Morse theory and stable pairs


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1 Richard A. SCGAS 2010
2 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado)
3 Outline Introduction 1 Introduction 2 3
4 Outline Introduction 1 Introduction 2 3
5 Cohomology of Kähler and hyperkähler quotients The goal is to compute the equivariant cohomology of symplectic (Kähler or hyperkähler) reductions. By the KempfNess, GuilleminSternberg theorem, examples arise in geometric invariant theory. Kirwan, AtiyahBott: In the symplectic case there is a perfect" Morse stratification. HyperKähler case still unknown.
6 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.
7 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 Gbundles Hom(π, G C ) // G C moduli of GHiggs bundles
8 Application to representation varieties Theorem (DaskalopoulosWeitsmanWilkin 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 t ) + (3 2g) ( 2 ) (22g 1)t 4g 4 (1 + t) 2g 2 + (1 t) 2g 2 2
9 Outline Introduction 1 Introduction 2 3
10 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 MarsdenWeinstein quotient µ 1 (0)/G is a symplectic variety What is the cohomology of µ 1 (0)/G?
11 Example Introduction Take S 2 with the action of S 1 by rotation in the xyplane. 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.
12 Equivariant cohomology Z topological space with an action by G Classifying space: EG BG is a contractible, principal Gbundle 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)
13 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 )
14 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
15 Kirwan, AtiyahBott 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 )
16 Two key steps MorseBott Lemma: H G (S β, α<β S α ) H G (ν β, ν β \ {0}) H λ β G (η β ) AtiyahBott 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)
17 Perfect equivariant Morse theory Theorem (Kirwan, AtiyahBott) 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.
18 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 = YangMills functional
19 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 YangMills 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) MorseBott lemma: Negative directions given by H 0,1 (L 1 L 2); λ d = dim is constant.
20 Theorem (AtiyahBott, 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 )
21 Outline Introduction 1 Introduction 2 3
22 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 τvortex equations: 1 FA + ΦΦ = τ I This is the moment map for the action on B pairs A Ω 0 (E).
23 Higgs bundles B higgs = {(A, Φ) : Φ Ω 1,0 (End E), A Φ = 0} Dimensional reduction of antiself 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 (CorletteDonaldson). Hyperkähler structure.
24 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. MorseBott isomorphism: Need to define a deformation retraction.
25 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})
26 Critical pair The set of pairs (A, 0), where A is minimal YangMills 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 MorseBott lemma holds. If d 4g 4, H 0 (E) jumps in dimension; describes BrillNoether loci. Can still compute the contribution from this critical set.
27 Theorem (DaskalopoulosWeitsmanWilkin) 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.
28 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.
29 Conclusion More examples, general construction? Proof of the MorseBott 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?
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