}}C) = CD −DC. That means u_{z_{k}} is a parabolic element if and only if there exist D_{1}, …, D_{n} ∈ 𝔤 such that ∂_{z_{k}}C_{j} = C_{j}D_{j} − D_{j}C_{j} for each j = 1, …, n. For the simplicity put C_{j} = C, E_{j} = E, and P_{j} = P. Since E = P^{−1}CP, ∂_{z_{k}}C = ∂_{z_{k}}(P EP^{−1}) = (∂_{z_{k}}P)EP^{−1} + P (∂_{z_{k}}E)P^{−1} + P E(∂_{z_{k}}P^{−1}) = (∂_{z_{k}}P)(P^{−1}CP)P^{−1} + P (∂_{z_{k}}E)P^{−1} + P (P^{−1}CP)(−P^{−1}(∂_{z_{k}}P)P^{−1}) = (∂_{z_{k}}P)P^{−1}C + P (∂_{z_{k}}E)P^{−1} − C(∂_{z_{k}}P)P^{−1}.

(⇐) Suppose ∂_{z_{k}}E = 0. Let D = −(∂_{z_{k}}P)P^{−1}. Then ∂_{z_{k}}C = CD − DC. Therefore u_{z_{k}} is a parabolic element.

(⇒) Suppose u_{z_{k}} is a parabolic element. Then there exists D̄ ∈ 𝔤 such that ∂_{z_{k}}C = CD̄ − D̄C. Let X = D̄ − D where D = −(∂_{z_{k}}P)P^{−1}. Then CX − XC = (CD̄ − D̄C) − (CD − DC) = P (∂_{z_{k}}E)P^{−1}. So

(28)  ∂_{z_{k}}E = P^{−1}CXP − P^{−1}XCP.

Since C = P EP^{−1}, Equation (28) becomes

(29)  ∂_{z_{k}}E = EX̄ − X̄E.

where X̄ = P^{−1}XP. Since E is a diagonal matrix, each diagonal entry of EX̄ −X̄E is zero. Therefore ∂_{z_{k}}E = 0 since ∂_{z_{k}}E is a diagonal matrix whose diagonal entries are all zero. q.e.d.

Now we calculate the dimension of H_{par}^{1}(π; 𝔤_{Adϕ}). Suppose [u] ∈ H_{par}^{1}(π; 𝔤_{Adϕ}). Then for each j, u|_{π_{j}} ∈ B^{1}(π_{j}; 𝔤_{Adϕ}) ; that is [u|_{π_{j}}] = 0 ∈ H^{1}(π_{j}; 𝔤_{Adϕ}). Thus u(C_{j}) lies in a fixed conjugacy class in G. Hence we don’t have any choice to select [u(C_{j})] in Hom(π, G)^{−−}/G. Recall π has n boundary generators that are positive hyperbolic. Since for each

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 39

positive hyperbolic element has 2-dimensional coordinate parameters (λ, τ) or (ℓ, m),

(30) dim H_{par}^{1}(π; 𝔤_{Adϕ}) = dim H^{1}(π; 𝔤_{Adϕ}) − 2n = 16g − 16 + 6n.

5.7. Example : A pair of pants Σ(0, 3). Let M be a pair of pants Σ(0, 3) ; that is a surface homeomorphic to S^{2} minus three open discs. Then π admits a representation < C_{1}, C_{2}, C_{3} | R = C_{1}C_{2}C_{3} = I > and dim 𝔓(M) = 16g − 16 + 8n = 8. Since each positive hyperbolic element C_{j} has two parameters (λ_{j}, τ_{j}) and π has three generators, we need two more parameters s, t. We call s, t the internal parameters of 𝔓(M). Goldman [8] expressed C_{1}, C_{2}, C_{3} using parameters \{λ_{1}, λ_{2}, λ_{3}, τ_{1}, τ_{2}, τ_{3}, s, t\} of 𝔓(M) where 0 < λ_{j} < 1, 2/\sqrt{λ_{j}} < τ_{j} < λ_{j} + λ_{j}^{−2}, s > 0 and t > 0. Since s and t are positive, we replace the coordinates s and t by e^{s} and e^{t}. Then the matrices C_{1}, C_{2} and C_{3} are represented by

(31)  C_{1} = ( \begin{matrix}\begin{matrix}e^{−s} & \sqrt{λ_{3}} \\ \sqrt{λ_{2}} & \sqrt{λ_{1}} \end{matrix} + τ_{1} & \begin{matrix}e^{−s+t} & \sqrt{λ_{3}} \\ \sqrt{λ_{2}} & \sqrt{λ_{1}} \end{matrix} & 0 \\ −\begin{matrix}e^{−s−t} & \sqrt{λ_{3}} \\ \sqrt{λ_{2}} & \sqrt{λ_{1}} \end{matrix} − \begin{matrix}e^{s−t} & \sqrt{λ_{2}} \\ \sqrt{λ_{3}} & \sqrt{λ_{1}} \end{matrix} − e^{−t}τ_{1} & −\begin{matrix}e^{−s} & \sqrt{λ_{3}} \\ \sqrt{λ_{2}} & \sqrt{λ_{1}} \end{matrix} & 0 \\ a_{31} & a_{32} & λ_{1} \end{matrix} )

where a_{31} = \frac{1}{2}e^{2s}λ_{3} + \frac{e^{−s−t}\sqrt{λ_{3}}}{2\sqrt{λ_{2}}\sqrt{λ_{1}}} + \frac{e^{s−t}\sqrt{λ_{2}}}{2\sqrt{λ_{3}}\sqrt{λ_{1}}} + \frac{e^{s−t}\sqrt{λ_{3}}\sqrt{λ_{1}}}{2 λ_{2}^{3/2}} + \frac{e^{3s−t}\sqrt{λ_{1}}}{2\sqrt{λ_{3}}\sqrt{λ_{2}}} + \frac{λ_{1}}{2} + \frac{1}{2}e^{2s−t}τ_{3}+\frac{e^{−t}λ_{3}τ_{3}}{2λ_{2}} +\frac{1}{2}e^{s}\sqrt{λ_{3}}\sqrt{λ_{2}}\sqrt{λ_{1}}τ_{2}+\frac{1}{2}e^{−t}τ_{1}+\frac{e^{2s−t}λ_{1}τ_{1}}{2λ_{2}} +\frac{e^{s−t}\sqrt{λ_{3}}\sqrt{λ_{1}}τ_{3}τ_{1}}{2\sqrt{λ_{2}}} and a_{32} = \frac{e^{−s}\sqrt{λ_{3}}}{2\sqrt{λ_{2}}\sqrt{λ_{1}}} + \frac{e^{s}\sqrt{λ_{3}}\sqrt{λ_{1}}}{2 λ_{2}^{3/2}} + \frac{λ_{3}τ_{3}}{2λ_{2}} ,

(32) C_{2} = ( \begin{matrix}−\begin{matrix}e^{−s} & \sqrt{λ_{1}} \\ \sqrt{λ_{3}} & \sqrt{λ_{2}} \end{matrix} & 0 & −\frac{2e^{−s}\sqrt{λ_{1}}}{\sqrt{λ_{3}}\sqrt{λ_{2}}} \\ \begin{matrix}e^{−s−t} & \sqrt{λ_{1}} \\ \sqrt{λ_{3}} & \sqrt{λ_{2}} \end{matrix} + \frac{\sqrt{λ_{2}}\sqrt{λ_{1}}}{e^{t−s}λ_{3}^{3/2}} + \frac{e^{−t}λ_{1}τ_{1}}{λ_{3}} & λ_{2} & b_{23} \\ \frac{e^{s}\sqrt{λ_{3}}}{2\sqrt{λ_{2}}\sqrt{λ_{1}}} + \frac{e^{−s}\sqrt{λ_{1}}}{2\sqrt{λ_{3}}\sqrt{λ_{2}}} + \frac{τ_{2}}{2} & 0 & \begin{matrix}e^{−s} & \sqrt{λ_{1}} \\ \sqrt{λ_{3}} & \sqrt{λ_{2}} \end{matrix} + τ_{2} \end{matrix} )

where b_{23} = 2λ_{2} + \frac{2e^{−s−t}\sqrt{λ_{1}}}{\sqrt{λ_{3}}\sqrt{λ_{2}}} + \frac{2\sqrt{λ_{2}}\sqrt{λ_{1}}}{e^{t−s}λ_{3}^{3/2}} + \frac{2λ_{1}τ_{1}}{e^{t}λ_{3}} ,

(33)  C_{3} = ( \begin{matrix}λ_{3} & e^{t}λ_{3} + \begin{matrix}e^{−s} & \sqrt{λ_{2}} \\ \sqrt{λ_{3}} & \sqrt{λ_{1}} \end{matrix} + \begin{matrix}e^{s} & \sqrt{λ_{1}} \\ \sqrt{λ_{3}} & \sqrt{λ_{2}} \end{matrix} + τ_{3} & \frac{2e^{−s}\sqrt{λ_{2}}}{\sqrt{λ_{3}}\sqrt{λ_{1}}} \\ 0 & \begin{matrix}e^{−s} & \sqrt{λ_{2}} \\ \sqrt{λ_{3}} & \sqrt{λ_{1}} \end{matrix} + τ_{3} & \frac{2e^{−s}\sqrt{λ_{2}}}{\sqrt{λ_{3}}\sqrt{λ_{1}}} \\ 0 & −\frac{e^{−s}\sqrt{λ_{2}}}{2\sqrt{λ_{3}}\sqrt{λ_{1}}} − \frac{e^{s}\sqrt{λ_{1}}}{2\sqrt{λ_{3}}\sqrt{λ_{2}}} − \frac{τ_{3}}{2} & −\begin{matrix}e^{−s} & \sqrt{λ_{2}} \\ \sqrt{λ_{3}} & \sqrt{λ_{1}} \end{matrix} \end{matrix} )

up to conjugation. For more detail see Goldman [8]. The eigenvalues λ_{j}, μ_{j}, ν_{j} of C_{j} are as follow.

\{λ_{1}, μ_{1} = \frac{\sqrt{λ_{1}} τ_{1} − \sqrt{−4 + λ_{1} τ_{1}^{2}}}{2 \sqrt{λ_{1}}} , ν_{1} = \frac{\sqrt{λ_{1}} τ_{1} + \sqrt{−4 + λ_{1} τ_{1}^{2}}}{2 \sqrt{λ_{1}}} \},

\{λ_{2}, μ_{2} = \frac{\sqrt{λ_{2}} τ_{2} − \sqrt{−4 + λ_{2} τ_{2}^{2}}}{2 \sqrt{λ_{2}}} , ν_{2} = \frac{\sqrt{λ_{2}} τ_{2} + \sqrt{−4 + λ_{2} τ_{2}^{2}}}{2 \sqrt{λ_{2}}} \},

\{λ_{3}, μ_{3} = \frac{\sqrt{λ_{3}} τ_{3} − \sqrt{−4 + λ_{3} τ_{3}^{2}}}{2 \sqrt{λ_{3}}} , ν_{3} = \frac{\sqrt{λ_{3}} τ_{3} + \sqrt{−4 + λ_{3} τ_{3}^{2}}}{2 \sqrt{λ_{3}}} \}.

From (30), dim H_{par}^{1}(π; 𝔤_{Adϕ}) = 8 − 6 = 2. By Proposition 5.7, u_{s} and u_{t} are parabolic elements since eigenvalues of C_{1}, C_{2} and C_{3} don’t involve s and t.

Now we calculate the symplectic form ω̃. Let’s compute ω̃(u_{s}, u_{t}). Suppose u_{s}|_{π_{1}} = dD_{1}, u_{s}|_{π_{2}} = dD_{2} and u_{s}|_{π_{3}} = dD_{3}. Since Z_{R} = (∂_{C_{1}}R, C_{1}) + (∂_{C_{2}}R, C_{2}) + (∂_{C_{3}}R, C_{3}) = (I, C_{1}) + (C_{1}, C_{2}) + (C_{1}C_{2}, C_{3}),

ω(u_{s}, u_{t}) = β_{∗}(u_{s}, u_{t})Z_{R} = β(u_{s}(I), I.u_{t}(C_{1}))

+β(u_{s}(C_{1}), C_{1}.u_{t}(C_{2})) + β(u_{s}(C_{1}C_{2}), C_{1}C_{2}.u_{t}(C_{3}))
= 0 + β(u_{s}(C_{1}), C_{1}.u_{t}(C_{2})) + β(u_{s}(C_{3}^{−1}), C_{3}^{−1}.u_{t}(C_{3}))
= β(u_{s}(C_{1}), C_{1}.u_{t}(C_{2})) + β(C_{3}.u_{s}(C_{3}^{−1}), u_{t}(C_{3}))
= β(u_{s}(C_{1}), C_{1}u_{t}(C_{2})C_{1}^{−1}) − β(u_{s}(C_{3})), u_{t}(C_{3}))

= tr{(∂_{s}C_{1})C_{1}^{−1}C_{1}(∂_{t}C_{2})C_{2}^{−1}C_{1}^{−1}} − tr{(∂_{s}C_{3})C_{3}^{−1}(∂_{t}C_{3})C_{3}^{−1}}

= tr{(∂_{s}C_{1})(∂_{t}C_{2})C_{3} + (∂_{s}C_{3})(∂_{t}C_{3}^{−1})}.

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 41

So ω̃(u_{s}, u_{t})

= ω(u_{s}, u_{t}) − β(D_{1}, u_{t}(C_{1})) − β(D_{2}, u_{t}(C_{2})) − β(D_{3}, u_{t}(C_{3}))
= tr{(∂_{s}C_{1})(∂_{t}C_{2})C_{3} + (∂_{s}C_{3})(∂_{t}C_{3}^{−1})
− D_{1}(∂_{t}C_{1})C_{1}^{−1} − D_{2}(∂_{t}C_{2})C_{2}^{−1} − D_{3}(∂_{t}C_{3})C_{3}^{−1}}.

Let P_{1} be a matrix such that P_{1}^{−1}C_{1}P_{1} is diagonal. Then by the proof of Proposition 5.7, −(∂_{s}P_{1})P_{1}^{−1} is one of D_{1}. Therefore we have the explicit formula for symplectic form ω̃(u_{s}, u_{t})

= tr{(∂_{s}C_{1})(∂_{t}C_{2})C_{3} + (∂_{s}C_{3})(∂_{t}C_{3}^{−1}) + (∂_{s}P_{1})P_{1}^{−1}(∂_{t}C_{1})C_{1}^{−1}
+ (∂_{s}P_{2})P_{2}^{−1}(∂_{t}C_{2})C_{2}^{−1} + (∂_{s}P_{3})P_{3}^{−1}(∂_{t}C_{3})C_{3}^{−1}}.

Put P_{j} the transpose of an eigenvector matrix of C_{j} for each j. Then we get an amazing result calculated by Mathematica.

ω̃(u_{s}, u_{t}) = −1.

Theorem 5.8. Suppose M is a pair of pants Σ(0, 3) and 𝔓(M) is the moduli space of convex real projective structures on M. Then the symplectic form on each parabolic leaf of 𝔓(M) is

ω̃ = dt ∧ ds.

Proof. See the detailed Mathematica computation in the appendix of Hong C. Kim’s dissertation [11]. q.e.d.

6. The global symplectic coordinates on 𝔓(M)

Let M be a closed oriented surface Σ(g, 0) with χ(M) = 2 − 2g < 0 and 𝔓(M) the moduli space of convex real projective structures on M. For a simply closed geodesic γ on M, we want to associate the twisting parameters θ_{γ}, β_{γ} ∈ ℝ on γ. Let A ∈ SL(3, ℝ) be the diagonal representation (1) of γ. For any u, v ∈ ℝ, we define g_{(u,v)} ∈ SL(3, ℝ) by

(34)  g_{(u,v)} = [ \begin{matrix}e^{− \frac{u}{2} − \frac{v}{3}} & 0 & 0 \\ 0 & e^{\frac{2}{3} v} & 0 \\ 0 & 0 & e^{\frac{u}{2} − \frac{v}{3}} \end{matrix} ]

then ℓ(g_{(u,v)}A) = |u + ℓ(A)| and m(g_{(u,v)}A) = v + m(A).

For u, v ∈ ℝ and γ a simply closed geodesic on M, we define Ψ_{γ(u,v)} : 𝔓(M) → 𝔓(M) by Ψ_{γ(u,v)}([h]) = [h′] where [h], [h′] ∈ Hom(π, G)/G such that h′ ≡ h except h′(x_{γ}) = g_{(u,v)}h(x_{γ})g_{(u,v)}^{−1} for x_{γ} ∈ π = π_{1}(M) corresponding to γ.

The flows Ψ_{γ(u,0)} and Ψ_{γ(0,v)} on 𝔓(M) are called the generalized twisting flows. Vector fields ∂/∂θ_{γ}, ∂/∂β_{γ} generated by Ψ_{γ(u,0)}, Ψ_{γ(0,v)} are called the generalized twisting vector fields whose potential functions are ℓ(γ) and m(γ) respectively.

Let Γ = {γ_{i}}_{i=1,...,3g−3} be a set of nontrivial homotopically distinct disjoint simply closed geodesics on M such that Γ decompose M as the disjoint union of 2g − 2 pairs of pants. At each γ_{i}, length parameters ℓ_{i}, m_{i} and twisting parameters θ_{i}, β_{i} are defined where ℓ_{i} ∈ ℝ_{+} and m_{i}, θ_{i}, β_{i} ∈ ℝ. Then the coordinate vector fields ∂/∂θ_{i}, ∂/∂β_{i} are the generalized twisting vector fields generated by the flows Ψ_{γ_{i}(u,0)}, Ψ_{γ_{i}(0,v)} and their potential functions are ℓ_{i}, m_{i} ; that is

(35)  ω(∂/∂θ_{i}, ) = −dℓ_{i}, ω(∂/∂β_{i}, ) = −dm_{i}.

A pair of pants P has a unique reflection ρ fixing (set wise) each boundary component. The fixed point set of ρ consists of the three geodesics connecting pairs of boundary components of P. The restriction of ρ to a boundary component γ of P is completely determined by its fixed point. Consider pairs of pants P_{1}, P_{2} with boundaries γ_{1}, γ_{2} and

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 43

unique reflections ρ_{1}, ρ_{2} respectively. Assume ℓ_{γ_{1}} = ℓ_{γ_{2}} and m_{γ_{1}} = −m_{γ_{2}} ; that is C_{γ_{1}} and C_{γ_{2}}^{−1} are conjugate where C_{γ_{i}} ∈ SL(3, ℝ) is the holonomy representation of γ_{i}. Then we can form the (geometric) sum P_{1} ∨ P_{2} by gluing γ_{1} and γ_{2} such that the real projective structures on P_{1} and P_{2} extend to P_{1} ∨ P_{2}. The pair (ρ_{1}, ρ_{2}) extends to a reflection ρ on P_{1} ∨ P_{2}. Hence given a surface M with a pants decomposition {P_{j}}, by varying the twisting parameters we may arrange that the reflection maps ρ_{j} of P_{j} piece together. Therefore the surface M has a reflection ρ fixing each boundary of {P_{j}}.

If we only allow the integer times of Dehn twist on the gluing processor of the boundaries of the partition {P_{j}}, then ρ is an orientation reversing map which preserving the real projective structure on M. The map ρ induces a map ρ̂ : 𝔓(M) → 𝔓(M) defined by ρ̂([h]) = [h ∘ ρ_{#}] where ρ_{#} : π_{1}(M) → π_{1}(M). Let ω be the symplectic form on 𝔓(M). Then ρ̂^{∗}ω = −ω since ρ̂ is orientation reversing.

Lemma 6.1. For the above ρ̂,

(1) ρ̂^{∗}dℓ_{i} = dℓ_{i}, ρ̂^{∗}dm_{i} = −dm_{i}

(2) ρ̂^{∗}dθ_{i} = −dθ_{i} + a_{i}dℓ_{i}, ρ̂^{∗}dβ_{i} = −dβ_{i} + b_{i}dm_{i} where a_{i}, b_{i} ∈ ℤ.

Proof. Suppose the eigenvalues of h(γ_{i}) are λ_{i} < μ_{i} < ν_{i}. Then the eigenvalues of (h ∘ ρ_{#})(γ_{i}) are ν_{i}^{−1} < μ_{i}^{−1} < λ_{i}^{−1} since [(h ∘ ρ_{#})(γ_{i})] = [h(γ_{i})^{−1}]. Therefore 1 holds. Since [h] and [h ∘ ρ_{#}] have the opposite orientations, the twisting parameters θ_{i}, β_{i} are measured with opposite senses. Furthermore there is an ambiguity of the integer times of Dehn twist for extending ρ_{j} to ρ. Therefore 2 holds. q.e.d.

From the duality formula (35), we can determine

ω(∂/∂θ_{k}, X) = −dℓ_{k}(X) = 0, ω(∂/∂β_{k}, Y) = −dm_{k}(Y) = 0

44 HONG CHAN KIM

where X and Y are vector fields such that X ∈ Vect(𝔓(M)) − < ∂/∂ℓ_{k} > and Y ∈ Vect(𝔓(M)) − < ∂/∂m_{k} > where 1 ≤ i ≤ 3g − 3 and 1 ≤ j ≤ 2g − 2.

Proposition 6.2. Let M be a closed surface Σ(g, 0) having an orientation reversing map ρ fixing the elements of a partition Γ = {γ_{i}} and preserving the real projective structure on M. Then for any i and j,

(36) ω(\frac{∂}{∂θ_{i}} , \frac{∂}{∂θ_{j}}) = ω(\frac{∂}{∂ℓ_{i}} , \frac{∂}{∂ℓ_{j}}) = ω(\frac{∂}{∂β_{i}} , \frac{∂}{∂β_{j}}) = ω(\frac{∂}{∂m_{i}} , \frac{∂}{∂m_{j}}) = 0.

Proof. The proof is based on the analogous properties of Fenchel-Nielsen coordinates on the Teichmüller space due to Wolpert [17]. First let us obtain a description for ρ̂_{∗} the adjoint of ρ̂^{∗}. From Lemma 6.1 we find that

(37) ρ̂^{∗}(∂/∂θ_{i}) = −∂/∂θ_{i}, ρ̂^{∗}(∂/∂ℓ_{i}) = ∂/∂ℓ_{i} + a_{i}∂/∂θ_{i},
(38) ρ̂^{∗}(∂/∂β_{i}) = ∂/∂β_{i}, ρ̂^{∗}(∂/∂m_{i}) = −∂/∂m_{i} + b_{i}∂/∂β_{i}.

Now we make the first calculation.

−ω(\frac{∂}{∂θ_{i}} , \frac{∂}{∂θ_{j}}) = ρ̂^{∗}ω(\frac{∂}{∂θ_{i}} , \frac{∂}{∂θ_{j}}) = ω(ρ̂_{∗} \frac{∂}{∂θ_{i}} , ρ̂_{∗} \frac{∂}{∂θ_{j}}) = ω(−\frac{∂}{∂θ_{i}} , −\frac{∂}{∂θ_{j}}).

Thus ω(∂/∂θ_{i}, ∂/∂θ_{j}) = 0. Similarly ω(∂/∂β_{i}, ∂/∂β_{j}) = 0.

The second argument for ω(∂/∂β_{i}, ∂/∂β_{j}) is as follow. Since ω is a skew-symmetry, without loss of generality assume i ≠ j.

− ω(∂/∂ℓ_{i}, ∂/∂ℓ_{j}) = ρ̂^{∗}ω(∂/∂ℓ_{i}, ∂/∂ℓ_{j}) = ω(ρ̂_{∗}∂/∂ℓ_{i}, ρ̂_{∗}∂/∂ℓ_{j})
= ω(∂/∂ℓ_{i} + a_{i}∂/∂θ_{i}, ∂/∂ℓ_{j} + a_{j}∂/∂θ_{j}) = ω(∂/∂ℓ_{i}, ∂/∂ℓ_{j})
+ a_{j}ω(∂/∂ℓ_{i}, ∂/∂θ_{j}) + a_{i}ω(∂/∂θ_{i}, ∂/∂ℓ_{j}) + a_{i}a_{j}ω(∂/∂θ_{i}, ∂/∂θ_{j})
= ω(∂/∂ℓ_{i}, ∂/∂ℓ_{j}) − a_{j}ω(∂/∂θ_{j}, ∂/∂ℓ_{i}) + a_{i}ω(∂/∂θ_{i}, ∂/∂ℓ_{j}) + 0
= ω(∂/∂ℓ_{i}, ∂/∂ℓ_{j}) + a_{j}dℓ_{j}(∂/∂ℓ_{i}) − a_{i}dℓ_{i}(∂/∂ℓ_{j})
= ω(∂/∂ℓ_{i}, ∂/∂ℓ_{j}) + a_{j}δ_{ji} − a_{i}δ_{ij} = ω(∂/∂ℓ_{i}, ∂/∂ℓ_{j}).

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 45

Thus ω(∂/∂ℓ_{i}, ∂/∂ℓ_{j}) = 0. Similarly ω(∂/∂m_{i}, ∂/∂m_{j}) = 0. q.e.d.

To show ω(∂/∂s_{j}, ∂/∂s_{k}) = 0 and ω(∂/∂t_{j}, ∂/∂t_{k}) = 0, we need a Mathematica computation. Now consider a surface M with genus 2. It can be formed by gluing boundaries of two punctured tori. A punctured torus can be obtained by gluing two boundaries γ_{1}, γ_{2} of a pair of pants such that ℓ_{1} = ℓ_{2} and m_{1} = −m_{2}. Let C_{1}, C_{2} ∈ SL(3, ℝ) be the holonomy representations of γ_{1}, γ_{2} respectively. Then there exists an invertible matrix Q such that C_{2} = QC_{1}^{−1}Q^{−1}.

Suppose that P_{j} ∈ SL(3, ℝ) such that P_{j}^{−1}C_{j}P_{j} = E_{j} the diagonal matrix (1). Let ℨ be the following subset of SL(3, ℝ) ;

\{[ \begin{matrix}0 & 0 & −1 \\ 0 & −1 & 0 \\ −1 & 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ −1 & 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 & −1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ]\} .

Since C_{2} and C_{1}^{−1} are conjugate, we have E_{2} = ZE_{1}^{−1}Z^{−1} for any element Z ∈ ℨ. Put Q = P_{2}ZP_{1}^{−1}. Then we get

QC_{1}^{−1}Q^{−1} = (P_{2}ZP_{1}^{−1})C_{1}^{−1}(P_{1}Z^{−1}P_{2}^{−1}) = P_{2}(ZE_{1}^{−1}Z^{−1})P_{2}^{−1} = C_{2}.

Suppose we have another P̄_{j} such that P̄_{j}^{−1}C_{j}P̄_{j} = E_{j} for each j. Then there exists a diagonal matrix D_{j} that P̄_{j} = P_{j}D_{j}. Therefore in general

Q = (P_{2}D_{2})Z(P_{1}D_{1})^{−1} = P_{2}(D_{2}ZD_{1}^{−1})P_{1}^{−1} = P_{2}ZDP_{1}^{−1}

where D is a diagonal matrix and Z ∈ ℨ.

Proposition 6.3. For any diagonal matrix D ∈ SL(3, ℝ) with nonnegative entries, there exists a unique element Z of ℨ such that Q = P_{2}ZDP_{1}^{−1} is positive hyperbolic.

Proof. By Goldman [8], Q is positive hyperbolic if and only if x > 0, y > 0 and f(x, y) = x^{2}y^{2} −4(x^{3} +y^{3})+18xy−27 > 0 where x = tr(Q)

46 HONG CHAN KIM

and y = tr(Q^{−1}). Suppose x > 4, y > 4 and \sqrt{4x} < y < \frac{x^{2}}{4} . Then y^{2} − 4x > 0 and x^{2} − 4y > 0. Hence f(x, y) = (x^{2} − 4y)(y^{2} − 4x) + 2(xy − 16) + 5 > 0. Through the Mathematica computation there exists a unique Z ∈ ℨ such that x > 0 and y > 0. At this case we can compute x > 4, y > 4 and \sqrt{4x} < y < \frac{x^{2}}{4} by using Mathematica. q.e.d.

Consider the geodesic γ connecting the boundary components γ_{1} and γ_{2} of a pair of pants. Then there exists a diagonal matrix D_{0} ∈ SL(3, ℝ) such that ℓ(P_{2}ZD_{0}P_{1}^{−1}) equals the length of the geodesic γ and m(P_{2}ZD_{0}P_{1}^{−1}) vanishes. Therefore Q = P_{2}ZD_{0}DP_{1}^{−1} where

D = [ \begin{matrix}e^{− \frac{θ}{2} − \frac{β}{3}} & 0 & 0 \\ 0 & e^{\frac{2}{3} β} & 0 \\ 0 & 0 & e^{\frac{θ}{2} − \frac{β}{3}} \end{matrix} ] .

Now we have a representation of a punctured torus Σ(1, 1).

A_{1} = C_{1}, B_{1} = Q = P_{2}ZD_{0}DP_{1}^{−1}, F_{1} = C_{3}

since A_{1}B_{1}A_{1}^{−1}B_{1}^{−1}F_{1} = C_{1}QC_{1}^{−1}Q^{−1}C_{3} = C_{1}C_{2}C_{3} = I.

Similarly we can glue two punctured tori M_{1} and M_{2} via two boundaries such that ℓ_{1} = ℓ_{2} and m_{1} = −m_{2}. Suppose M_{i} has a relation A_{i}B_{i}A_{i}^{−1}B_{i}^{−1}F_{i} = I. Let Q = P_{6}ZDP_{3}^{−1} where P_{3}^{−1}F_{1}P_{3} = E_{3} and P_{6}^{−1}F_{2}P_{6} = E_{6} the diagonal matrices (1). Then F_{2} = QF_{1}^{−1}Q^{−1}. Therefore the (geometric) sum M = M_{1} ∨ M_{2} is a closed surface with genus 2 having a relation Ā_{1}B̄_{1}Ā_{1}^{−1}B̄_{1}^{−1}Ā_{2}B̄_{2}Ā_{2}^{−1}B̄_{2}^{−1} = I where Ā_{1} = A_{1}, B̄_{1} = B_{1}, Ā_{2} = Q^{−1}A_{2}Q and B̄_{2} = Q^{−1}B_{2}Q.

Suppose that a closed surface M with genus 2 has a relation

R = A_{1}B_{1}A_{1}^{−1}B_{1}^{−1}A_{2}B_{2}A_{2}^{−1}B_{2}^{−1} = I.

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 47

Then its fundamental cycle is

Z_{R} = (I − A_{1}B_{1}A_{1}^{−1}, A_{1}) + (A_{1} − A_{1}B_{1}A_{1}^{−1}B_{1}^{−1}, B_{1})
+ (A_{1}B_{1}A_{1}^{−1}B_{1}^{−1} − RB_{2}, A_{2}) + (A_{1}B_{1}A_{1}^{−1}B_{1}^{−1}A_{2} − R, B_{2}).

Hence the coefficient of its symplectic form ω is

ω(u, v) = β(u(A_{1}B_{1}^{−1}A_{1}^{−1}), v(A_{1})) − β(u(A_{1}^{−1}), v(B_{1}))
+ β(u([A_{1}, B_{1}]^{−1}), v(B_{1})) − β(u([A_{2}, B_{2}]), v(A_{2}))
+ β(u(B_{2}^{−1}), v(A_{2})) − β(u(B_{2}A_{2}B_{2}^{−1}), v(B_{2})).

Proposition 6.4. For each j and k,

(39)  ω(∂/∂s_{j}, ∂/∂s_{k}) = 0, ω(∂/∂t_{j}, ∂/∂t_{k}) = 0.

Proof. By the definition of u_{z_{k}} in (27), ω(u_{s_{j}}, u_{s_{k}}) becomes

tr \{\frac{∂(A_{1}B_{1}^{−1}A_{1}^{−1})}{∂s_{j}}(A_{1}B_{1}A_{1}^{−1})\frac{∂A_{1}}{∂s_{k}} A_{1}^{−1} − \frac{∂A_{1}^{−1}}{∂s_{j}} A_{1} \frac{∂B_{1}}{∂s_{k}} B_{1}^{−1}

+\frac{∂F_{1}}{∂s_{j}} F_{1}^{−1} \frac{∂B_{1}}{∂s_{k}} B_{1}^{−1} − \frac{∂F_{2}^{−1}}{∂s_{j}} F_{2} \frac{∂A_{2}^{−1}}{∂s_{k}} A_{2}

+\frac{∂B_{2}^{−1}}{∂s_{j}} B_{2} \frac{∂A_{2}}{∂s_{k}} A_{2}^{−1} − \frac{∂(B_{2}A_{2}B_{2}^{−1})}{∂s_{j}}(B_{2}A_{2}^{−1}B_{2}^{−1})\frac{∂B_{2}}{∂s_{k}} B_{2}^{−1}\} .

Through the Mathematica calculation, we have the desired result. q.e.d.

Let {P_{j}}_{j=1,...,2g−2} be the disjoint union of pairs of pants decomposed by a partition Γ = {γ_{i}}_{i=1,...,3g−3}. Let s_{j}, t_{j} be the internal parameters of each pair of pants P_{j} where s_{j}, t_{j} ∈ ℝ. Then {ℓ_{i}, m_{i}, θ_{i}, β_{i}, s_{j}, t_{j}} is a global coordinate on 𝔓(M) for 1 ≤ i ≤ 3g − 3 and 1 ≤ j ≤ 2g − 2. By Proposition 6.2, 6.4, Equation (35) and Theorem 5.8, the symplectic form ω on 𝔓(M) is represented by

ω = ∑_{i=1}^{3g−3} dℓ_{i} ∧ dθ_{i} + ∑_{i=1}^{3g−3} dm_{i} ∧ dβ_{i} + ∑_{j=1}^{2g−2} f_{j}ds_{j} ∧ dt_{j}

48 HONG CHAN KIM

where f_{j} is a smooth function on 𝔓(M) such that f_{j}|_{𝔏(M)} ≡ −1 for each parabolic leaf 𝔏(M) fixing ℓ_{i}, m_{i}, θ_{i}, β_{i}. So f_{j} = f_{j}(ℓ_{i}, m_{i}, θ_{i}, β_{i}) ; that is f_{j} is independent of parameters s_{j}, t_{j}.

Theorem 6.5. Let M be a closed smooth surface Σ(g, 0). Then the symplectic form on the moduli space 𝔓(M) of convex real projective structures is

ω = ∑_{i=1}^{3g−3} dℓ_{i} ∧ dθ_{i} + ∑_{i=1}^{3g−3} dm_{i} ∧ dβ_{i} + ∑_{j=1}^{2g−2} dt_{j} ∧ ds_{j}

where ℓ_{i}, m_{i} are length parameters, θ_{i}, β_{i} are twisting parameters, and s_{j}, t_{j} are internal parameters on 𝔓(M). Therefore 𝔓(M) is symplectomorphic to ℝ^{16g−16}.

Proof. Suppose the symplectic form is

ω = ∑_{i=1}^{3g−3} dℓ_{i} ∧ dθ_{i} + ∑_{i=1}^{3g−3} dm_{i} ∧ dβ_{i} + ∑_{j=1}^{2g−2} f_{j} ds_{j} ∧ dt_{j}

where f_{j} = f_{j}(ℓ_{i}, m_{i}, θ_{i}, β_{i}) ∈ C^{∞}(𝔓(M)). Then

ω(∂/∂s_{j}, ·) = f_{j} dt_{j}.

Let X_{t_{j}} be the Hamiltonian vector fields of t_{j}. Then ∂/∂s_{j} = f_{j}X_{t_{j}}. Since the symplectic form ω is invariant under Hamiltonian vector fields, L_{X_{t_{j}}} ω = 0. Hence

L_{∂/∂s_{j}}ω = L_{f_{j}X_{t_{j}}} ω = f_{j}L_{X_{t_{j}}} ω = 0.

So

0 = L_{∂/∂s_{j}}ω = ι_{∂/∂s_{j}}dω + dι_{∂/∂s_{j}}ω = 0 + d(ω(∂/∂s_{j}, ·))
= d(f_{j} dt_{j}) = df_{j} ∧ dt_{j} + f_{j} d^{2}t_{j} = df_{j} ∧ dt_{j}.

THE SYMPLECTIC GLOBAL COORDINATES ON THE MODULI SPACE 49

Therefore f_{j} is independent of all parameters except t_{j}. Since f_{j} is independent of t_{j}, f_{j} is independent of all parameters. We conclude f_{j} = −1 since f_{j}|_{𝔏(M)} = −1. q.e.d.

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50 HONG CHAN KIM

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Department of Mathematics, University of Maryland, College Park, Maryland 20742

Current address: Korea Institute for Advanced Study, 207-43 Cheongryangridong, Dongdaemun-gu, Seoul 130-012, Korea

    E-mail address: hongchan@kias.re.kr