The complex symplectic geometry of the deformation space of complex projective structures

Loustau, Brice

doi:10.2140/gt.2015.19.1737

Cited by 17 publications

(41 citation statements)

References 40 publications

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“…How is ω σ affected by the choice of the "zero section" σ? A small computation (see [Lou14]) shows that: Proposition 2.2. For any two sections σ 1 and σ 2 to p : CP(S) → T (S),…”

Section: Cotangent Affine Symplectic Structuresmentioning

confidence: 98%

“…Using results of [Lou14] and an ad hoc notion of renormalized for almost-Fuchsian manifolds, we compare the Goldman symplectic structure ω G of the character variety restricted to AF(S) with the symplectic structure ω H on H and, indirectly, the canonical symplectic structure ω can on the cotangent bundle to Teichmüller space T * T (S). We show in particular 1 : Theorem 4.9.…”

Section: Introductionmentioning

confidence: 99%

“…This is thoroughly addressed in [Lou14]. Recall that there is a natural "forgetful" projection p : CP(S) → T (S) which assigns to a complex projective structure on S its underlying complex structure.…”

Section: Introductionmentioning

confidence: 99%

“…As a consequence there is an identification τ σ : CP(S) ∼ → T * T (S), but it is not canonical: it depends on the choice of a "zero section" σ : T (S) → CP(S). The result proven in [Lou14] that we will use is:…”

Section: Introductionmentioning

confidence: 99%

“…Note: The study and comparison of the affine cotangent symplectic structures on CP(S), Goldman's symplectic structure on X (S) and the "complex Hamiltonian picture" of QF(S) is addressed in [Lou14]. Both articles are based on the author's PhD thesis ([Lou11]).…”

Section: Introductionmentioning

confidence: 99%

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Minimal surfaces and symplectic structures of moduli spaces

Loustau

2015

Geom Dedicata

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Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X (S, P SL(2, C)) and the affine cotangent symplectic structure on the space of complex projective structures CP(S) given by the Schwarzian parametrization. This is done in restriction to the moduli space of almost-Fuchsian structures by involving a notion of renormalized volume, used to relate the geometry of a minimal surface in a hyperbolic 3-manifold to the geometry of its ideal conformal boundary.

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“…How is ω σ affected by the choice of the "zero section" σ? A small computation (see [Lou14]) shows that: Proposition 2.2. For any two sections σ 1 and σ 2 to p : CP(S) → T (S),…”

Section: Cotangent Affine Symplectic Structuresmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimal surfaces and symplectic structures of moduli spaces

Loustau

2015

Geom Dedicata

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On Kawai theorem for orbifold Riemann surfaces

Takhtajan

2018

Math. Ann.

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We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmüller space to the PSL(2, C)-character variety, which allows to evaluate explicitly the pullback of Goldman symplectic form in the spirit of Riemann bilinear relations. As a corollary, we obtain a generalization of Goldman's theorem that the pullback of Goldman symplectic form on the PSL(2, R)-character variety is a symplectic form of the Weil-Petersson metric on the Teichmüller space.

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Symplectic geometry of the moduli space of projective structures in homological coordinates

2017

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We introduce a natural symplectic structure on the moduli space of quadratic differentials with simple zeros and describe its Darboux coordinate systems in terms of so-called homological coordinates. We then show that this structure coincides with the canonical Poisson structure on the cotangent bundle of the moduli space of Riemann surfaces, and therefore the homological coordinates provide a new system of Darboux coordinates. We define a natural family of commuting "homological flows" on the moduli space of quadratic differentials and find the corresponding action-angle variables.The space of projective structures over the moduli space can be identified with the cotangent bundle upon selection of a reference projective connection that varies holomorphically and thus can be naturally endowed with a symplectic structure. Different choices of projective connections of this kind (Bergman, Schottky, Wirtinger) give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations: the corresponding generating functions are found. We also study the monodromy representation of the Schwarzian equation associated with a projective connection, and we show that the natural symplectic structure on the the space of projective connections induces the Goldman Poisson structure on the character variety. Combined with results of Kawai, this result shows the symplectic equivalence between the embeddings of the cotangent bundle into the space of projective structures given by the Bers and Bergman projective connections.

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The complex symplectic geometry of the deformation space of complex projective structures

Cited by 17 publications

References 40 publications

Minimal surfaces and symplectic structures of moduli spaces

Minimal surfaces and symplectic structures of moduli spaces

On Kawai theorem for orbifold Riemann surfaces

Symplectic geometry of the moduli space of projective structures in homological coordinates

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