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Coupled equations for K ahler metrics and YangMills connections Mario Garc a-Fern andez Joint work with: Luis Alvarez-C onsul and Oscar Garc a-Prada ICMAT (CSIC-UAM-UC3M-UCM) Bath (27 Nov 2009) LAC, MGF & OGP

  1. Moment maps ( X , ω ) = symplectic manifold, G = Lie group with Lie algebra g , G × X → X , left G -action preserving ω . Suppose that ∃ a G -equivariant moment map i.e. ∃ µ : X → g ∗ such that µ ( g · x ) = Ad ( g ) − 1 · µ ( x ) , d � µ, ζ � = ω ( Y ζ , · ) and for all g ∈ G and ζ ∈ g , where Y ζ | x = d dt t =0 exp( t ζ ) · x ∈ T x X . Symplectic quotient (Marsden & Weinstein ’74): If we have a “good” action then µ − 1 (0) / G inherits a natural symplectic structure. ahler quotient (Guillemin & Stenberg ’82): If ( X , J , ω ) is K¨ ahler and K¨ we have a “good” action of G � ( X , ω, J ) then µ − 1 (0) / G inherits a natural K¨ ahler structure. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 4 / 20

  2. Moment maps ( X , ω ) = symplectic manifold, G = Lie group with Lie algebra g , G × X → X , left G -action preserving ω . Suppose that ∃ a G -equivariant moment map i.e. ∃ µ : X → g ∗ such that µ ( g · x ) = Ad ( g ) − 1 · µ ( x ) , d � µ, ζ � = ω ( Y ζ , · ) and for all g ∈ G and ζ ∈ g , where Y ζ | x = d dt t =0 exp( t ζ ) · x ∈ T x X . Symplectic quotient (Marsden & Weinstein ’74): If we have a “good” action then µ − 1 (0) / G inherits a natural symplectic structure. ahler quotient (Guillemin & Stenberg ’82): If ( X , J , ω ) is K¨ ahler and K¨ we have a “good” action of G � ( X , ω, J ) then µ − 1 (0) / G inherits a natural K¨ ahler structure. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 4 / 20

  3. Moment maps ( X , ω ) = symplectic manifold, G = Lie group with Lie algebra g , G × X → X , left G -action preserving ω . Suppose that ∃ a G -equivariant moment map i.e. ∃ µ : X → g ∗ such that µ ( g · x ) = Ad ( g ) − 1 · µ ( x ) , d � µ, ζ � = ω ( Y ζ , · ) and for all g ∈ G and ζ ∈ g , where Y ζ | x = d dt t =0 exp( t ζ ) · x ∈ T x X . Symplectic quotient (Marsden & Weinstein ’74): If we have a “good” action then µ − 1 (0) / G inherits a natural symplectic structure. ahler quotient (Guillemin & Stenberg ’82): If ( X , J , ω ) is K¨ ahler and K¨ we have a “good” action of G � ( X , ω, J ) then µ − 1 (0) / G inherits a natural K¨ ahler structure. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 4 / 20

  4. Moment maps ( X , ω ) = symplectic manifold, G = Lie group with Lie algebra g , G × X → X , left G -action preserving ω . Suppose that ∃ a G -equivariant moment map i.e. ∃ µ : X → g ∗ such that µ ( g · x ) = Ad ( g ) − 1 · µ ( x ) , d � µ, ζ � = ω ( Y ζ , · ) and for all g ∈ G and ζ ∈ g , where Y ζ | x = d dt t =0 exp( t ζ ) · x ∈ T x X . Symplectic quotient (Marsden & Weinstein ’74): If we have a “good” action then µ − 1 (0) / G inherits a natural symplectic structure. ahler quotient (Guillemin & Stenberg ’82): If ( X , J , ω ) is K¨ ahler and K¨ we have a “good” action of G � ( X , ω, J ) then µ − 1 (0) / G inherits a natural K¨ ahler structure. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 4 / 20

  5. Moment maps ( X , ω ) = symplectic manifold, G = Lie group with Lie algebra g , G × X → X , left G -action preserving ω . Suppose that ∃ a G -equivariant moment map i.e. ∃ µ : X → g ∗ such that µ ( g · x ) = Ad ( g ) − 1 · µ ( x ) , d � µ, ζ � = ω ( Y ζ , · ) and for all g ∈ G and ζ ∈ g , where Y ζ | x = d dt t =0 exp( t ζ ) · x ∈ T x X . Symplectic quotient (Marsden & Weinstein ’74): If we have a “good” action then µ − 1 (0) / G inherits a natural symplectic structure. ahler quotient (Guillemin & Stenberg ’82): If ( X , J , ω ) is K¨ ahler and K¨ we have a “good” action of G � ( X , ω, J ) then µ − 1 (0) / G inherits a natural K¨ ahler structure. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 4 / 20

  6. Example 1: The Hermite–Yang–Mills equations ( X , ω, J , g ) smooth compact K¨ ahler manifold: ω symplectic structure, J complex structure and g metric. E G -bundle over X , A connection on E , F A curvature of A A = { connections A on E } G = { automorphisms g : E → E covering the identity on X } � A . The infinite-dimensional manifold A has a K¨ ahler structure ( ω A , I A , g A ) preserved by G . � ( a 0 ∧ a 1 ) ∧ ω n − 1 , I A a 0 = − a 0 ( J · ) with a j ∈ Ω 1 ( ad E ) . ω A ( a 0 , a 1 ) = X Moment map (Atiyah–Bott (’83) & Donaldson): µ A : A → (Lie G ) ∗ � ( ζ ∧ F A ) ∧ ω n − 1 � µ A ( A ) , ζ � = ζ ∈ ad E ≡ Lie G . X G � A 1 , 1 = { A ∈ A : F 0 , 2 = 0 } ≡ holomorphic struct. on E c = E × G G c A HYM equations: F 0 , 2 Λ ω F A = z , = 0 , z ∈ z ( centre of g ) . A LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 5 / 20

  7. Example 1: The Hermite–Yang–Mills equations ( X , ω, J , g ) smooth compact K¨ ahler manifold: ω symplectic structure, J complex structure and g metric. E G -bundle over X , A connection on E , F A curvature of A A = { connections A on E } G = { automorphisms g : E → E covering the identity on X } � A . The infinite-dimensional manifold A has a K¨ ahler structure ( ω A , I A , g A ) preserved by G . � ( a 0 ∧ a 1 ) ∧ ω n − 1 , I A a 0 = − a 0 ( J · ) with a j ∈ Ω 1 ( ad E ) . ω A ( a 0 , a 1 ) = X Moment map (Atiyah–Bott (’83) & Donaldson): µ A : A → (Lie G ) ∗ � ( ζ ∧ F A ) ∧ ω n − 1 � µ A ( A ) , ζ � = ζ ∈ ad E ≡ Lie G . X G � A 1 , 1 = { A ∈ A : F 0 , 2 = 0 } ≡ holomorphic struct. on E c = E × G G c A HYM equations: F 0 , 2 Λ ω F A = z , = 0 , z ∈ z ( centre of g ) . A LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 5 / 20

  8. Example 1: The Hermite–Yang–Mills equations ( X , ω, J , g ) smooth compact K¨ ahler manifold: ω symplectic structure, J complex structure and g metric. E G -bundle over X , A connection on E , F A curvature of A A = { connections A on E } G = { automorphisms g : E → E covering the identity on X } � A . The infinite-dimensional manifold A has a K¨ ahler structure ( ω A , I A , g A ) preserved by G . � ( a 0 ∧ a 1 ) ∧ ω n − 1 , I A a 0 = − a 0 ( J · ) with a j ∈ Ω 1 ( ad E ) . ω A ( a 0 , a 1 ) = X Moment map (Atiyah–Bott (’83) & Donaldson): µ A : A → (Lie G ) ∗ � ( ζ ∧ F A ) ∧ ω n − 1 � µ A ( A ) , ζ � = ζ ∈ ad E ≡ Lie G . X G � A 1 , 1 = { A ∈ A : F 0 , 2 = 0 } ≡ holomorphic struct. on E c = E × G G c A HYM equations: F 0 , 2 Λ ω F A = z , = 0 , z ∈ z ( centre of g ) . A LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 5 / 20

  9. Example 1: The Hermite–Yang–Mills equations ( X , ω, J , g ) smooth compact K¨ ahler manifold: ω symplectic structure, J complex structure and g metric. E G -bundle over X , A connection on E , F A curvature of A A = { connections A on E } G = { automorphisms g : E → E covering the identity on X } � A . The infinite-dimensional manifold A has a K¨ ahler structure ( ω A , I A , g A ) preserved by G . � ( a 0 ∧ a 1 ) ∧ ω n − 1 , I A a 0 = − a 0 ( J · ) with a j ∈ Ω 1 ( ad E ) . ω A ( a 0 , a 1 ) = X Moment map (Atiyah–Bott (’83) & Donaldson): µ A : A → (Lie G ) ∗ � ( ζ ∧ F A ) ∧ ω n − 1 � µ A ( A ) , ζ � = ζ ∈ ad E ≡ Lie G . X G � A 1 , 1 = { A ∈ A : F 0 , 2 = 0 } ≡ holomorphic struct. on E c = E × G G c A HYM equations: F 0 , 2 Λ ω F A = z , = 0 , z ∈ z ( centre of g ) . A LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 5 / 20

  10. Example 1: The Hermite–Yang–Mills equations ( X , ω, J , g ) smooth compact K¨ ahler manifold: ω symplectic structure, J complex structure and g metric. E G -bundle over X , A connection on E , F A curvature of A A = { connections A on E } G = { automorphisms g : E → E covering the identity on X } � A . The infinite-dimensional manifold A has a K¨ ahler structure ( ω A , I A , g A ) preserved by G . � ( a 0 ∧ a 1 ) ∧ ω n − 1 , I A a 0 = − a 0 ( J · ) with a j ∈ Ω 1 ( ad E ) . ω A ( a 0 , a 1 ) = X Moment map (Atiyah–Bott (’83) & Donaldson): µ A : A → (Lie G ) ∗ � ( ζ ∧ F A ) ∧ ω n − 1 � µ A ( A ) , ζ � = ζ ∈ ad E ≡ Lie G . X G � A 1 , 1 = { A ∈ A : F 0 , 2 = 0 } ≡ holomorphic struct. on E c = E × G G c A HYM equations: F 0 , 2 Λ ω F A = z , = 0 , z ∈ z ( centre of g ) . A LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 5 / 20

  11. Example 1: The Hermite–Yang–Mills equations ( X , ω, J , g ) smooth compact K¨ ahler manifold: ω symplectic structure, J complex structure and g metric. E G -bundle over X , A connection on E , F A curvature of A A = { connections A on E } G = { automorphisms g : E → E covering the identity on X } � A . The infinite-dimensional manifold A has a K¨ ahler structure ( ω A , I A , g A ) preserved by G . � ( a 0 ∧ a 1 ) ∧ ω n − 1 , I A a 0 = − a 0 ( J · ) with a j ∈ Ω 1 ( ad E ) . ω A ( a 0 , a 1 ) = X Moment map (Atiyah–Bott (’83) & Donaldson): µ A : A → (Lie G ) ∗ � ( ζ ∧ F A ) ∧ ω n − 1 � µ A ( A ) , ζ � = ζ ∈ ad E ≡ Lie G . X G � A 1 , 1 = { A ∈ A : F 0 , 2 = 0 } ≡ holomorphic struct. on E c = E × G G c A HYM equations: F 0 , 2 Λ ω F A = z , = 0 , z ∈ z ( centre of g ) . A LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 5 / 20

  12. Example 1: The Hermite–Yang–Mills equations ( X , ω, J , g ) smooth compact K¨ ahler manifold: ω symplectic structure, J complex structure and g metric. E G -bundle over X , A connection on E , F A curvature of A A = { connections A on E } G = { automorphisms g : E → E covering the identity on X } � A . The infinite-dimensional manifold A has a K¨ ahler structure ( ω A , I A , g A ) preserved by G . � ( a 0 ∧ a 1 ) ∧ ω n − 1 , I A a 0 = − a 0 ( J · ) with a j ∈ Ω 1 ( ad E ) . ω A ( a 0 , a 1 ) = X Moment map (Atiyah–Bott (’83) & Donaldson): µ A : A → (Lie G ) ∗ � ( ζ ∧ F A ) ∧ ω n − 1 � µ A ( A ) , ζ � = ζ ∈ ad E ≡ Lie G . X G � A 1 , 1 = { A ∈ A : F 0 , 2 = 0 } ≡ holomorphic struct. on E c = E × G G c A HYM equations: F 0 , 2 Λ ω F A = z , = 0 , z ∈ z ( centre of g ) . A LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 5 / 20

  13. Example 1: The Hermite–Yang–Mills equations ( X , ω, J , g ) smooth compact K¨ ahler manifold: ω symplectic structure, J complex structure and g metric. E G -bundle over X , A connection on E , F A curvature of A A = { connections A on E } G = { automorphisms g : E → E covering the identity on X } � A . The infinite-dimensional manifold A has a K¨ ahler structure ( ω A , I A , g A ) preserved by G . � ( a 0 ∧ a 1 ) ∧ ω n − 1 , I A a 0 = − a 0 ( J · ) with a j ∈ Ω 1 ( ad E ) . ω A ( a 0 , a 1 ) = X Moment map (Atiyah–Bott (’83) & Donaldson): µ A : A → (Lie G ) ∗ � ( ζ ∧ F A ) ∧ ω n − 1 � µ A ( A ) , ζ � = ζ ∈ ad E ≡ Lie G . X G � A 1 , 1 = { A ∈ A : F 0 , 2 = 0 } ≡ holomorphic struct. on E c = E × G G c A HYM equations: F 0 , 2 Λ ω F A = z , = 0 , z ∈ z ( centre of g ) . A LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 5 / 20

  14. Example 2: The constant scalar curvature equation ( X , ω ) smooth compact symplectic manifold of K¨ ahler type. J = { complex structures on X compatible with ω } H = { Hamiltonian symplectomorphisms of ( X , ω ) } � J The infinite-dimensional (singular) manifold J has a K¨ ahler structure ( ω J , I J , g J ) preserved by H . Given b j ∈ T J J ⊂ Ω 0 ( End TX ), � tr( J · b 0 · b 1 ) ω n ω J | J ( b 0 , b 1 ) = n ! , I J b 0 = Jb 0 . X Moment map (Fujiki(1992)–Donaldson(1997)): µ J : J → (Lie H ) ∗ � S ) ω n φ ( S J − ˆ � µ J ( J ) , φ � = − n ! X � ω n 1 φ ∈ C ∞ ( X ) / R ∼ ˆ = Lie H S = S J Vol ( X ) n ! X S J = ˆ J ∈ J . CscK equation: S , LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 6 / 20

  15. Example 2: The constant scalar curvature equation ( X , ω ) smooth compact symplectic manifold of K¨ ahler type. J = { complex structures on X compatible with ω } H = { Hamiltonian symplectomorphisms of ( X , ω ) } � J The infinite-dimensional (singular) manifold J has a K¨ ahler structure ( ω J , I J , g J ) preserved by H . Given b j ∈ T J J ⊂ Ω 0 ( End TX ), � tr( J · b 0 · b 1 ) ω n ω J | J ( b 0 , b 1 ) = n ! , I J b 0 = Jb 0 . X Moment map (Fujiki(1992)–Donaldson(1997)): µ J : J → (Lie H ) ∗ � S ) ω n φ ( S J − ˆ � µ J ( J ) , φ � = − n ! X � ω n 1 φ ∈ C ∞ ( X ) / R ∼ ˆ = Lie H S = S J Vol ( X ) n ! X S J = ˆ J ∈ J . CscK equation: S , LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 6 / 20

  16. Example 2: The constant scalar curvature equation ( X , ω ) smooth compact symplectic manifold of K¨ ahler type. J = { complex structures on X compatible with ω } H = { Hamiltonian symplectomorphisms of ( X , ω ) } � J The infinite-dimensional (singular) manifold J has a K¨ ahler structure ( ω J , I J , g J ) preserved by H . Given b j ∈ T J J ⊂ Ω 0 ( End TX ), � tr( J · b 0 · b 1 ) ω n ω J | J ( b 0 , b 1 ) = n ! , I J b 0 = Jb 0 . X Moment map (Fujiki(1992)–Donaldson(1997)): µ J : J → (Lie H ) ∗ � S ) ω n φ ( S J − ˆ � µ J ( J ) , φ � = − n ! X � ω n 1 φ ∈ C ∞ ( X ) / R ∼ ˆ = Lie H S = S J Vol ( X ) n ! X S J = ˆ J ∈ J . CscK equation: S , LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 6 / 20

  17. Example 2: The constant scalar curvature equation ( X , ω ) smooth compact symplectic manifold of K¨ ahler type. J = { complex structures on X compatible with ω } H = { Hamiltonian symplectomorphisms of ( X , ω ) } � J The infinite-dimensional (singular) manifold J has a K¨ ahler structure ( ω J , I J , g J ) preserved by H . Given b j ∈ T J J ⊂ Ω 0 ( End TX ), � tr( J · b 0 · b 1 ) ω n ω J | J ( b 0 , b 1 ) = n ! , I J b 0 = Jb 0 . X Moment map (Fujiki(1992)–Donaldson(1997)): µ J : J → (Lie H ) ∗ � S ) ω n φ ( S J − ˆ � µ J ( J ) , φ � = − n ! X � ω n 1 φ ∈ C ∞ ( X ) / R ∼ ˆ = Lie H S = S J Vol ( X ) n ! X S J = ˆ J ∈ J . CscK equation: S , LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 6 / 20

  18. Example 2: The constant scalar curvature equation ( X , ω ) smooth compact symplectic manifold of K¨ ahler type. J = { complex structures on X compatible with ω } H = { Hamiltonian symplectomorphisms of ( X , ω ) } � J The infinite-dimensional (singular) manifold J has a K¨ ahler structure ( ω J , I J , g J ) preserved by H . Given b j ∈ T J J ⊂ Ω 0 ( End TX ), � tr( J · b 0 · b 1 ) ω n ω J | J ( b 0 , b 1 ) = n ! , I J b 0 = Jb 0 . X Moment map (Fujiki(1992)–Donaldson(1997)): µ J : J → (Lie H ) ∗ � S ) ω n φ ( S J − ˆ � µ J ( J ) , φ � = − n ! X � ω n 1 φ ∈ C ∞ ( X ) / R ∼ ˆ = Lie H S = S J Vol ( X ) n ! X S J = ˆ J ∈ J . CscK equation: S , LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 6 / 20

  19. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  20. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  21. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  22. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  23. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  24. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  25. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  26. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  27. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  28. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  29. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  30. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  31. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  32. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  33. Coupled equations for K¨ ahler metrics and connections ( X , ω ), G , E , J and A as before. Phase space: J × A . Group of symmetries: 1 → G → � G → H → 1, with � G � J × A . 4 α 1 Symplectic structure: ω α = α 0 ω J + ( n − 1)! ω A , 0 � = α 0 , α 1 ∈ R . Remarks: • J × A has an integrable complex structure that fibers over ( J , I J ) , ahler if α 1 given by I ( J , A ) ( b , a ) = ( Jb , − a ( J · )) and ω α is K¨ α 0 > 0!!! • Why � G ? Geometry: It preserves I , ω α and the complex submanifold P = { ( J , A ) ∈ J × A : A ∈ A 1 , 1 J }≡ K¨ ahler structure on X with fixed ω + holomorphic structure on E c over X . Physics: Natural group of symmetries for ( J , A ) (grav. field + gauge G ⊂ Diff ( E ) G “biggest” subgroup preserving ω α field) ⇒ Diff ( E ) G . � and I . • Why ω α ? For simplicity (following cscK & HYM). Problem 1: We find a solution if � G � J × A is Hamiltonian. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 7 / 20

  34. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  35. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  36. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  37. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  38. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  39. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  40. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  41. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  42. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  43. Lie group extensions and Hamiltonian actions Question: Is � G � ( J × A , ω α ) Hamiltonian? Recall: 1 → G → � G → H → 1 and the � G -action is symplectic. It is enough to prove that � G � A is Hamiltonian. General fact for extensions: If G � A is Hamiltonian and W � = ∅ , W := � G -equivariant smooth maps θ : A → W where W ⊂ Hom (Lie � G , Lie G ) affine space of vector space splittings of 0 → Lie G → Lie � G → Lie H → 0 . then, � G � A is Hamiltonian ⇔ ∃ a � G -equivariant map σ θ : A → (Lie H ) ∗ ω A ( Y θ ⊥ φ , · ) = � µ G , ( d θ ) φ � + d � σ θ , φ � , for all φ ∈ Lie H , where θ ⊥ = Id − θ : Lie H → Lie � G and Y θ ⊥ φ is the inf. action on A . G ∼ Example: If A = {·} , W � = ∅ ⇒ Lie � = Lie G ⋊ Lie H . but ... In our case: the vertical projection θ A : TE → VE defined by any connection A ∈ A defines an element θ : A → W in W . Finally, � ω ( F A ∧ F A ) − c ′ ) · ω n X φ (Λ 2 � σ θ ( A ) , φ � = − n ! . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 8 / 20

  44. Coupled equations for K¨ ahler metrics and connections This proves that ... Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] For any α 0 and α 1 there exists a � G -equivariant moment map µ α : J × A → Lie � G ∗ G , covering φ ∈ C ∞ ( X ) / R ∼ for the � G -action. If ζ ∈ Lie � = Lie H then, � � � · ω n φ ( α 0 S J + α 1 Λ 2 � µ α ( J , A ) , ζ � = − ω ( F A ∧ F A ) − c ) − 4 α 1 ( θ A ζ, Λ ω F A ) n ! X The � G -action preserves the complex submanifold P = { ( J , A ) ∈ J × A : G ∗ and the conditions A ∈ A 1 , 1 J } . ⇒ µ α : P → Lie � µ α ( J , A ) = 0 , ( J , A ) ∈ P defines ( completely! ) coupled equations for ( ω, J , g , A ) that can be written as follows (after a suitable shift by z ∈ z , the center of g ):  Definition: Λ ω F A = z ,  F 0 , 2 J (1) = 0 , A  α 0 S g + α 1 Λ 2 ω ( F A ∧ F A ) = c . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 9 / 20

  45. Coupled equations for K¨ ahler metrics and connections This proves that ... Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] For any α 0 and α 1 there exists a � G -equivariant moment map µ α : J × A → Lie � G ∗ G , covering φ ∈ C ∞ ( X ) / R ∼ for the � G -action. If ζ ∈ Lie � = Lie H then, � � � · ω n φ ( α 0 S J + α 1 Λ 2 � µ α ( J , A ) , ζ � = − ω ( F A ∧ F A ) − c ) − 4 α 1 ( θ A ζ, Λ ω F A ) n ! X The � G -action preserves the complex submanifold P = { ( J , A ) ∈ J × A : G ∗ and the conditions A ∈ A 1 , 1 J } . ⇒ µ α : P → Lie � µ α ( J , A ) = 0 , ( J , A ) ∈ P defines ( completely! ) coupled equations for ( ω, J , g , A ) that can be written as follows (after a suitable shift by z ∈ z , the center of g ):  Definition: Λ ω F A = z ,  F 0 , 2 J (1) = 0 , A  α 0 S g + α 1 Λ 2 ω ( F A ∧ F A ) = c . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 9 / 20

  46. Coupled equations for K¨ ahler metrics and connections This proves that ... Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] For any α 0 and α 1 there exists a � G -equivariant moment map µ α : J × A → Lie � G ∗ G , covering φ ∈ C ∞ ( X ) / R ∼ for the � G -action. If ζ ∈ Lie � = Lie H then, � � � · ω n φ ( α 0 S J + α 1 Λ 2 � µ α ( J , A ) , ζ � = − ω ( F A ∧ F A ) − c ) − 4 α 1 ( θ A ζ, Λ ω F A ) n ! X The � G -action preserves the complex submanifold P = { ( J , A ) ∈ J × A : G ∗ and the conditions A ∈ A 1 , 1 J } . ⇒ µ α : P → Lie � µ α ( J , A ) = 0 , ( J , A ) ∈ P defines ( completely! ) coupled equations for ( ω, J , g , A ) that can be written as follows (after a suitable shift by z ∈ z , the center of g ):  Definition: Λ ω F A = z ,  F 0 , 2 J (1) = 0 , A  α 0 S g + α 1 Λ 2 ω ( F A ∧ F A ) = c . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 9 / 20

  47. Coupled equations for K¨ ahler metrics and connections This proves that ... Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] For any α 0 and α 1 there exists a � G -equivariant moment map µ α : J × A → Lie � G ∗ G , covering φ ∈ C ∞ ( X ) / R ∼ for the � G -action. If ζ ∈ Lie � = Lie H then, � � � · ω n φ ( α 0 S J + α 1 Λ 2 � µ α ( J , A ) , ζ � = − ω ( F A ∧ F A ) − c ) − 4 α 1 ( θ A ζ, Λ ω F A ) n ! X The � G -action preserves the complex submanifold P = { ( J , A ) ∈ J × A : G ∗ and the conditions A ∈ A 1 , 1 J } . ⇒ µ α : P → Lie � µ α ( J , A ) = 0 , ( J , A ) ∈ P defines ( completely! ) coupled equations for ( ω, J , g , A ) that can be written as follows (after a suitable shift by z ∈ z , the center of g ):  Definition: Λ ω F A = z ,  F 0 , 2 J (1) = 0 , A  α 0 S g + α 1 Λ 2 ω ( F A ∧ F A ) = c . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 9 / 20

  48. Coupled equations for K¨ ahler metrics and connections This proves that ... Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] For any α 0 and α 1 there exists a � G -equivariant moment map µ α : J × A → Lie � G ∗ G , covering φ ∈ C ∞ ( X ) / R ∼ for the � G -action. If ζ ∈ Lie � = Lie H then, � � � · ω n φ ( α 0 S J + α 1 Λ 2 � µ α ( J , A ) , ζ � = − ω ( F A ∧ F A ) − c ) − 4 α 1 ( θ A ζ, Λ ω F A ) n ! X The � G -action preserves the complex submanifold P = { ( J , A ) ∈ J × A : G ∗ and the conditions A ∈ A 1 , 1 J } . ⇒ µ α : P → Lie � µ α ( J , A ) = 0 , ( J , A ) ∈ P defines ( completely! ) coupled equations for ( ω, J , g , A ) that can be written as follows (after a suitable shift by z ∈ z , the center of g ):  Definition: Λ ω F A = z ,  F 0 , 2 J (1) = 0 , A  α 0 S g + α 1 Λ 2 ω ( F A ∧ F A ) = c . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 9 / 20

  49. Coupled equations for K¨ ahler metrics and connections This proves that ... Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] For any α 0 and α 1 there exists a � G -equivariant moment map µ α : J × A → Lie � G ∗ G , covering φ ∈ C ∞ ( X ) / R ∼ for the � G -action. If ζ ∈ Lie � = Lie H then, � � � · ω n φ ( α 0 S J + α 1 Λ 2 � µ α ( J , A ) , ζ � = − ω ( F A ∧ F A ) − c ) − 4 α 1 ( θ A ζ, Λ ω F A ) n ! X The � G -action preserves the complex submanifold P = { ( J , A ) ∈ J × A : G ∗ and the conditions A ∈ A 1 , 1 J } . ⇒ µ α : P → Lie � µ α ( J , A ) = 0 , ( J , A ) ∈ P defines ( completely! ) coupled equations for ( ω, J , g , A ) that can be written as follows (after a suitable shift by z ∈ z , the center of g ):  Definition: Λ ω F A = z ,  F 0 , 2 J (1) = 0 , A  α 0 S g + α 1 Λ 2 ω ( F A ∧ F A ) = c . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 9 / 20

  50. Coupled equations for K¨ ahler metrics and connections This proves that ... Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] For any α 0 and α 1 there exists a � G -equivariant moment map µ α : J × A → Lie � G ∗ G , covering φ ∈ C ∞ ( X ) / R ∼ for the � G -action. If ζ ∈ Lie � = Lie H then, � � � · ω n φ ( α 0 S J + α 1 Λ 2 � µ α ( J , A ) , ζ � = − ω ( F A ∧ F A ) − c ) − 4 α 1 ( θ A ζ, Λ ω F A ) n ! X The � G -action preserves the complex submanifold P = { ( J , A ) ∈ J × A : G ∗ and the conditions A ∈ A 1 , 1 J } . ⇒ µ α : P → Lie � µ α ( J , A ) = 0 , ( J , A ) ∈ P defines ( completely! ) coupled equations for ( ω, J , g , A ) that can be written as follows (after a suitable shift by z ∈ z , the center of g ):  Definition: Λ ω F A = z ,  F 0 , 2 J (1) = 0 , A  α 0 S g + α 1 Λ 2 ω ( F A ∧ F A ) = c . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 9 / 20

  51. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  52. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  53. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  54. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  55. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  56. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  57. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  58. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  59. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  60. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  61. Why HYM and cscK? HYM: 1. Construction of moduli spaces with K¨ ahler structure ⇒ ⇒ Donaldson’s invariants for smooth 4-manifolds (1990). 2. Special solutions of the Yang–Mills equation : critical points of the Yang-Mills functional A → � F A � 2 (physicists interested).The Hitchin–Kobayashi correspondence (Donaldson and Uhlenbeck–Yau) relating the existence of solutions to the HYM equation with the Mumford stability of bundles ⇒ algebraic criterion for finding YM connections. CscK: 1. Calabi’s problem (1954, 1982) : Find preferred metrics in K¨ ahler geometry. Three natural notions (that can be seen as uniformizers of the complex structure): K¨ ahler–Einstein metrics ⇒ cscK metrics ⇒ extremal metrics ≡ � X S 2 ≡ critical points of the Calabi Functional g → g vol g , for K¨ ahler ahler class. CscK metrics ≡ absolute minimizers. metrics g in a fixed K¨ 2. Moduli problem for projective varieties :Yau-Tian-Donaldson’s conjecture relating existence of cscK metrics on a compact complex manifold with the stability of the manifold ⇒ numerical approximation of K¨ ahler–Einstein metrics and Weyl–Petterson metrics on moduli spaces. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 10 / 20

  62. Variational interpretation of the coupled equations Given real constants α 0 and α 1 ∈ R consider the following functional. � ( α 0 S g − 2 α 1 | F A | 2 ) 2 · vol g + 2 α 1 · � F A � 2 , CYM ( g , A ) = (2) X where g is a Riemannian metric on X , A is a connection on E and vol g is the volume form of g . Note that J ∋ J → g = ω ( · , J · ), fixing ω . Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] The solutions to the coupled equations (1) on J × A are the absolute minimizers of CYM : J × A → R (after suitable re-scaling of the coupling constants). g = π ∗ g + t · g V ( θ A · , θ A · ) on Tot ( E ), with Given a pair ( g , A ), consider ˆ t = 2 α 1 g ) → ( X , g ) is a Riemannian submersion with α 0 > 0.Then ( Tot ( E ) , ˆ totally geodesic fibers and so g = S g − 2 α 1 | F A | 2 S ˆ α 0 Therefore CYM = C + YM and if ( X , J , ω, g , A ), with F 0 , 2 = 0, is a A solution to the coupled equations (1) then S ˆ g = const . Moreover, if A is irreducible ˆ g Einstein ⇒ (1) ⇒ S ˆ g = const . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 11 / 20

  63. Variational interpretation of the coupled equations Given real constants α 0 and α 1 ∈ R consider the following functional. � ( α 0 S g − 2 α 1 | F A | 2 ) 2 · vol g + 2 α 1 · � F A � 2 , CYM ( g , A ) = (2) X where g is a Riemannian metric on X , A is a connection on E and vol g is the volume form of g . Note that J ∋ J → g = ω ( · , J · ), fixing ω . Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] The solutions to the coupled equations (1) on J × A are the absolute minimizers of CYM : J × A → R (after suitable re-scaling of the coupling constants). g = π ∗ g + t · g V ( θ A · , θ A · ) on Tot ( E ), with Given a pair ( g , A ), consider ˆ t = 2 α 1 g ) → ( X , g ) is a Riemannian submersion with α 0 > 0.Then ( Tot ( E ) , ˆ totally geodesic fibers and so g = S g − 2 α 1 | F A | 2 S ˆ α 0 Therefore CYM = C + YM and if ( X , J , ω, g , A ), with F 0 , 2 = 0, is a A solution to the coupled equations (1) then S ˆ g = const . Moreover, if A is irreducible ˆ g Einstein ⇒ (1) ⇒ S ˆ g = const . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 11 / 20

  64. Variational interpretation of the coupled equations Given real constants α 0 and α 1 ∈ R consider the following functional. � ( α 0 S g − 2 α 1 | F A | 2 ) 2 · vol g + 2 α 1 · � F A � 2 , CYM ( g , A ) = (2) X where g is a Riemannian metric on X , A is a connection on E and vol g is the volume form of g . Note that J ∋ J → g = ω ( · , J · ), fixing ω . Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] The solutions to the coupled equations (1) on J × A are the absolute minimizers of CYM : J × A → R (after suitable re-scaling of the coupling constants). g = π ∗ g + t · g V ( θ A · , θ A · ) on Tot ( E ), with Given a pair ( g , A ), consider ˆ t = 2 α 1 g ) → ( X , g ) is a Riemannian submersion with α 0 > 0.Then ( Tot ( E ) , ˆ totally geodesic fibers and so g = S g − 2 α 1 | F A | 2 S ˆ α 0 Therefore CYM = C + YM and if ( X , J , ω, g , A ), with F 0 , 2 = 0, is a A solution to the coupled equations (1) then S ˆ g = const . Moreover, if A is irreducible ˆ g Einstein ⇒ (1) ⇒ S ˆ g = const . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 11 / 20

  65. Variational interpretation of the coupled equations Given real constants α 0 and α 1 ∈ R consider the following functional. � ( α 0 S g − 2 α 1 | F A | 2 ) 2 · vol g + 2 α 1 · � F A � 2 , CYM ( g , A ) = (2) X where g is a Riemannian metric on X , A is a connection on E and vol g is the volume form of g . Note that J ∋ J → g = ω ( · , J · ), fixing ω . Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] The solutions to the coupled equations (1) on J × A are the absolute minimizers of CYM : J × A → R (after suitable re-scaling of the coupling constants). g = π ∗ g + t · g V ( θ A · , θ A · ) on Tot ( E ), with Given a pair ( g , A ), consider ˆ t = 2 α 1 g ) → ( X , g ) is a Riemannian submersion with α 0 > 0.Then ( Tot ( E ) , ˆ totally geodesic fibers and so g = S g − 2 α 1 | F A | 2 S ˆ α 0 Therefore CYM = C + YM and if ( X , J , ω, g , A ), with F 0 , 2 = 0, is a A solution to the coupled equations (1) then S ˆ g = const . Moreover, if A is irreducible ˆ g Einstein ⇒ (1) ⇒ S ˆ g = const . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 11 / 20

  66. Variational interpretation of the coupled equations Given real constants α 0 and α 1 ∈ R consider the following functional. � ( α 0 S g − 2 α 1 | F A | 2 ) 2 · vol g + 2 α 1 · � F A � 2 , CYM ( g , A ) = (2) X where g is a Riemannian metric on X , A is a connection on E and vol g is the volume form of g . Note that J ∋ J → g = ω ( · , J · ), fixing ω . Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] The solutions to the coupled equations (1) on J × A are the absolute minimizers of CYM : J × A → R (after suitable re-scaling of the coupling constants). g = π ∗ g + t · g V ( θ A · , θ A · ) on Tot ( E ), with Given a pair ( g , A ), consider ˆ t = 2 α 1 g ) → ( X , g ) is a Riemannian submersion with α 0 > 0.Then ( Tot ( E ) , ˆ totally geodesic fibers and so g = S g − 2 α 1 | F A | 2 S ˆ α 0 Therefore CYM = C + YM and if ( X , J , ω, g , A ), with F 0 , 2 = 0, is a A solution to the coupled equations (1) then S ˆ g = const . Moreover, if A is irreducible ˆ g Einstein ⇒ (1) ⇒ S ˆ g = const . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 11 / 20

  67. Variational interpretation of the coupled equations Given real constants α 0 and α 1 ∈ R consider the following functional. � ( α 0 S g − 2 α 1 | F A | 2 ) 2 · vol g + 2 α 1 · � F A � 2 , CYM ( g , A ) = (2) X where g is a Riemannian metric on X , A is a connection on E and vol g is the volume form of g . Note that J ∋ J → g = ω ( · , J · ), fixing ω . Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] The solutions to the coupled equations (1) on J × A are the absolute minimizers of CYM : J × A → R (after suitable re-scaling of the coupling constants). g = π ∗ g + t · g V ( θ A · , θ A · ) on Tot ( E ), with Given a pair ( g , A ), consider ˆ t = 2 α 1 g ) → ( X , g ) is a Riemannian submersion with α 0 > 0.Then ( Tot ( E ) , ˆ totally geodesic fibers and so g = S g − 2 α 1 | F A | 2 S ˆ α 0 Therefore CYM = C + YM and if ( X , J , ω, g , A ), with F 0 , 2 = 0, is a A solution to the coupled equations (1) then S ˆ g = const . Moreover, if A is irreducible ˆ g Einstein ⇒ (1) ⇒ S ˆ g = const . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 11 / 20

  68. Variational interpretation of the coupled equations Given real constants α 0 and α 1 ∈ R consider the following functional. � ( α 0 S g − 2 α 1 | F A | 2 ) 2 · vol g + 2 α 1 · � F A � 2 , CYM ( g , A ) = (2) X where g is a Riemannian metric on X , A is a connection on E and vol g is the volume form of g . Note that J ∋ J → g = ω ( · , J · ), fixing ω . Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] The solutions to the coupled equations (1) on J × A are the absolute minimizers of CYM : J × A → R (after suitable re-scaling of the coupling constants). g = π ∗ g + t · g V ( θ A · , θ A · ) on Tot ( E ), with Given a pair ( g , A ), consider ˆ t = 2 α 1 g ) → ( X , g ) is a Riemannian submersion with α 0 > 0.Then ( Tot ( E ) , ˆ totally geodesic fibers and so g = S g − 2 α 1 | F A | 2 S ˆ α 0 Therefore CYM = C + YM and if ( X , J , ω, g , A ), with F 0 , 2 = 0, is a A solution to the coupled equations (1) then S ˆ g = const . Moreover, if A is irreducible ˆ g Einstein ⇒ (1) ⇒ S ˆ g = const . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 11 / 20

  69. Variational interpretation of the coupled equations Given real constants α 0 and α 1 ∈ R consider the following functional. � ( α 0 S g − 2 α 1 | F A | 2 ) 2 · vol g + 2 α 1 · � F A � 2 , CYM ( g , A ) = (2) X where g is a Riemannian metric on X , A is a connection on E and vol g is the volume form of g . Note that J ∋ J → g = ω ( · , J · ), fixing ω . Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] The solutions to the coupled equations (1) on J × A are the absolute minimizers of CYM : J × A → R (after suitable re-scaling of the coupling constants). g = π ∗ g + t · g V ( θ A · , θ A · ) on Tot ( E ), with Given a pair ( g , A ), consider ˆ t = 2 α 1 g ) → ( X , g ) is a Riemannian submersion with α 0 > 0.Then ( Tot ( E ) , ˆ totally geodesic fibers and so g = S g − 2 α 1 | F A | 2 S ˆ α 0 Therefore CYM = C + YM and if ( X , J , ω, g , A ), with F 0 , 2 = 0, is a A solution to the coupled equations (1) then S ˆ g = const . Moreover, if A is irreducible ˆ g Einstein ⇒ (1) ⇒ S ˆ g = const . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 11 / 20

  70. Variational interpretation of the coupled equations Given real constants α 0 and α 1 ∈ R consider the following functional. � ( α 0 S g − 2 α 1 | F A | 2 ) 2 · vol g + 2 α 1 · � F A � 2 , CYM ( g , A ) = (2) X where g is a Riemannian metric on X , A is a connection on E and vol g is the volume form of g . Note that J ∋ J → g = ω ( · , J · ), fixing ω . Proposition [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] The solutions to the coupled equations (1) on J × A are the absolute minimizers of CYM : J × A → R (after suitable re-scaling of the coupling constants). g = π ∗ g + t · g V ( θ A · , θ A · ) on Tot ( E ), with Given a pair ( g , A ), consider ˆ t = 2 α 1 g ) → ( X , g ) is a Riemannian submersion with α 0 > 0.Then ( Tot ( E ) , ˆ totally geodesic fibers and so g = S g − 2 α 1 | F A | 2 S ˆ α 0 Therefore CYM = C + YM and if ( X , J , ω, g , A ), with F 0 , 2 = 0, is a A solution to the coupled equations (1) then S ˆ g = const . Moreover, if A is irreducible ˆ g Einstein ⇒ (1) ⇒ S ˆ g = const . LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 11 / 20

  71. First examples of solutions We fix a compact complex manifold ( X , J ) and a G -bundle over X . Consider the equations for ( ω, A ), with ω ∈ [ ω ] and A ∈ A 1 , 1 . Trivial examples: The system of equations (1) decouples when dim C X = 1 since ( F A ∧ F A ) = 0. Solutions = stable holomorphic bundles over ( X , J ). If E = L , or if E es projectively flat, with c 1 ( E ) = λ [ ω ] then the coupled equations admit decoupled solutions: cscK + HYM. Remark: In both cases ∃ a solution to F A = λω , which implies Lie � G = Lie G ⋉ Lie H . Less trivial examples: The coupled equations (1) have solutions on Homogenous holomorphic bundles E c over homogeneous K¨ ahler manifolds if the bundle comes from an irreducible representation ( ≡ ∃ HYM connection). Proof: invariant structures and representation theory. Solutions are given by simultaneous solutions for the cases α 1 = 0 , α 0 � = 0 and α 0 = 0 , α 1 � = 0. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 12 / 20

  72. First examples of solutions We fix a compact complex manifold ( X , J ) and a G -bundle over X . Consider the equations for ( ω, A ), with ω ∈ [ ω ] and A ∈ A 1 , 1 . Trivial examples: The system of equations (1) decouples when dim C X = 1 since ( F A ∧ F A ) = 0. Solutions = stable holomorphic bundles over ( X , J ). If E = L , or if E es projectively flat, with c 1 ( E ) = λ [ ω ] then the coupled equations admit decoupled solutions: cscK + HYM. Remark: In both cases ∃ a solution to F A = λω , which implies Lie � G = Lie G ⋉ Lie H . Less trivial examples: The coupled equations (1) have solutions on Homogenous holomorphic bundles E c over homogeneous K¨ ahler manifolds if the bundle comes from an irreducible representation ( ≡ ∃ HYM connection). Proof: invariant structures and representation theory. Solutions are given by simultaneous solutions for the cases α 1 = 0 , α 0 � = 0 and α 0 = 0 , α 1 � = 0. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 12 / 20

  73. First examples of solutions We fix a compact complex manifold ( X , J ) and a G -bundle over X . Consider the equations for ( ω, A ), with ω ∈ [ ω ] and A ∈ A 1 , 1 . Trivial examples: The system of equations (1) decouples when dim C X = 1 since ( F A ∧ F A ) = 0. Solutions = stable holomorphic bundles over ( X , J ). If E = L , or if E es projectively flat, with c 1 ( E ) = λ [ ω ] then the coupled equations admit decoupled solutions: cscK + HYM. Remark: In both cases ∃ a solution to F A = λω , which implies Lie � G = Lie G ⋉ Lie H . Less trivial examples: The coupled equations (1) have solutions on Homogenous holomorphic bundles E c over homogeneous K¨ ahler manifolds if the bundle comes from an irreducible representation ( ≡ ∃ HYM connection). Proof: invariant structures and representation theory. Solutions are given by simultaneous solutions for the cases α 1 = 0 , α 0 � = 0 and α 0 = 0 , α 1 � = 0. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 12 / 20

  74. First examples of solutions We fix a compact complex manifold ( X , J ) and a G -bundle over X . Consider the equations for ( ω, A ), with ω ∈ [ ω ] and A ∈ A 1 , 1 . Trivial examples: The system of equations (1) decouples when dim C X = 1 since ( F A ∧ F A ) = 0. Solutions = stable holomorphic bundles over ( X , J ). If E = L , or if E es projectively flat, with c 1 ( E ) = λ [ ω ] then the coupled equations admit decoupled solutions: cscK + HYM. Remark: In both cases ∃ a solution to F A = λω , which implies Lie � G = Lie G ⋉ Lie H . Less trivial examples: The coupled equations (1) have solutions on Homogenous holomorphic bundles E c over homogeneous K¨ ahler manifolds if the bundle comes from an irreducible representation ( ≡ ∃ HYM connection). Proof: invariant structures and representation theory. Solutions are given by simultaneous solutions for the cases α 1 = 0 , α 0 � = 0 and α 0 = 0 , α 1 � = 0. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 12 / 20

  75. First examples of solutions We fix a compact complex manifold ( X , J ) and a G -bundle over X . Consider the equations for ( ω, A ), with ω ∈ [ ω ] and A ∈ A 1 , 1 . Trivial examples: The system of equations (1) decouples when dim C X = 1 since ( F A ∧ F A ) = 0. Solutions = stable holomorphic bundles over ( X , J ). If E = L , or if E es projectively flat, with c 1 ( E ) = λ [ ω ] then the coupled equations admit decoupled solutions: cscK + HYM. Remark: In both cases ∃ a solution to F A = λω , which implies Lie � G = Lie G ⋉ Lie H . Less trivial examples: The coupled equations (1) have solutions on Homogenous holomorphic bundles E c over homogeneous K¨ ahler manifolds if the bundle comes from an irreducible representation ( ≡ ∃ HYM connection). Proof: invariant structures and representation theory. Solutions are given by simultaneous solutions for the cases α 1 = 0 , α 0 � = 0 and α 0 = 0 , α 1 � = 0. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 12 / 20

  76. First examples of solutions We fix a compact complex manifold ( X , J ) and a G -bundle over X . Consider the equations for ( ω, A ), with ω ∈ [ ω ] and A ∈ A 1 , 1 . Trivial examples: The system of equations (1) decouples when dim C X = 1 since ( F A ∧ F A ) = 0. Solutions = stable holomorphic bundles over ( X , J ). If E = L , or if E es projectively flat, with c 1 ( E ) = λ [ ω ] then the coupled equations admit decoupled solutions: cscK + HYM. Remark: In both cases ∃ a solution to F A = λω , which implies Lie � G = Lie G ⋉ Lie H . Less trivial examples: The coupled equations (1) have solutions on Homogenous holomorphic bundles E c over homogeneous K¨ ahler manifolds if the bundle comes from an irreducible representation ( ≡ ∃ HYM connection). Proof: invariant structures and representation theory. Solutions are given by simultaneous solutions for the cases α 1 = 0 , α 0 � = 0 and α 0 = 0 , α 1 � = 0. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 12 / 20

  77. An existence criterion In the previous examples the K¨ ahler metric on ( X , J ) is always cscK. Are there any examples of solutions ( ω, A ) with ω non cscK? Theorem [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] Let ( X , L ) be a compact polarised manifold, G c be a complex reductive Lie group and E c be a holomorphic G c -bundle over X . If there exists a cscK metric ω ∈ c 1 ( L ), X has finite automorphism group and E c is stable with respect to L then, given a pair of positive real constants α 0 , α 1 > 0 with small ratio 0 < α 1 α 0 << 1, there exists a solution ( ω α , A α ) to (1) with these coupling constants and ω α ∈ c 1 ( L ). Proof: Deformation argument using the Implicit Function Theorem in Banach spaces (either fixing ω and moving J or viceversa). Idea (fixing ω ): G c that extends the � suppose � G has a complexification � G -action on P . G ∗ : ζ → µ α ( e i ζ ). Then, Consider the map L : Lie � G → Lie � � dL 0 ( ζ 0 , ζ 1 � = ω α ( Y ζ 1 , I Y ζ 0 ) , where Y ζ j is the infinitesimal action of ζ j on P . If � G I ⊂ Aut ( E c ) is finite G c does not exist ... dL 0 is an isomorphism. But � LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 13 / 20

  78. An existence criterion In the previous examples the K¨ ahler metric on ( X , J ) is always cscK. Are there any examples of solutions ( ω, A ) with ω non cscK? Theorem [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] Let ( X , L ) be a compact polarised manifold, G c be a complex reductive Lie group and E c be a holomorphic G c -bundle over X . If there exists a cscK metric ω ∈ c 1 ( L ), X has finite automorphism group and E c is stable with respect to L then, given a pair of positive real constants α 0 , α 1 > 0 with small ratio 0 < α 1 α 0 << 1, there exists a solution ( ω α , A α ) to (1) with these coupling constants and ω α ∈ c 1 ( L ). Proof: Deformation argument using the Implicit Function Theorem in Banach spaces (either fixing ω and moving J or viceversa). Idea (fixing ω ): G c that extends the � suppose � G has a complexification � G -action on P . G ∗ : ζ → µ α ( e i ζ ). Then, Consider the map L : Lie � G → Lie � � dL 0 ( ζ 0 , ζ 1 � = ω α ( Y ζ 1 , I Y ζ 0 ) , where Y ζ j is the infinitesimal action of ζ j on P . If � G I ⊂ Aut ( E c ) is finite G c does not exist ... dL 0 is an isomorphism. But � LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 13 / 20

  79. An existence criterion In the previous examples the K¨ ahler metric on ( X , J ) is always cscK. Are there any examples of solutions ( ω, A ) with ω non cscK? Theorem [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] Let ( X , L ) be a compact polarised manifold, G c be a complex reductive Lie group and E c be a holomorphic G c -bundle over X . If there exists a cscK metric ω ∈ c 1 ( L ), X has finite automorphism group and E c is stable with respect to L then, given a pair of positive real constants α 0 , α 1 > 0 with small ratio 0 < α 1 α 0 << 1, there exists a solution ( ω α , A α ) to (1) with these coupling constants and ω α ∈ c 1 ( L ). Proof: Deformation argument using the Implicit Function Theorem in Banach spaces (either fixing ω and moving J or viceversa). Idea (fixing ω ): G c that extends the � suppose � G has a complexification � G -action on P . G ∗ : ζ → µ α ( e i ζ ). Then, Consider the map L : Lie � G → Lie � � dL 0 ( ζ 0 , ζ 1 � = ω α ( Y ζ 1 , I Y ζ 0 ) , where Y ζ j is the infinitesimal action of ζ j on P . If � G I ⊂ Aut ( E c ) is finite G c does not exist ... dL 0 is an isomorphism. But � LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 13 / 20

  80. An existence criterion In the previous examples the K¨ ahler metric on ( X , J ) is always cscK. Are there any examples of solutions ( ω, A ) with ω non cscK? Theorem [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] Let ( X , L ) be a compact polarised manifold, G c be a complex reductive Lie group and E c be a holomorphic G c -bundle over X . If there exists a cscK metric ω ∈ c 1 ( L ), X has finite automorphism group and E c is stable with respect to L then, given a pair of positive real constants α 0 , α 1 > 0 with small ratio 0 < α 1 α 0 << 1, there exists a solution ( ω α , A α ) to (1) with these coupling constants and ω α ∈ c 1 ( L ). Proof: Deformation argument using the Implicit Function Theorem in Banach spaces (either fixing ω and moving J or viceversa). Idea (fixing ω ): G c that extends the � suppose � G has a complexification � G -action on P . G ∗ : ζ → µ α ( e i ζ ). Then, Consider the map L : Lie � G → Lie � � dL 0 ( ζ 0 , ζ 1 � = ω α ( Y ζ 1 , I Y ζ 0 ) , where Y ζ j is the infinitesimal action of ζ j on P . If � G I ⊂ Aut ( E c ) is finite G c does not exist ... dL 0 is an isomorphism. But � LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 13 / 20

  81. An existence criterion In the previous examples the K¨ ahler metric on ( X , J ) is always cscK. Are there any examples of solutions ( ω, A ) with ω non cscK? Theorem [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] Let ( X , L ) be a compact polarised manifold, G c be a complex reductive Lie group and E c be a holomorphic G c -bundle over X . If there exists a cscK metric ω ∈ c 1 ( L ), X has finite automorphism group and E c is stable with respect to L then, given a pair of positive real constants α 0 , α 1 > 0 with small ratio 0 < α 1 α 0 << 1, there exists a solution ( ω α , A α ) to (1) with these coupling constants and ω α ∈ c 1 ( L ). Proof: Deformation argument using the Implicit Function Theorem in Banach spaces (either fixing ω and moving J or viceversa). Idea (fixing ω ): G c that extends the � suppose � G has a complexification � G -action on P . G ∗ : ζ → µ α ( e i ζ ). Then, Consider the map L : Lie � G → Lie � � dL 0 ( ζ 0 , ζ 1 � = ω α ( Y ζ 1 , I Y ζ 0 ) , where Y ζ j is the infinitesimal action of ζ j on P . If � G I ⊂ Aut ( E c ) is finite G c does not exist ... dL 0 is an isomorphism. But � LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 13 / 20

  82. An existence criterion In the previous examples the K¨ ahler metric on ( X , J ) is always cscK. Are there any examples of solutions ( ω, A ) with ω non cscK? Theorem [—, L. ´ Alvarez C´ onsul, O. Garc´ ıa Prada] Let ( X , L ) be a compact polarised manifold, G c be a complex reductive Lie group and E c be a holomorphic G c -bundle over X . If there exists a cscK metric ω ∈ c 1 ( L ), X has finite automorphism group and E c is stable with respect to L then, given a pair of positive real constants α 0 , α 1 > 0 with small ratio 0 < α 1 α 0 << 1, there exists a solution ( ω α , A α ) to (1) with these coupling constants and ω α ∈ c 1 ( L ). Proof: Deformation argument using the Implicit Function Theorem in Banach spaces (either fixing ω and moving J or viceversa). Idea (fixing ω ): G c that extends the � suppose � G has a complexification � G -action on P . G ∗ : ζ → µ α ( e i ζ ). Then, Consider the map L : Lie � G → Lie � � dL 0 ( ζ 0 , ζ 1 � = ω α ( Y ζ 1 , I Y ζ 0 ) , where Y ζ j is the infinitesimal action of ζ j on P . If � G I ⊂ Aut ( E c ) is finite G c does not exist ... dL 0 is an isomorphism. But � LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 13 / 20

  83. Examples Example: Let X be a high degree hypersurface of P 3 . Then, ∃ KE metric ω ∈ c 1 ( X ) (in particular cscK) (Aubin & Yau). Moreover, c 1 ( X ) < 0 ⇒ Aut ( X ) finite. Let E be a smooth SU (2)-bundle over X with second Chern number � 1 k = X tr F A ∧ F A ∈ Z , where A is a connection on E . If k ≫ 0, the 8 π 2 moduli space M k of Anti-Self-Dual (ASD) connections A on E with respect to ω is non-empty, non-compact but admits a compactification.Let A be a connection that determines a point in M k . Then, A is irreducible and so we can apply our Theorem obtaining solutions ( ω α , A α ) to (1) for small 0 < α = α 1 α 0 . How can we assure that ω α is not cscK? Recall that the scalar equation in (1) is equivalent to S ω α − α | F A α | 2 = const . Since ( ω α , A α ) → ( ω, A ) uniformly as α → 0 it is enough to take A such that | F A | 2 is not a constant function on X .Take A near to the boundary of the moduli space (bubbling). Can we make this argument explicit?Locally yes. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 14 / 20

  84. Examples Example: Let X be a high degree hypersurface of P 3 . Then, ∃ KE metric ω ∈ c 1 ( X ) (in particular cscK) (Aubin & Yau). Moreover, c 1 ( X ) < 0 ⇒ Aut ( X ) finite. Let E be a smooth SU (2)-bundle over X with second Chern number � 1 k = X tr F A ∧ F A ∈ Z , where A is a connection on E . If k ≫ 0, the 8 π 2 moduli space M k of Anti-Self-Dual (ASD) connections A on E with respect to ω is non-empty, non-compact but admits a compactification.Let A be a connection that determines a point in M k . Then, A is irreducible and so we can apply our Theorem obtaining solutions ( ω α , A α ) to (1) for small 0 < α = α 1 α 0 . How can we assure that ω α is not cscK? Recall that the scalar equation in (1) is equivalent to S ω α − α | F A α | 2 = const . Since ( ω α , A α ) → ( ω, A ) uniformly as α → 0 it is enough to take A such that | F A | 2 is not a constant function on X .Take A near to the boundary of the moduli space (bubbling). Can we make this argument explicit?Locally yes. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 14 / 20

  85. Examples Example: Let X be a high degree hypersurface of P 3 . Then, ∃ KE metric ω ∈ c 1 ( X ) (in particular cscK) (Aubin & Yau). Moreover, c 1 ( X ) < 0 ⇒ Aut ( X ) finite. Let E be a smooth SU (2)-bundle over X with second Chern number � 1 k = X tr F A ∧ F A ∈ Z , where A is a connection on E . If k ≫ 0, the 8 π 2 moduli space M k of Anti-Self-Dual (ASD) connections A on E with respect to ω is non-empty, non-compact but admits a compactification.Let A be a connection that determines a point in M k . Then, A is irreducible and so we can apply our Theorem obtaining solutions ( ω α , A α ) to (1) for small 0 < α = α 1 α 0 . How can we assure that ω α is not cscK? Recall that the scalar equation in (1) is equivalent to S ω α − α | F A α | 2 = const . Since ( ω α , A α ) → ( ω, A ) uniformly as α → 0 it is enough to take A such that | F A | 2 is not a constant function on X .Take A near to the boundary of the moduli space (bubbling). Can we make this argument explicit?Locally yes. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 14 / 20

  86. Examples Example: Let X be a high degree hypersurface of P 3 . Then, ∃ KE metric ω ∈ c 1 ( X ) (in particular cscK) (Aubin & Yau). Moreover, c 1 ( X ) < 0 ⇒ Aut ( X ) finite. Let E be a smooth SU (2)-bundle over X with second Chern number � 1 k = X tr F A ∧ F A ∈ Z , where A is a connection on E . If k ≫ 0, the 8 π 2 moduli space M k of Anti-Self-Dual (ASD) connections A on E with respect to ω is non-empty, non-compact but admits a compactification.Let A be a connection that determines a point in M k . Then, A is irreducible and so we can apply our Theorem obtaining solutions ( ω α , A α ) to (1) for small 0 < α = α 1 α 0 . How can we assure that ω α is not cscK? Recall that the scalar equation in (1) is equivalent to S ω α − α | F A α | 2 = const . Since ( ω α , A α ) → ( ω, A ) uniformly as α → 0 it is enough to take A such that | F A | 2 is not a constant function on X .Take A near to the boundary of the moduli space (bubbling). Can we make this argument explicit?Locally yes. LAC, MGF & OGP (ICMAT) K¨ ahler & Yang–Mills Bath (27 Nov 2009) 14 / 20

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