KempfNess type theorems and Nahms equations Maxence Mayrand - PowerPoint PPT Presentation

Kempf–Ness type theorems and Nahm’s equations Maxence Mayrand University of Toronto December 7, 2019 Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 1 / 15

Setup of Kempf–Ness type theorems Let ( M , ω, I , L , � · � ) be a Hodge manifold , i.e. ( M , ω, I ) K¨ ahler manifold (not necessarily compact); i 2 π F , F curvature of unitary holomorphic line bundle L → M ω = with hermitian metric � · � (prequantization). Example (standard) M ⊆ CP n , ω = ω FS | M , L = O (1) | M . M ⊆ C n , ω = ω flat | M , L = M × C . Example (non-standard) ⇒ M ⊆ CP n projective. Kodaira: compact + Hodge = But ω � = ω FS | M in general. M ⊆ C n with K¨ ahler potential f : M → R , i.e. ω = 2 i ∂ ¯ ∂ f . → C N in general. ∄ isometry M ֒ Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 2 / 15

Setup of Kempf–Ness type theorems Input ( M , ω, I , L , � · � ) Hodge manifold, L → M complex algebraic; G compact Lie group; G C � L such that G preserves � · � . Then, G � M preserving ( ω, I ) and there is a canonical moment map 1 µ ( p )( x ) = d � 2 π log � e itx · ˆ µ : M − → g ∗ , p � , � dt � t =0 p ∈ L ∗ \ { 0 } , ˆ for x ∈ g , p ∈ M , ˆ p �→ p . Output Two types of quotients: 1 Symplectic quotient: µ − 1 (0) / G (stratified symplectic space) 2 GIT quotient: M / / (complex algebraic variety) L G C Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 3 / 15

Kempf–Ness type theorems We have µ − 1 (0) ⊆ M L -ss , so there is a map µ − 1 (0) / G − → M / / L G C . (1) A Kempf–Ness type theorem is a condition which implies (1) is an isomorphism, i.e. a homeomorphism respecting the natural stratifications; the symplectic structures on the strata of the LHS and the complex structures on the strata of the RHS give K¨ ahler structures. ⇒ (1) is ∼ Example. M compact = =. [Kirwan 1984] for the case M ⊆ CP n with ω = ω FS | M . [Sjamaar 1994] for the general case ( M ⊆ CP n but ω � = ω FS | M ). If M is non-compact, we have to be more careful. We will discuss the case of affine varieties with ω = 2 i ∂ ¯ ∂ f in detail. Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 4 / 15

Complex analytic version of the Kempf–Ness theorem First step: Complex analytic version. M µ -ss := { p ∈ M : G C · p ∩ µ − 1 (0) � = ∅} ⊆ M . G C -invariant analytically semistable points open Theorem (Guillemin–Sternberg 1982, Kirwan 1984, Sjamaar 1994, Heinzner–Loose 1994) There is a categorial quotient in the category of complex analytic spaces for G C � M µ -ss , denoted M µ -ss / / G C . Moreover, µ − 1 (0) M µ -ss ∼ = µ − 1 (0) / G M µ -ss / / G C . Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 5 / 15

Complex analytic version of the Kempf–Ness theorem Recall: GIT quotient L G C = M L -ss / M / / / G C . categorical quot. algebraic varieties Luna 1976: Underlying complex analytic space L G C = M L -ss / M / / / G C . categorical quot. complex spaces By previous theorem, = M µ -ss / µ − 1 (0) / G ∼ / G C categorical quot. complex spaces so, by uniqueness of categorical quotients, Kempf–Ness holds if M µ -ss = M L -ss analytic semistability = algebraic semistability Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 6 / 15

The general Kempf–Ness theorem Theorem (Kempf–Ness 1979, Mumford, Guillemin–Sternberg 1982, Ness 1984, Kirwan 1984, Sjamaar 1994, Heinzner–Loose 1994, ...) ( M , ω, I , L , � · � ) Hodge manifold G C � L, G preserves � · � Then, G � ( M , ω, I ) with canonical moment map µ : M → g ∗ . We have µ − 1 (0) ⊆ M L -ss so there is a map µ − 1 (0) / G − → M / / L G C . (2) Suppose: (i) Algebraic Condition: ( M , L ) satisfies the geometric criterion : M L -ss = { p ∈ M : ∃ ˆ p ∈ L ∗ \ { 0 } , ˆ p ⊆ L ∗ \ { 0 }} p �→ p , G C · ˆ e.g. M is projective, affine, or projective-over-affine. (ii) Analytic Condition: � · � 2 : L ∗ → R is proper on closed G C -orbits disjoint from the zero-section. Then, M µ -ss = M L -ss so (2) is an isomorphism. ⇒ (i) & (ii). So µ − 1 (0) / G ∼ Example. M compact = = M / / L G C . Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 7 / 15

The case of affine varieties (I) Kempf–Ness 1979 M ⊆ C n complex affine G C � M via G C → GL ( n , C ) ω = ω flat | M L = M × C , G C � L , g · ( p , z ) = ( g · p , z ). = ⇒ µ = µ std µ std ( p )( x ) = − 1 µ std : M − → g ∗ , 2 Im � xp , p � , ( p ∈ M , x ∈ g ) . Kempf–Ness holds, so std (0) / G ∼ µ − 1 = Spec C [ M ] G C Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 8 / 15

The case of affine varieties (II) King 1994 M ⊆ C n complex affine G C � M via G C → GL ( n , C ) ω = ω flat | M L χ = M × C , χ : G C → C ∗ g · ( p , z ) = ( g · p , χ ( g ) z ) , = ⇒ µ = µ std − ξ ξ := i 2 π d χ ∈ g ∗ Kempf–Ness holds, so � ∞ � µ − 1 ( ξ ) / G ∼ C [ M ] G C ,χ n � = M / / L χ G C = Proj n =0 Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 9 / 15

The case of affine varieties (III) Azad–Loeb 1993 M ⊆ C n complex affine G C � M via G C → GL ( n , C ) ( f = � · � 2 recovers (I)). ω = 2 i ∂ ¯ f : M → R , ∂ f , G -invariant L = M × C , g · ( p , z ) = ( g · p , z ) = ⇒ µ = µ f , where µ f ( p )( x ) = df ( Ix # µ f : M − → g ∗ , p ) , ( p ∈ M , x ∈ g ) . Kempf-Ness holds if f is proper and bounded below . In that case, (0) / G ∼ µ − 1 = Spec C [ M ] G C . f Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 10 / 15

The case of affine varieties (IV) M ⊆ C n complex affine G C � M via G C → GL ( n , C ) ω = 2 i ∂ ¯ f : M → R , ∂ f , G -invariant L χ = M × C , χ : G C → C ∗ g · ( p , z ) = ( g · p , χ ( g ) z ) , = ⇒ µ = µ f − ξ Kempf–Ness can fail even if f is proper and bounded below: Example C ∗ � C ∗ with f ( z ) = 1 + (log | z | 2 ) 2 and χ ( z ) = z 3 . Then, � µ − 1 ( ξ ) / G = ∅ , L χ G C = { pt } . M / / f Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 11 / 15

The case of affine varieties Theorem If C [ M ] ⊆ o ( e f ) i.e. u ( p ) ∀ polynomial u : M → C , lim e f ( p ) = 0 , p →∞ then the Kempf–Ness theorem holds, so � ∞ � ( ξ ) / G ∼ C [ M ] G C ,χ n µ − 1 � = Proj , f n =0 where µ f ( p )( x ) = df ( Ix # i p ) and ξ = 2 π d χ ∈ g ∗ . For example, C [ x ] ⊆ o ( x log x ) = o ( e (log x ) 2 ). The example with Nahm’s equations will look like this, i.e. f ( x ) ∼ (log | x | ) 2 . Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 12 / 15

Example from Nahm’s equations Nahm’s equations: 1D reduction of the self-dual Yang–Mills equations. A = ( A 0 , A 1 , A 2 , A 3 ) : I ⊆ R − → g ⊗ H ˙ A 1 + [ A 0 , A 1 ] + [ A 2 , A 3 ] = 0 ˙ A 2 + [ A 0 , A 2 ] + [ A 3 , A 1 ] = 0 ˙ A 3 + [ A 0 , A 3 ] + [ A 1 , A 2 ] = 0 . Natural action by gauge transformations: G := { g : I → G } � { solutions to Nahm’s eqs. } I = [0 , 1], G 0 = { g ∈ G : g (0) = g (1) = 1 } . M := { solutions to Nahm’s eqs } / G 0 Theorem (Kronheimer 1988) M is a hyperk¨ ahler manifold; ( M , g , I , J , K ) . M ∼ = T ∗ G C , biholomorphism with respect to I. Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 13 / 15

Example from Nahm’s equations Theorem (Dancer–Swann 1996) G × G � T ∗ G C preserves hyperk¨ ahler structure. There is a hyperk¨ ahler moment map � A 1 (0) � A 2 (0) A 3 (0) → ( g ∗ × g ∗ ) 3 , µ : T ∗ G C − µ ( A ) = . − A 1 (1) − A 2 (1) − A 3 (1) For all closed subgroup H ⊆ G × G and χ 1 , χ 2 , χ 3 : H → S 1 , ξ H := µ − 1 T ∗ G C / / / h ( ξ ) / H 2 π ( d χ 1 , d χ 2 , d χ 3 ) ∈ ( h ∗ ) 3 and i is a stratified hyperk¨ ahler space, where ξ = i ∗ µ ( g ∗ × g ∗ ) 3 h ( h ∗ ) 3 . µ h : M Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 14 / 15

Example from Nahm’s equations T ∗ G C is a complex affine variety → C N in general ∄ isometric T ∗ G C ֒ ω 1 = 2 i ∂ ¯ ∂ f and µ 1 = µ f , where � 1 f ( A ) = 1 2 � A 1 � 2 + � A 2 � 2 + � A 3 � 2 � f : T ∗ G C − → R , � 4 0 µ C := µ 2 + i µ 3 : T ∗ G C → g ∗ C × g ∗ C is complex algebraic Theorem We have C [ T ∗ G C ] ⊆ o ( e f ) . Hence, for all H ⊆ G × G and χ 1 , χ 2 , χ 3 : H → S 1 , � ∞ � ξ H ∼ C ( ξ 2 + i ξ 3 )] H C ,χ n � C [ µ − 1 T ∗ G C / / / = Proj . 1 n =0 Maxence Mayrand (UofT) Kempf-Ness and Nahm December 7, 2019 15 / 15