Richard A. SCGAS 2010
Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado)
Outline Introduction 1 Introduction 2 3
Outline Introduction 1 Introduction 2 3
Cohomology of Kähler and hyperkähler quotients The goal is to compute the equivariant cohomology of symplectic (Kähler or hyperkähler) reductions. By the Kempf-Ness, Guillemin-Sternberg theorem, examples arise in geometric invariant theory. Kirwan, Atiyah-Bott: In the symplectic case there is a perfect" Morse stratification. HyperKähler case still unknown.
Infinite dimensional examples: Higgs bundles (Hitchin) Stable pairs (Bradlow) Quiver varieties (Nakajima) These involve symplectic reduction in the presence of singularities. Key points: this poses no (additional) analytic difficulties. Singularities can cause the Morse stratification to lose perfection." Computations of cohomology are (sometimes) still possible.
Application to representation varieties M = a closed Riemann surface g 2 π = π 1 (M, ) G = a compact connected Lie group G C = its complexification (e.g. G = U(n), G C = GL(n, C)) Representation varieties: Hom(π, G)/G moduli of G-bundles Hom(π, G C ) // G C moduli of G-Higgs bundles
Application to representation varieties Theorem (Daskalopoulos-Weitsman--Wilkin 09) The Poincaré polynomial is given by P SL(2,C) t (Hom(π, SL(2, C)) = (1 + t3 ) 2g (1 + t) 2g t 2g+2 (1 t 2 )(1 t 4 ) t 4g 4 + t2g+2 (1 + t) 2g (1 t 2 )(1 t 4 ) + (1 t)2g t 4g 4 4(1 + t 2 ) + (1 + t)2g t 4g 4 ( 2g 2(1 t 2 ) t + 1 + 1 t 2 1 1 ) + (3 2g) ( 2 ) + 1 2 (22g 1)t 4g 4 (1 + t) 2g 2 + (1 t) 2g 2 2
Outline Introduction 1 Introduction 2 3
Symplectic reduction (X, ω) symplectic manifold (compact) G compact, connected Lie group, acting symplectically µ : X g a moment map (dµ ξ ( ) = ω(ξ, )) the Marsden-Weinstein quotient µ 1 (0)/G is a symplectic variety What is the cohomology of µ 1 (0)/G?
Example Introduction Take S 2 with the action of S 1 by rotation in the xy-plane. A moment map is given by the height: µ(x, y, z) = z + const. If µ = z, then µ 1 (0) is the equator with a free action of S 1. If µ = z + 1, then µ 1 (0) is a point with a trivial S 1 action. In both cases, µ 1 (0)/S 1 is a point.
Equivariant cohomology Z topological space with an action by G Classifying space: EG BG is a contractible, principal G-bundle Equivariant cohomology: H G (Z ) = H (Z G EG) If the action is free: H G (Z ) = H (Z /G) If the action is trivial: H G (Z ) = H (Z ) H (BG)
Perfect equivariant Morse theory S 1 acts freely on the sphere S, so BS 1 = CP Therefore P t (BS 1 ) = 1 + t 2 + t 4 + = 1 1 t 2 Example of the sphere: Pt S1 (S 2 ) = P t (H (S 2 ) H (BS 1 )) = 1 + t2 1 t 2 Sum over critical points of f = µ 2 : P S1 t (S 2 ) = p j crit. pt. t λ j P S1 t (p j )
Example Introduction f = µ 2, µ = z: two critical points of index 2, plus the minimum: 1 + t2 1 t 2 + t2 1 t 2 = 1 + t2 1 t 2 f = µ 2, µ = z + 1: two critical points, one of index 2 and one of index zero: 1 1 t 2 + t2 1 t 2 = 1 + t2 1 t 2
Kirwan, Atiyah-Bott For the general case, study the gradient flow f = µ 2. Critical sets η β are characterized in terms of isotropy in G. Gradient flow smooth stratification X = β I S β ; with normal bundles ν β. The corresponding long exact sequence splits H G (S β, α<β S α ) H G (S β) H G ( α<βs α ) Compute change at each step from H G (S β, α<β S α )
Two key steps Morse-Bott Lemma: H G (S β, α<β S α ) H G (ν β, ν β \ {0}) H λ β G (η β ) Atiyah-Bott Lemma: criterion for multiplication by the equivariant Euler class to be injective. H p G (S β, α<β S α ) H p G (S β) = H p G (ν β, ν β \ {0}) H p G (η β)
Perfect equivariant Morse theory Theorem (Kirwan, Atiyah-Bott) P G t (µ 1 (0)) = P G t (X) β t λ β P G t (η β ) Theorem (Kirwan surjectivity) The map on cohomology HG (X) H G (µ 1 (0)) induced from inclusion µ 1 (0) X is surjective.
Vector bundles on Riemann surfaces M a Riemann surface. A = {unitary connections A on hermitian bundle E M} G = gauge group of unitary endomorphisms of E. µ : A Lie(G) is given by A F A µ 2 = Yang-Mills functional
Minimum is the space of projectively flat connections (i.e. representation variety) 1 FA = const. I The flow converges and the Morse stratification is smooth (Daskalopoulos) Higher critical sets correspond to split Yang-Mills connections, i.e. representations to smaller groups. For example, E = L 1 L 2, d = deg L 1 > deg L 2 : η d = Jac(M) Jac(M) Morse-Bott lemma: Negative directions given by H 0,1 (L 1 L 2); λ d = dim is constant.
Theorem (Atiyah-Bott, Daskalopoulos) P SU(2) t (Hom(π, SU(2)) = P(BG) d=0 t λ d Pt S1 (Jac d (M)) = (1 + t3 ) 2g t 2g+2 (1 + t) 2g (1 t 2 )(1 t 4 )
Outline Introduction 1 Introduction 2 3
Holomorphic pairs B pairs = {(A, Φ) : Φ Ω 0 (E), A Φ = 0} Higher rank version of Sym d (C) Moduli space of Bradlow pairs corresponds to solutions of the τ-vortex equations: 1 FA + ΦΦ = τ I This is the moment map for the action on B pairs A Ω 0 (E).
Higgs bundles B higgs = {(A, Φ) : Φ Ω 1,0 (End E), A Φ = 0} Dimensional reduction of anti-self dual equations. Moduli space corresponds to solutions of the Hitchin equations: F A + [Φ, Φ ] = 0 Homeomorphic to the space of flat GL(n, C) connections (Corlette-Donaldson). Hyperkähler structure.
Singularities Singularities because of the jump in dim ker A. Kuranishi model: {Slice} H 1 (deformation complex) Negative directions: ν β is the intersection of negative directions with the image of the slice. Morse-Bott isomorphism: Need to define a deformation retraction.
Critical Higgs bundle ( ) Φ1 0 E = L 1 L 2, A = A 1 A 2, Φ = 0 Φ 2 Negative directions ν: (a, ϕ) strictly lower triangular. a H 0,1 (L 1 L 2), ϕ H 1,0 (L 1 L 2) deg(l 1 L 2) < 0 deg(l 1 L 2 K M ) is not necessarily negative Can still prove HG (X d, X d 1 ) HG (ν d, ν d \ {0})
Critical pair The set of pairs (A, 0), where A is minimal Yang-Mills is a critical set of B pairs. ν = H 0 (E) If d > 4g 4, then H 0 (E) is constant in dimension, and the Morse-Bott lemma holds. If d 4g 4, H 0 (E) jumps in dimension; describes Brill-Noether loci. Can still compute the contribution from this critical set.
Theorem (Daskalopoulos-Weitsman--Wilkin) For the case of Higgs bundles, Kirwan surjectivity holds for GL(2, C) but fails for SL(2, C). Theorem (-Wilkin) For stable pairs, Kirwan surjectivity holds, even though the Morse stratification fails to be perfect.
Higher degree embedding Theorem (MacDonald) The embedding Sym d M Sym d+1 M of symmetric products of Riemann surfaces induces a surjection in cohomology. Theorem (-Wilkin) The same result holds for rank 2 semistable pairs. These are important in showing splitting of the long exact sequences.
Conclusion More examples, general construction? Proof of the Morse-Bott lemma in general? (Kuranishi model) Hyperkähler reduction using the sum of the squares of the moment map? Finite dimensional hyperkähler example where Kirwan surjectivity fails?