Introduction to the geometry of moduli spaces of Higgs bundles - PowerPoint PPT Presentation

Introduction to the geometry of moduli spaces of Higgs bundles Jochen Heinloth (Universität Duisburg-Essen) 1 / 18

What are these moduli spaces? Fix: C / k smooth projective curve/compact Riemann surface 2 / 18

What are these moduli spaces? Fix: C / k smooth projective curve/compact Riemann surface G (= GL n ) a reductive group. Bun n space of all GL n -bundles (vector bundles on C ). 2 / 18

What are these moduli spaces? Fix: C / k smooth projective curve/compact Riemann surface G (= GL n ) a reductive group. Bun n space of all GL n -bundles (vector bundles on C ). Higgs-bundles - M Dol T ∗ Bun n = Higgs n = � ( E , θ : E → E ⊗ Ω C ) � ⊇ Higgs d , sst n M Dol := ( Higgs d , sst ) coarse n 2 / 18

What are these moduli spaces? Fix: C / k smooth projective curve/compact Riemann surface G (= GL n ) a reductive group. Bun n space of all GL n -bundles (vector bundles on C ). Higgs-bundles - M Dol T ∗ Bun n = Higgs n = � ( E , θ : E → E ⊗ Ω C ) � ⊇ Higgs d , sst n M Dol := ( Higgs d , sst ) coarse n Connections - M DR Con n := � ( E , ∇ ) |∇ connection on E� → Bun n M DR := Con coarse n 2 / 18

What are these moduli spaces? Fix: C / k smooth projective curve/compact Riemann surface G (= GL n ) a reductive group. Bun n space of all GL n -bundles (vector bundles on C ). Higgs-bundles - M Dol T ∗ Bun n = Higgs n = � ( E , θ : E → E ⊗ Ω C ) � ⊇ Higgs d , sst n M Dol := ( Higgs d , sst ) coarse n Connections - M DR Con n := � ( E , ∇ ) |∇ connection on E� → Bun n M DR := Con coarse n Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation 2 / 18

What are these moduli spaces? Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Connections - M DR M DR := Con coarse n Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation 3 / 18

What are these moduli spaces? Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Connections - M DR M DR := Con coarse n Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation Toy example: GL 1 M Dol = T ∗ Pic ∼ = C g × Pic. M DR = affine bundle over Pic. M Betti ∼ = ( C ∗ ) 2 g 3 / 18

What are these moduli spaces? Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Connections - M DR M DR := Con coarse n Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation Toy example: GL 1 M Dol = T ∗ Pic ∼ = C g × Pic. M DR = affine bundle over Pic. M Betti ∼ = ( C ∗ ) 2 g ≃ ( R × S 1 ) 2 g 3 / 18

What are these moduli spaces? Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Connections - M DR M DR := Con coarse n Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation Toy example: GL 1 M Dol = T ∗ Pic ∼ = C g × Pic. M DR = affine bundle over Pic. = ( C ∗ ) 2 g ≃ ( R × S 1 ) 2 g ≃ R 2 g × ( S 1 ) 2 g ≃ M Dol M Betti ∼ 3 / 18

What are these moduli spaces? Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Connections - M DR M DR := Con coarse n Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation 4 / 18

What are these moduli spaces? Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Connections - M DR M DR := Con coarse n Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation Some variants Can look at bundles with level structures � Repr of π 1 ( C − pts ) with prescribed monodromy at punctures. G non-split, e.g. for k = R . G / C family of groups over C . 4 / 18

Questions (Hitchin) Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation 5 / 18

Questions (Hitchin) Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation What is the global topology (e.g. cohomology) of M ∗ ( C ) ? 5 / 18

Questions (Hitchin) Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation What is the global topology (e.g. cohomology) of M ∗ ( C ) ? How are the extra structures on H ∗ ( M ? ) induced by algebraic structure of M Dol , M Betti related? 5 / 18

Questions (Hitchin) Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation What is the global topology (e.g. cohomology) of M ∗ ( C ) ? How are the extra structures on H ∗ ( M ? ) induced by algebraic structure of M Dol , M Betti related? How are the results for different groups related? 5 / 18

Questions (Hitchin) Higgs-bundles - M Dol M Dol := � ( E , θ : E → E ⊗ Ω C ) � sst , coarse Representations - M Betti M Betti := { ( A i , B i ) ∈ GL 2 g n | � g i = 1 [ A i , B i ] = 1 } / conjugation What is the global topology (e.g. cohomology) of M ∗ ( C ) ? How are the extra structures on H ∗ ( M ? ) induced by algebraic structure of M Dol , M Betti related? How are the results for different groups related? First results: n = 2 Hitchin, n = 3 Gothen: Computed e.g. H ∗ ( M Dol ) . 5 / 18

Plan: M Betti - Method used by Hausel–Rodriguez-Villegas 1 M Dol - Two geometric methods 2 P=W – A conjecture relating the extra structure on H ∗ ’s 3 6 / 18

Point-counting on M Betti Part I 7 / 18

Point-counting on M Betti Part I Weil conjectures allow to deduce H ∗ ( X ( C )) from counting X ( F q ) if X is smooth projective. 7 / 18

Point-counting on M Betti Part I Weil conjectures allow to deduce H ∗ ( X ( C )) from counting X ( F q ) if X is smooth projective. Warning: This does not apply to M Betti ! 7 / 18

Point-counting on M Betti Part I 8 / 18

Point-counting on M Betti Part I Frobenius knew ( G finite group, F : G k → G ): 1 # { g ∈ G k | F ( g ) = 1 } = � � χ ( 1 ) χ ( F ( g )) # G g χ ∈ Irr G 8 / 18

Point-counting on M Betti Part I Frobenius knew ( G finite group, F : G k → G ): 1 # { g ∈ G k | F ( g ) = 1 } = � � χ ( 1 ) χ ( F ( g )) # G g χ ∈ Irr G � 1 g = 1 because � χ ∈ Irr G χ ( 1 ) χ ( g ) = 0 otherwise ( Irr G - Irreducible representations, χ characters, ρ χ corresp. representation) 8 / 18

Point-counting on M Betti Part I Frobenius knew ( G finite group, F : G k → G ): 1 # { g ∈ G k | F ( g ) = 1 } = � � χ ( 1 ) χ ( F ( g )) # G g χ ∈ Irr G � 1 g = 1 because � χ ∈ Irr G χ ( 1 ) χ ( g ) = 0 otherwise For M Betti F = � [ A i , B i ] C this simplifies: ( Irr G - Irreducible representations, χ characters, ρ χ corresp. representation) 8 / 18

Point-counting on M Betti Part I Frobenius knew ( G finite group, F : G k → G ): 1 # { g ∈ G k | F ( g ) = 1 } = � � χ ( 1 ) χ ( F ( g )) # G g χ ∈ Irr G � 1 g = 1 because � χ ∈ Irr G χ ( 1 ) χ ( g ) = 0 otherwise For M Betti F = � [ A i , B i ] C this simplifies: A ∈ G ρ χ ( ABA − 1 ) = � ( Irr G - Irreducible representations, χ characters, ρ χ corresp. representation) 8 / 18

Point-counting on M Betti Part I Frobenius knew ( G finite group, F : G k → G ): 1 # { g ∈ G k | F ( g ) = 1 } = � � χ ( 1 ) χ ( F ( g )) # G g χ ∈ Irr G � 1 g = 1 because � χ ∈ Irr G χ ( 1 ) χ ( g ) = 0 otherwise For M Betti F = � [ A i , B i ] C this simplifies: A ∈ G ρ χ ( ABA − 1 ) =# G χ ( B ) � χ ( 1 ) Id ( Irr G - Irreducible representations, χ characters, ρ χ corresp. representation) 8 / 18

Point-counting on M Betti Part I Frobenius knew ( G finite group, F : G k → G ): 1 # { g ∈ G k | F ( g ) = 1 } = � � χ ( 1 ) χ ( F ( g )) # G g χ ∈ Irr G � 1 g = 1 because � χ ∈ Irr G χ ( 1 ) χ ( g ) = 0 otherwise For M Betti F = � [ A i , B i ] C this simplifies: A ∈ G ρ χ ( ABA − 1 ) =# G χ ( B ) � χ ( 1 ) Id A ρ χ ( ABA − 1 B − 1 ) = � � B ( Irr G - Irreducible representations, χ characters, ρ χ corresp. representation) 8 / 18

Point-counting on M Betti Part I Frobenius knew ( G finite group, F : G k → G ): 1 # { g ∈ G k | F ( g ) = 1 } = � � χ ( 1 ) χ ( F ( g )) # G g χ ∈ Irr G � 1 g = 1 because � χ ∈ Irr G χ ( 1 ) χ ( g ) = 0 otherwise For M Betti F = � [ A i , B i ] C this simplifies: A ∈ G ρ χ ( ABA − 1 ) =# G χ ( B ) � χ ( 1 ) Id χ ( B ) χ ( B − 1 ) A ρ χ ( ABA − 1 B − 1 ) =# G � � � Id = B B χ ( 1 ) ( Irr G - Irreducible representations, χ characters, ρ χ corresp. representation) 8 / 18