Quasi-split real groups and the Hitchin map International Meeting - PowerPoint PPT Presentation

Higgs bundles Abelianization The HKR section What’s next? Quasi-split real groups and the Hitchin map International Meeting AMS/EMS/SPM Special session Higgs bundles and character varieties Porto, June 2015 Ana Peón-Nieto Mathematisches Institut Ruprecht–Karls Universität Heidelberg A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? Index G -Higgs bundles and the Hitchin map 1 Abelianization 2 The Hitchin–Kostant–Rallis section 3 What’s next? 4 A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. H C � m C isotropy representation A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. H C � m C isotropy representation ( O ( n ) � sym ( n , R ) ). A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. H C � m C isotropy representation ( O ( n ) � sym ( n , R ) ). Definition A G-Higgs bundle on X is a pair ( E , φ ) A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. H C � m C isotropy representation ( O ( n ) � sym ( n , R ) ). Definition A G-Higgs bundle on X is a pair ( E , φ ) with E → X a holomorphic principal H C -bundle A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. H C � m C isotropy representation ( O ( n ) � sym ( n , R ) ). Definition A G-Higgs bundle on X is a pair ( E , φ ) with E → X a holomorphic principal H C -bundle and φ ∈ H 0 ( X , E ( m C ) ⊗ K ) the Higgs field. A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. H C � m C isotropy representation ( O ( n ) � sym ( n , R ) ). Definition A G-Higgs bundle on X is a pair ( E , φ ) with E → X a holomorphic principal H C -bundle and φ ∈ H 0 ( X , E ( m C ) ⊗ K ) the Higgs field. Examples A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. H C � m C isotropy representation ( O ( n ) � sym ( n , R ) ). Definition A G-Higgs bundle on X is a pair ( E , φ ) with E → X a holomorphic principal H C -bundle and φ ∈ H 0 ( X , E ( m C ) ⊗ K ) the Higgs field. Examples 1. GL ( n , C ) R : E rk n vector bundle, φ : E → E ⊗ K . A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. H C � m C isotropy representation ( O ( n ) � sym ( n , R ) ). Definition A G-Higgs bundle on X is a pair ( E , φ ) with E → X a holomorphic principal H C -bundle and φ ∈ H 0 ( X , E ( m C ) ⊗ K ) the Higgs field. Examples 1. GL ( n , C ) R : E rk n vector bundle, φ : E → E ⊗ K . = E ∗ rk n v.b.+symmetric form, φ symmetric. 2. GL ( n , R ) : E ∼ A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? G-Higgs bundles X connected smooth projective curve/ C , g ( X ) ≥ 2. G real reductive Lie group ( GL ( n , R ) , U ( n ) ). H ≤ G maximal compact subgroup ( O ( n ) , U ( n ) ). g = h ⊕ m Cartan decomposition, θ Cartan involution. H C � m C isotropy representation ( O ( n ) � sym ( n , R ) ). Definition A G-Higgs bundle on X is a pair ( E , φ ) with E → X a holomorphic principal H C -bundle and φ ∈ H 0 ( X , E ( m C ) ⊗ K ) the Higgs field. Examples 1. GL ( n , C ) R : E rk n vector bundle, φ : E → E ⊗ K . = E ∗ rk n v.b.+symmetric form, φ symmetric. 2. GL ( n , R ) : E ∼ γ β 3. U ( p , q ) : E = V ⊕ W , φ = ( β, γ ) : V �→ W ⊗ K , W �→ V ⊗ K . A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? The Hitchin map A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? The Hitchin map Definition The Hitchin map is a morphism B G := H 0 ( X , ⊕ i K d i ) h G : M ( G ) → A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? The Hitchin map Definition The Hitchin map is a morphism B G := H 0 ( X , ⊕ i K d i ) h G : M ( G ) → ( E , φ ) �→ ( p 1 ( φ ) , . . . , p r ( φ )) A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? The Hitchin map Definition The Hitchin map is a morphism B G := H 0 ( X , ⊕ i K d i ) h G : M ( G ) → ( E , φ ) �→ ( p 1 ( φ ) , . . . , p r ( φ )) M ( G ) moduli space of ps. Higgs bundles A.Peón-Nieto Quasi-split real groups and the Hitchin map

Higgs bundles Abelianization The HKR section What’s next? The Hitchin map Definition The Hitchin map is a morphism B G := H 0 ( X , ⊕ i K d i ) h G : M ( G ) → ( E , φ ) �→ ( p 1 ( φ ) , . . . , p r ( φ )) M ( G ) moduli space of ps. Higgs bundles( ∼ = Hom ( π 1 , G ) // G) A.Peón-Nieto Quasi-split real groups and the Hitchin map