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