How to describe a momentpolytope using a line bundle

Timothy Goldberg

Friday, February 15, 2008Lie Groups SeminarCornell University

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AbstractGiven a Hamiltonian group action, proving that a momentimage is a convex polytope is sometimes much easier thanfiguring out exactly what the polytope is. But in certaincases, where the manifold is Kahler and admits a compati-ble holomorphic positive line bundle L, the sections of L cansometimes lead to descriptions of these moment polytopes.I will discuss several results about this, proved by M. Brion,V. Guillemin and R. Sjamaar, L. O’Shea and R. Sjamaar, P.Schutzdeller, and myself.

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Starring:

(M,ω) compact, connected Kahler manifold

L holomorphic, Hermitian line bundle over M

G compact, connected Lie group

GC complexification of G

Assume:

ω = 12πi

curv(∇).

G,GC act holomorphically on (M,L) by bundleautomorphisms preserving ∇.

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Where’s the moment map?

∀ξ ∈ g, have two operators on C∞(M,L): ∇(ξM),L(ξ)

Fact: (B. Kostant, 1970)L(ξ) −∇(ξM) is multiplication by an iR-valued function.

Define φ : g → C∞(M) by

φ(ξ) :=1

2πi(L(ξ) −∇(ξM)) .

Can show φ is G-equivariant and d (φ(ξ)) = ω (ξM, ·).

Defines a moment map!

Φ : M → g∗, Φ(x)ξ = φ(ξ)(x)

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Also starring:

X ⊂M closed, irreducible, analytic subvariety

T ⊂ G maximal torus

t∗+ ⊂ t∗ closed positive Weyl chamber

Λ ⊂ t∗ weight lattice

Λ+ ⊂ t∗+ dominant weight set

ΛQ ⊂ t∗ rational points, ΛQ := Λ⊗Q

B ⊂ GC Borel subgroup corresponding to t∗+

N = [B, B] maximal unipotent subgroup

Example:

G = SU(n), GC = GL(n; C),T = {diagonal elements of G},B = {upper triangular elements of GC},N = {elements of B with 1’s on the diagonal}

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Kodaira Embedding Theorem: (M,L) is ample.

⇒ ∃ embedding M ↪→ some CPN(holomorphic but not usually Kahler)

So we are really working with algebraic things,not just analytic.

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The torus case

Moment map for T y M:

ΦT : MΦ // g∗ // // t∗ .

Theorem: (Atiyah, 1982)If X is preserved by TC, then ΦT(X) is the convex hull ofΦT(X

T).

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T y Γ(M,L), so

Γ(M,L) =⊕λ∈Λ

Γ(M,L)λ, weight space decomposition.

(Note that dimC Γ(M,L) < ∞.)

Alternative Statement of Atiyah’s Theorem:Let j : X ↪→ M be inclusion. If X is preserved by TC, thenΦT(X) is the convex hull of

{λ ∈ Λ | ∃s ∈ Γ(M,L)λ, j∗s 6= 0} .

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The general case

Definition: The highest weight polytope C(X) of X con-tains λ ∈ ΛQ such that:

• ∃r > 0 with rλ ∈ Λ+, and

• ∃s ∈ Γ(M,Lr)rλ, N-invariant with j∗s 6= 0.

I.e. rλ is a highest weight for the G-representation spaceΓ(M,Lr), and ∃ an eigensection for rλ not vanishingidentically on X.

Put ∆(X) := Φ(X) ∩ t∗+.

Theorem: (M. Brion, 1986)If X is GC-invariant, then C(X) = ∆(X) ∩ΛQ andC(X) = ∆(X).

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Theorem: (V. Guillemin and R. Sjamaar, 2006)Even if X is only B-invariant, then Brion’s result still holds.

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What does Γ(M,L) have to do with Φ?

Magic Formula:

Suppose h ⊂ gC = complex Lie subalgebra,s ∈ Γ(M,L) transforms under h by a character χ : h → C.

Then ∀ξ ∈ h,

L(ξM)‖s‖2 = (2Reχ(ξ) + 4πφ(Im ξ)) ‖s‖2.

(For (M,Lr), just replace φ with rφ.)

Example:Let h = gC, χ = 0. Then ∀ξ ∈ g,

L ((iξ)M) ‖s‖2 = 4πφ(ξ)‖s‖2.

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SemistabilityDefinition: x ∈M is:

• algebraically semistable if ∃ r > 0 and s ∈ Γ(M,Lr)Gwith s(x) 6= 0.

• analytically semistable if 0 ∈ Φ(GC · x

).

Theorem:algebraic semistability ⇐⇒ analytic semistability

(⇒) follows from Magic Formula.(⇐) is hard. (V. Guillemin and S. Sternberg in 1982, F.Kirwan in 1984, L. Ness in 1984, R. Sjamaar in 1995.)

From this we can prove:

0 ∈ C(X) ⇐⇒ 0 ∈ Φ(X).

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The shifting trickGiven λ0 ∈ Λ+, want to construct (M ′, ω ′), L ′, X ′, andΦ ′ : M ′ → g∗ such that

• 0 ∈ ∆(X) ⇐⇒ λ0 ∈ ∆(X ′) and

• 0 ∈ C(X) ⇐⇒ λ0 ∈ C(X ′).

Choose λ0 ∈ Λ+.

Gλ0= stabilizer of λ0 w.r.t. coadjoint action G y g∗

Pλ0= parabolic subgroup of GC corresponding to λ0

Mλ0:= GC/Pλ0

(flag variety)≈ G/Gλ0

(homogenous space)≈ G · λ0 (coadjoint orbit)

L−λ0:= (GC × C−λ0

) /Pλ0

Symplecticω ′ comes from G ·λ0, andΦλ0: Mλ0

→ g∗ is thenegative of the inclusion

Mλ0≈ G · λ0 ↪→ g∗.

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Borel-Weil Theorem:Γ(Mλ0

, L−λ0) ∼= V(λ0)

∗ := dual of irreducible representationof G with highest weight λ0.

Put

M ′ = M×Mλ0, L ′ = L� L−λ0

, X ′ = X×Mλ0,

ω ′ = ω+ωλ0, Φ ′ = Φ+Φλ0

,

and let G y M ′ diagonally.

Γ(M ′, L ′)G ∼= (Γ(M,L)⊗ Γ(Mλ0, L−λ0

))G

∼= (Γ(M,L)⊗ V(λ0)∗)G

∼= Hom (V(λ0), Γ(M,L))G

∼= Γ(M,L)Nλ0.

Φ(x) = λ0 ⇐⇒Φ ′(x, λ) = Φ(x) +Φλ0

(λ0) = Φ(x) − λ0 = 0.

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Real resultsSuppose

• (τ, β) y (M,L) and σ y GC are antiholomorphicinvolutions,

• β preserves ∇,

• σ preserves G, T , and B.

σ y G σ y g, g∗

Compatibility with Hamiltonian action:

• Distibution: ∀g ∈ G, x ∈M, τ(g · x) = σ(g) · τ(x).

• Anti-equivariance: ∀x ∈M, Φ (τ(x)) = −σ (Φ(x)).

σ, τ ! complex conjugationMτ, Gσ ! “real parts” of M, G.

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Note:

x ∈Mτ ⇒ Φ(x) = Φ (τ(x)) = −σ (Φ(x))⇒ σ (Φ(x)) = −Φ(x)

SoΦ (Mτ) ⊂ g∗−1.

Theorem: (L. O’Shea and R. Sjamaar, 2000)Suppose X is preserved by GC and τ, and Xτ contains asmooth point. Then

∆(Xτ) = ∆(X) ∩ g∗−1.

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Theorem: (G., 2007) O’Shea and Sjamaar’s theorem holdseven if X is only preserved by B and τ.

Mantra:The real part of the moment polytope is (ought to be) themoment polytope of the real part.

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Strategy of proofs:

1. Show ∆(Xτ) ∩ΛQ = C(Xτ) ∩ g∗−1.

(Involves variant on the shifting trick.)

2. Show C(Xτ) ∩ g∗−1 = C(X) ∩ g∗−1.

(Xτ contains a Lagrangian submanifold,⇒ Xτ Zariski dense in X.)

3. Use previous results about C(X) and ∆(X).

Shifting trick variant:Need involution α y Mλ0

≈ G · λ0.Since this λ0 is in g∗−1, can define α := −σ y G · λ0.

Proposition: (G · λ0)α = Gσ · λ0.

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THE END

Thank you for listening.

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