The hyperk hler geometry of the deformation space of ...
Transcript of The hyperk hler geometry of the deformation space of ...
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The hyperkähler geometry of the deformation space ofcomplex projective structures on a surface
Brice Loustau
August 3, 2012
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Outline
1 Complex projective structures
2 The character variety
3 The Schwarzian parametrization
4 The minimal surface parametrization
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
1 Complex projective structures
2 The character variety
3 The Schwarzian parametrization
4 The minimal surface parametrization
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
What is a complex projective structure?
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
What is a complex projective structure?
Let S be a closed oriented surface of genus g > 2.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
What is a complex projective structure?
Let S be a closed oriented surface of genus g > 2.
Definition
A complex projective structure on S is a (G ,X )-structure on S
where the model space is X = CP1 and the Lie group of
transformations of X is G = PSL2(C).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
What is a complex projective structure?
Let S be a closed oriented surface of genus g > 2.
Definition
A complex projective structure on S is a (G ,X )-structure on S
where the model space is X = CP1 and the Lie group of
transformations of X is G = PSL2(C).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
CP(S) and Teichmüller space T (S)
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
CP(S) and Teichmüller space T (S)
Definition
CP(S) is the deformation space of all complex projective structureson S :
CP(S) = {all CP1-structures on S}/Diff +
0 (S) .
A point Z ∈ CP(S) is called a marked complex projective surface.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
CP(S) and Teichmüller space T (S)
Definition
CP(S) is the deformation space of all complex projective structureson S :
CP(S) = {all CP1-structures on S}/Diff +
0 (S) .
A point Z ∈ CP(S) is called a marked complex projective surface.
CP(S) is a complex manifold of dimension dimC CP(S) = 6g − 6.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
CP(S) and Teichmüller space T (S)
Definition
CP(S) is the deformation space of all complex projective structureson S :
CP(S) = {all CP1-structures on S}/Diff +
0 (S) .
A point Z ∈ CP(S) is called a marked complex projective surface.
CP(S) is a complex manifold of dimension dimC CP(S) = 6g − 6.
Note: A complex projective atlas is in particular a complex atlas onS (transition functions are holomorphic).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
CP(S) and Teichmüller space T (S)
Definition
CP(S) is the deformation space of all complex projective structureson S :
CP(S) = {all CP1-structures on S}/Diff +
0 (S) .
A point Z ∈ CP(S) is called a marked complex projective surface.
CP(S) is a complex manifold of dimension dimC CP(S) = 6g − 6.
Note: A complex projective atlas is in particular a complex atlas onS (transition functions are holomorphic).
Definition
There is a forgetful map p : CP(S) → T (S) where
T (S) = {all complex structures on S}/Diff+0 (S)
is the Teichmüller space of S .
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Fuchsian and quasifuchsian structures
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Fuchsian and quasifuchsian structures
If any Kleinian group Γ (i.e. discrete subgroup of PSL2(C)) actsfreely and properly on some open subset U of CP
1, the quotientinherits a complex projective structure.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Fuchsian and quasifuchsian structures
If any Kleinian group Γ (i.e. discrete subgroup of PSL2(C)) actsfreely and properly on some open subset U of CP
1, the quotientinherits a complex projective structure.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Fuchsian structures
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Fuchsian structures
In particular, any Riemann surface X can be equipped with acompatible CP
1-structure by the uniformization theorem:
X = H2
where Γ ⊂ PSL2(R) is a Fuchsian group.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Fuchsian structures
In particular, any Riemann surface X can be equipped with acompatible CP
1-structure by the uniformization theorem:
X = H2
where Γ ⊂ PSL2(R) is a Fuchsian group.
Note: This defines a Fuchsian section σF : T (S) → CP(S).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Quasifuchsian structures
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Quasifuchsian structures
By Bers’ simultaneous uniformization theorem, given two complexstructures (X+,X−) ∈ T (S)× T (S), there exists a unique Kleiniangroup Γ such that:
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Quasifuchsian structures
By Bers’ simultaneous uniformization theorem, given two complexstructures (X+,X−) ∈ T (S)× T (S), there exists a unique Kleiniangroup Γ such that:
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
1 Complex projective structures
2 The character variety
3 The Schwarzian parametrization
4 The minimal surface parametrization
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Holonomy
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Holonomy
Any complex projective structure Z ∈ CP(S) defines a holonomy
representation ρ : π1(S) → G = PSL2(C).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Holonomy
Any complex projective structure Z ∈ CP(S) defines a holonomy
representation ρ : π1(S) → G = PSL2(C).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The character variety
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The character variety
Holonomy defines a map
hol : CP(S) → X (S ,G) ;
where X (S ,G) = Hom(π1(S),G)//G is the character variety of S .
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The character variety
Holonomy defines a map
hol : CP(S) → X (S ,G) ;
where X (S ,G) = Hom(π1(S),G)//G is the character variety of S .hol is a local biholomorphism.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The character variety
Holonomy defines a map
hol : CP(S) → X (S ,G) ;
where X (S ,G) = Hom(π1(S),G)//G is the character variety of S .hol is a local biholomorphism.
By a general construction of Goldman, the character variety X (S ,G)enjoys a natural complex symplectic structure ωG .
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The character variety
Holonomy defines a map
hol : CP(S) → X (S ,G) ;
where X (S ,G) = Hom(π1(S),G)//G is the character variety of S .hol is a local biholomorphism.
By a general construction of Goldman, the character variety X (S ,G)enjoys a natural complex symplectic structure ωG .
Abusing notations, we also let ωG denote the complex symplecticstructure on CP(S) obtained by pulling back ωG by the holonomymap hol : CP(S) → X (S ,G).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The character variety (continued)
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The character variety (continued)
Theorem (Goldman)
The restriction of the complex symplectic structure on the Fuchsianslice F(S) is the Weil-Petersson Kähler form:
σ∗
F (ωG ) = ωWP .
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The character variety (continued)
Theorem (Goldman)
The restriction of the complex symplectic structure on the Fuchsianslice F(S) is the Weil-Petersson Kähler form:
σ∗
F (ωG ) = ωWP .
Theorem (Platis, L)
Complex Fenchel-Nielsen coordinates (li , τi ) associated to any pantsdecomposition are canonical coordinates for the symplectic structure:
ωG =∑
i
dli ∧ dτi .
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Hitchin-Kobayashi correspondence
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Hitchin-Kobayashi correspondence
Theorem (Hitchin, Simpson, Corlette, Donaldson)
Fix a complex structure X on S . There is a real-analytic bijection
HX : X 0(S ,G)∼→ M0
Dol(X ,G)
where M0
Dol(X ,G) is the moduli space of topologically trivialpolystable Higgs bundles on X .
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Hitchin-Kobayashi correspondence
Theorem (Hitchin, Simpson, Corlette, Donaldson)
Fix a complex structure X on S . There is a real-analytic bijection
HX : X 0(S ,G)∼→ M0
Dol(X ,G)
where M0
Dol(X ,G) is the moduli space of topologically trivialpolystable Higgs bundles on X .
Note: HX is not holomorphic, in fact:
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
Hitchin-Kobayashi correspondence
Theorem (Hitchin, Simpson, Corlette, Donaldson)
Fix a complex structure X on S . There is a real-analytic bijection
HX : X 0(S ,G)∼→ M0
Dol(X ,G)
where M0
Dol(X ,G) is the moduli space of topologically trivialpolystable Higgs bundles on X .
Note: HX is not holomorphic, in fact:
Theorem (Hitchin)
There is a natural hyperkähler structure (g , I , J,K) on M0
Dol(X ,G).The map HX is holomorphic with respect to J. It is also asymplectomorphism for the appropriate symplectic structures.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
1 Complex projective structures
2 The character variety
3 The Schwarzian parametrization
4 The minimal surface parametrization
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The cotangent hyperkähler structure
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The cotangent hyperkähler structure
Recall that if M is any complex manifold, its holomorphic cotangentbundle T ∗M is equipped with a canonical complex symplecticstructure ωcan.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The cotangent hyperkähler structure
Recall that if M is any complex manifold, its holomorphic cotangentbundle T ∗M is equipped with a canonical complex symplecticstructure ωcan.
Theorem (Feix, Kaledin)
If M is a real-analytic Kähler manifold, then there exists a uniquehyperkähler structure in a neighborhood of the zero section in T ∗M
such that:
• it refines the complex symplectic structure
• it extends the Kähler structure off the zero section
• the U(1)-action in the fibers is isometric.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization
Recall that there is a canonical holomorphic projectionp : CP(S) → T (S).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization
Recall that there is a canonical holomorphic projectionp : CP(S) → T (S).
The Schwarzian derivative is an operator on maps betweenprojective surfaces such that:
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization
Recall that there is a canonical holomorphic projectionp : CP(S) → T (S).
The Schwarzian derivative is an operator on maps betweenprojective surfaces such that:
• It turns a fiber p−1(X ) into a complex affine space modeledon the vector space H0(X ,K2) = T ∗
XT (S).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization
Recall that there is a canonical holomorphic projectionp : CP(S) → T (S).
The Schwarzian derivative is an operator on maps betweenprojective surfaces such that:
• It turns a fiber p−1(X ) into a complex affine space modeledon the vector space H0(X ,K2) = T ∗
XT (S).
• Globally, CP(S) ≈σ T ∗T (S) but this identification depends onthe choice of a “zero section” σ : T (S) → CP(S).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization
Recall that there is a canonical holomorphic projectionp : CP(S) → T (S).
The Schwarzian derivative is an operator on maps betweenprojective surfaces such that:
• It turns a fiber p−1(X ) into a complex affine space modeledon the vector space H0(X ,K2) = T ∗
XT (S).
• Globally, CP(S) ≈σ T ∗T (S) but this identification depends onthe choice of a “zero section” σ : T (S) → CP(S).
For each choice of σ, we thus get a symplectic structure ωσ on thewhole space CP(S) (pulling back ωcan) and a hyperkähler structureon some neighborhood of the Fuchsian slice.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Theorem (L)
CP(S) ≈σ T ∗T (S) is a complex symplectomorphism iffd(σ − σF ) = ωWP (on T (S)).
![Page 50: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/50.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Theorem (L)
CP(S) ≈σ T ∗T (S) is a complex symplectomorphism iffd(σ − σF ) = ωWP (on T (S)).
Using results of McMullen (also Takhtajan-Teo, Krasnov-Schlenker):
Theorem (Kawai, L)
If σ is a (generalized) Bers section, CP(S) ≈σ T ∗T (S) is a complexsymplectomorphism.
![Page 51: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/51.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Theorem (L)
CP(S) ≈σ T ∗T (S) is a complex symplectomorphism iffd(σ − σF ) = ωWP (on T (S)).
Using results of McMullen (also Takhtajan-Teo, Krasnov-Schlenker):
Theorem (Kawai, L)
If σ is a (generalized) Bers section, CP(S) ≈σ T ∗T (S) is a complexsymplectomorphism.
![Page 52: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/52.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Consequences:
![Page 53: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/53.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Consequences:
• Fibers of p and Bers slices are Lagrangian complexsubmanifolds.
![Page 54: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/54.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Consequences:
• Fibers of p and Bers slices are Lagrangian complexsubmanifolds.
• (generalized) Quasifuchsian reciprocity.
![Page 55: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/55.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Consequences:
• Fibers of p and Bers slices are Lagrangian complexsubmanifolds.
• (generalized) Quasifuchsian reciprocity.
• If σ is elected among Bers sections,
![Page 56: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/56.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Consequences:
• Fibers of p and Bers slices are Lagrangian complexsubmanifolds.
• (generalized) Quasifuchsian reciprocity.
• If σ is elected among Bers sections,
• The hyperkähler stucture we get on CP(S) refines the complexsymplectic structure,
![Page 57: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/57.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Consequences:
• Fibers of p and Bers slices are Lagrangian complexsubmanifolds.
• (generalized) Quasifuchsian reciprocity.
• If σ is elected among Bers sections,
• The hyperkähler stucture we get on CP(S) refines the complexsymplectic structure,
• but the new complex structure J depends on the choice of theBers section. In other words the bunch of hyperkähler structureswe get is parametrized by T (S).
![Page 58: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/58.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Consequences:
• Fibers of p and Bers slices are Lagrangian complexsubmanifolds.
• (generalized) Quasifuchsian reciprocity.
• If σ is elected among Bers sections,
• The hyperkähler stucture we get on CP(S) refines the complexsymplectic structure,
• but the new complex structure J depends on the choice of theBers section. In other words the bunch of hyperkähler structureswe get is parametrized by T (S).
• This is similar to the situation we saw with theHitchin-Kobayashi correspondence.
![Page 59: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/59.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The Schwarzian parametrization(continued)
Consequences:
• Fibers of p and Bers slices are Lagrangian complexsubmanifolds.
• (generalized) Quasifuchsian reciprocity.
• If σ is elected among Bers sections,
• The hyperkähler stucture we get on CP(S) refines the complexsymplectic structure,
• but the new complex structure J depends on the choice of theBers section. In other words the bunch of hyperkähler structureswe get is parametrized by T (S).
• This is similar to the situation we saw with theHitchin-Kobayashi correspondence.Quiz : what is a significant difference though?
![Page 60: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/60.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
1 Complex projective structures
2 The character variety
3 The Schwarzian parametrization
4 The minimal surface parametrization
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization
The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is aneighborhood of the Fuchsian slice such that if Z ∈ AF(S), thehyperbolic 3-manifold associated to Z contains a unique minimalsurface Σ.
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization
The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is aneighborhood of the Fuchsian slice such that if Z ∈ AF(S), thehyperbolic 3-manifold associated to Z contains a unique minimalsurface Σ.
The Gauss-Codazzi equations satisfied by the second fundamentalform IIΣ are equivalent to the fact that IIΣ is the real part of aunique holomorphic quadratic ϕ.
![Page 64: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/64.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization
The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is aneighborhood of the Fuchsian slice such that if Z ∈ AF(S), thehyperbolic 3-manifold associated to Z contains a unique minimalsurface Σ.
The Gauss-Codazzi equations satisfied by the second fundamentalform IIΣ are equivalent to the fact that IIΣ is the real part of aunique holomorphic quadratic ϕ.
This defines a mapAF(S) → T ∗T (S)
Z 7→ ([IΣ], ϕ).
![Page 65: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/65.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization
The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is aneighborhood of the Fuchsian slice such that if Z ∈ AF(S), thehyperbolic 3-manifold associated to Z contains a unique minimalsurface Σ.
The Gauss-Codazzi equations satisfied by the second fundamentalform IIΣ are equivalent to the fact that IIΣ is the real part of aunique holomorphic quadratic ϕ.
This defines a mapAF(S) → T ∗T (S)
Z 7→ ([IΣ], ϕ).
It is a diffeomorphism of AF(S) onto some neighborhood of thezero section of T ∗T (S).
![Page 66: The hyperk hler geometry of the deformation space of ...](https://reader031.fdocuments.in/reader031/viewer/2022012518/61926b15d736f131e415c5d9/html5/thumbnails/66.jpg)
The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization
The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is aneighborhood of the Fuchsian slice such that if Z ∈ AF(S), thehyperbolic 3-manifold associated to Z contains a unique minimalsurface Σ.
The Gauss-Codazzi equations satisfied by the second fundamentalform IIΣ are equivalent to the fact that IIΣ is the real part of aunique holomorphic quadratic ϕ.
This defines a mapAF(S) → T ∗T (S)
Z 7→ ([IΣ], ϕ).
It is a diffeomorphism of AF(S) onto some neighborhood of thezero section of T ∗T (S).
Again, one can use this “minimal surface parametrization” to pullback the hyperkähler structure of T ∗T (S) on CP(S).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization(continued)
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization(continued)
The notion of renormalized volume of almost-Fuchsian manifoldsdefines a function W on AF(S).
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization(continued)
The notion of renormalized volume of almost-Fuchsian manifoldsdefines a function W on AF(S).
Using arguments of Krasnov-Schlenker to compute the variation ofW under an infinitesimal deformation of the metric, one shows:
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The hyperkählergeometry of CP(S)
Brice Loustau
Complex projectivestructures
The character variety
The Schwarzianparametrization
The minimal surfaceparametrization
The minimal surface parametrization(continued)
The notion of renormalized volume of almost-Fuchsian manifoldsdefines a function W on AF(S).
Using arguments of Krasnov-Schlenker to compute the variation ofW under an infinitesimal deformation of the metric, one shows:
Theorem (L)
The minimal surface parametrization AF(S)∼→ T ∗T (S) is a real
symplectomorphism (for the appropriate symplectic structures).