The hyperkhler geometry of the deformation space of The character - PowerPoint PPT Presentation

The hyperkähler geometry of CP ( S ) Brice Loustau Complex projective structures The hyperkähler geometry of the deformation space of The character variety complex projective structures on a surface The Schwarzian parametrization The minimal surface parametrization Brice Loustau August 3, 2012

The hyperkähler geometry of CP ( S ) Outline Brice Loustau Complex projective 1 Complex projective structures structures The character variety 2 The character variety The Schwarzian parametrization The minimal surface parametrization 3 The Schwarzian parametrization 4 The minimal surface parametrization

The hyperkähler geometry of CP ( S ) 1 Complex projective structures Brice Loustau 2 The character variety Complex projective structures The character variety 3 The Schwarzian parametrization The Schwarzian parametrization The minimal surface 4 The minimal surface parametrization parametrization

The hyperkähler geometry of CP ( S ) What is a complex projective structure? Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) What is a complex projective structure? Brice Loustau Let S be a closed oriented surface of genus g � 2. Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) What is a complex projective structure? Brice Loustau Let S be a closed oriented surface of genus g � 2. Complex projective structures The character variety Definition The Schwarzian parametrization A complex projective structure on S is a ( G , X ) -structure on S The minimal surface where the model space is X = C P 1 and the Lie group of parametrization transformations of X is G = PSL 2 ( C ) .

The hyperkähler geometry of CP ( S ) What is a complex projective structure? Brice Loustau Let S be a closed oriented surface of genus g � 2. Complex projective structures The character variety Definition The Schwarzian parametrization A complex projective structure on S is a ( G , X ) -structure on S The minimal surface where the model space is X = C P 1 and the Lie group of parametrization transformations of X is G = PSL 2 ( C ) .

The hyperkähler geometry of CP ( S ) CP ( S ) and Teichmüller space T ( S ) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) CP ( S ) and Teichmüller space T ( S ) Brice Loustau Complex projective Definition structures CP ( S ) is the deformation space of all complex projective structures The character variety on S : The Schwarzian parametrization CP ( S ) = { all C P 1 -structures on S } / Diff + 0 ( S ) . The minimal surface parametrization A point Z ∈ CP ( S ) is called a marked complex projective surface.

The hyperkähler geometry of CP ( S ) CP ( S ) and Teichmüller space T ( S ) Brice Loustau Complex projective Definition structures CP ( S ) is the deformation space of all complex projective structures The character variety on S : The Schwarzian parametrization CP ( S ) = { all C P 1 -structures on S } / Diff + 0 ( S ) . The minimal surface parametrization A point Z ∈ CP ( S ) is called a marked complex projective surface. CP ( S ) is a complex manifold of dimension dim C CP ( S ) = 6 g − 6.

The hyperkähler geometry of CP ( S ) CP ( S ) and Teichmüller space T ( S ) Brice Loustau Complex projective Definition structures CP ( S ) is the deformation space of all complex projective structures The character variety on S : The Schwarzian parametrization CP ( S ) = { all C P 1 -structures on S } / Diff + 0 ( S ) . The minimal surface parametrization A point Z ∈ CP ( S ) is called a marked complex projective surface. CP ( S ) is a complex manifold of dimension dim C CP ( S ) = 6 g − 6. Note : A complex projective atlas is in particular a complex atlas on S (transition functions are holomorphic).

The hyperkähler geometry of CP ( S ) CP ( S ) and Teichmüller space T ( S ) Brice Loustau Complex projective Definition structures CP ( S ) is the deformation space of all complex projective structures The character variety on S : The Schwarzian parametrization CP ( S ) = { all C P 1 -structures on S } / Diff + 0 ( S ) . The minimal surface parametrization A point Z ∈ CP ( S ) is called a marked complex projective surface. CP ( S ) is a complex manifold of dimension dim C CP ( S ) = 6 g − 6. Note : A complex projective atlas is in particular a complex atlas on S (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 .

The hyperkähler geometry of CP ( S ) Fuchsian and quasifuchsian structures Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) Fuchsian and quasifuchsian structures Brice Loustau If any Kleinian group Γ ( i.e. discrete subgroup of PSL 2 ( C ) ) acts Complex projective freely and properly on some open subset U of C P 1 , the quotient structures inherits a complex projective structure. The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) Fuchsian and quasifuchsian structures Brice Loustau If any Kleinian group Γ ( i.e. discrete subgroup of PSL 2 ( C ) ) acts Complex projective freely and properly on some open subset U of C P 1 , the quotient structures inherits a complex projective structure. The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) Fuchsian structures Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) Fuchsian structures Brice Loustau Complex projective structures In particular, any Riemann surface X can be equipped with a compatible C P 1 -structure by the uniformization theorem: The character variety The Schwarzian parametrization X = H 2 The minimal surface parametrization where Γ ⊂ PSL 2 ( R ) is a Fuchsian group.

The hyperkähler geometry of CP ( S ) Fuchsian structures Brice Loustau Complex projective structures In particular, any Riemann surface X can be equipped with a compatible C P 1 -structure by the uniformization theorem: The character variety The Schwarzian parametrization X = H 2 The minimal surface parametrization where Γ ⊂ PSL 2 ( R ) is a Fuchsian group. Note : This defines a Fuchsian section σ F : T ( S ) → CP ( S ) .

The hyperkähler geometry of CP ( S ) Quasifuchsian structures Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) Quasifuchsian structures Brice Loustau Complex projective structures By Bers’ simultaneous uniformization theorem, given two complex structures ( X + , X − ) ∈ T ( S ) × T ( S ) , there exists a unique Kleinian The character variety group Γ such that: The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) Quasifuchsian structures Brice Loustau Complex projective structures By Bers’ simultaneous uniformization theorem, given two complex structures ( X + , X − ) ∈ T ( S ) × T ( S ) , there exists a unique Kleinian The character variety group Γ such that: The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) 1 Complex projective structures Brice Loustau 2 The character variety Complex projective structures The character variety 3 The Schwarzian parametrization The Schwarzian parametrization The minimal surface 4 The minimal surface parametrization parametrization

The hyperkähler geometry of CP ( S ) Holonomy Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) Holonomy Brice Loustau Any complex projective structure Z ∈ CP ( S ) defines a holonomy Complex projective representation ρ : π 1 ( S ) → G = PSL 2 ( C ) . structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) Holonomy Brice Loustau Any complex projective structure Z ∈ CP ( S ) defines a holonomy Complex projective representation ρ : π 1 ( S ) → G = PSL 2 ( C ) . structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) The character variety Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of CP ( S ) The character variety Brice Loustau Holonomy defines a map Complex projective structures hol : CP ( S ) → X ( S , G ) ; The character variety The Schwarzian where X ( S , G ) = Hom ( π 1 ( S ) , G ) // G is the character variety of S . parametrization The minimal surface parametrization