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IntroductionCohomology of symplectic quotients
Approach in a singular setting
Morse theory and stable pairs
Richard A. Wentworth
SCGAS 2010
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Joint with
Georgios Daskalopoulos (Brown University)
Jonathan Weitsman (Northeastern University)
Graeme Wilkin (University of Colorado)
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Outline
1 Introduction
2 Cohomology of symplectic quotients
3 Approach in a singular setting
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Outline
1 Introduction
2 Cohomology of symplectic quotients
3 Approach in a singular setting
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Cohomology of Kähler and hyperKähler quotients
The goal is to compute the equivariant cohomology ofsymplectic (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.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Infinite dimensional examples:
Higgs bundles (Hitchin)Stable pairs (Bradlow)Quiver varieties (Nakajima)
These involve symplectic reduction in the presence ofsingularities.
Key points:
this poses no (additional) analytic difficulties.Singularities can cause the Morse stratification to lose“perfection."Computations of cohomology are (sometimes) stillpossible.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Application to representation varieties
M = a closed Riemann surface g ≥ 2π = π1(M, ∗)G = a compact connected Lie groupGC = its complexification (e.g. G = U(n), GC = GL(n,C))Representation varieties:
Hom(π,G)/G ∼ moduli of G-bundles
Hom(π,GC)//
GC ∼ moduli of G-Higgs bundles
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Application to representation varieties
Theorem (Daskalopoulos-Weitsman-Wentworth-Wilkin ’09)The Poincaré polynomial is given by
PSL(2,C)t (Hom(π,SL(2,C)) =
(1 + t3)2g − (1 + t)2g t2g+2
(1− t2)(1− t4)
− t4g−4 +t2g+2(1 + t)2g
(1− t2)(1− t4)+
(1− t)2g t4g−4
4(1 + t2)
+(1 + t)2g t4g−4
2(1− t2)
(2g
t + 1+
1t2 − 1
− 12
+ (3− 2g)
)+ 1
2(22g − 1)t4g−4(
(1 + t)2g−2 + (1− t)2g−2 − 2)
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Outline
1 Introduction
2 Cohomology of symplectic quotients
3 Approach in a singular setting
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Symplectic reduction
(X , ω) symplectic manifold (compact)G compact, connected Lie group, acting symplecticallyµ : X → g∗ a moment map (dµξ(·) = ω(ξ], ·))the Marsden-Weinstein quotient µ−1(0)/G is a symplecticvarietyWhat is the cohomology of µ−1(0)/G?
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Example
Take S2 with the action of S1 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 S1.If µ = z + 1, then µ−1(0) is a point with a trivial S1 action.In both cases, µ−1(0)/S1 is a point.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Equivariant cohomology
Z topological space with an action by GClassifying space: EG→ BG is a contractible, principalG-bundleEquivariant 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)
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Perfect equivariant Morse theory
S1 acts freely on the sphere S∞, so BS1 = CP∞
Therefore Pt (BS1) = 1 + t2 + t4 + · · · =1
1− t2
Example of the sphere:
PS1
t (S2) ∼= Pt (H∗(S2)⊗ H∗(BS1)) =1 + t2
1− t2
Sum over critical points of f = µ2:
PS1
t (S2) =∑
pj crit. pt.
tλj PS1
t (pj)
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Example
f = µ2, µ = z: two critical points of index 2, plus theminimum:
1 +t2
1− t2 +t2
1− t2 =1 + t2
1− t2
f = µ2, µ = z + 1: two critical points, one of index 2 andone of index zero:
11− t2 +
t2
1− t2 =1 + t2
1− t2
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Kirwan, Atiyah-Bott
For the general case, study the gradient flow f = ‖µ‖2.Critical sets ηβ are characterized in terms of isotropy in G.Gradient flow smooth stratification X = ∪β∈ISβ; withnormal bundles νβ.The corresponding long exact sequence splits
· · · −→ H∗G(Sβ,∪α<βSα) −→ H∗G(Sβ) −→ H∗G(∪α<βSα) −→ · · ·
Compute change at each step from H∗G(Sβ,∪α<βSα)
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Two key steps
Morse-Bott Lemma:
H∗G(Sβ,∪α<βSα) ' H∗G(νβ, νβ \ 0) ' H∗−λβ
G (ηβ)
Atiyah-Bott Lemma: criterion for multiplication by theequivariant Euler class to be injective.
· · · // HpG(Sβ,∪α<βSα)
∼=
// HpG(Sβ) //
· · ·
HpG(νβ, νβ \ 0) // Hp
G(ηβ)
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Perfect equivariant Morse theory
Theorem (Kirwan, Atiyah-Bott)
PGt (µ−1(0)) = PG
t (X )−∑β
tλβ PGt (ηβ)
Theorem (Kirwan surjectivity)The map on cohomology
H∗G(X ) −→ H∗G(µ−1(0))
induced from inclusion µ−1(0) → X is surjective.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Vector bundles on Riemann surfaces
M a Riemann surface.A = unitary connections A on hermitian bundle E → MG = gauge group of unitary endomorphisms of E .µ : A → Lie(G) is given by A 7→ FA
‖µ‖2 = Yang-Mills functional
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
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-Millsconnections, i.e. representations to smaller groups. Forexample, E = L1 ⊕ L2, d = deg L1 > deg L2:
ηd = Jac(M)× Jac(M)
Morse-Bott lemma: Negative directions given byH0,1(L∗1 ⊗ L2); λd = dim is constant.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Theorem (Atiyah-Bott, Daskalopoulos)
PSU(2)t (Hom(π,SU(2)) = P(BG)−
∞∑d=0
tλd PS1
t (Jacd (M))
=(1 + t3)2g − t2g+2(1 + t)2g
(1− t2)(1− t4)
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Outline
1 Introduction
2 Cohomology of symplectic quotients
3 Approach in a singular setting
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Holomorphic pairs
Bpairs = (A,Φ) : Φ ∈ Ω0(E) , ∂AΦ = 0Higher rank version of Symd (C)
Moduli space of Bradlow pairs corresponds to solutions ofthe τ -vortex equations:
√−1 ∗ FA + ΦΦ∗ = τ · I
This is the moment map for the action onBpairs ⊂ A× Ω0(E).
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Higgs bundles
Bhiggs = (A,Φ) : Φ ∈ Ω1,0(End E) , ∂AΦ = 0Dimensional reduction of anti-self dual equations.Moduli space corresponds to solutions of the Hitchinequations:
FA + [Φ,Φ∗] = 0
Homeomorphic to the space of flat GL(n,C) connections(Corlette-Donaldson).Hyperkähler structure.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Singularities
Singularities because of the jump in dim ker ∂A.Kuranishi model: Slice → H1(deformation complex)
Negative directions: νβ is the intersection of negativedirections with the image of the slice.Morse-Bott isomorphism: Need to define a deformationretraction.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Critical Higgs bundle
E = L1 ⊕ L2, A = A1 ⊕ A2, Φ =
(Φ1 00 Φ2
)Negative directions ν: (a, ϕ) strictly lower triangular.
a ∈ H0,1(L∗1 ⊗ L2) , ϕ ∈ H1,0(L∗1 ⊗ L2)
deg(L∗1 ⊗ L2) < 0deg(L∗1 ⊗ L2 ⊗ KM) is not necessarily negativeCan still prove H∗G(Xd ,Xd−1) ' H∗G(νd , νd \ 0)
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Critical pair
The set of pairs (A,0), where A is minimal Yang-Mills is acritical set of Bpairs.ν = H0(E)
If d > 4g − 4, then H0(E) is constant in dimension, and theMorse-Bott lemma holds.If d ≤ 4g − 4, H0(E) jumps in dimension; describesBrill-Noether loci. Can still compute the contribution fromthis critical set.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Theorem (Daskalopoulos-Weitsman-Wentworth-Wilkin)For the case of Higgs bundles, Kirwan surjectivity holds forGL(2,C) but fails for SL(2,C).
Theorem (Wentworth-Wilkin)For stable pairs, Kirwan surjectivity holds, even though theMorse stratification fails to be perfect.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Higher degree embedding
Theorem (MacDonald)
The embedding SymdM → Symd+1M of symmetric products ofRiemann surfaces induces a surjection in cohomology.
Theorem (Wentworth-Wilkin)The same result holds for rank 2 semistable pairs.
These are important in showing splitting of the long exactsequences.
Wentworth Morse theory and stable pairs
IntroductionCohomology of symplectic quotients
Approach in a singular setting
Conclusion
More examples, general construction?Proof of the Morse-Bott lemma in general? (Kuranishimodel)Hyperkähler reduction using the sum of the squares of themoment map?Finite dimensional hyperkähler example where Kirwansurjectivity fails?
Wentworth Morse theory and stable pairs