GITand µ-GITrobbin/GIT_notes.pdf1. GITandµ-GIT Valentina Georgoulas Joel W. Robbin Dietmar A....

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1. GIT and μ-GIT Valentina Georgoulas Joel W. Robbin Dietmar A. Salamon ETH Z¨ urich UW Madison ETH Z¨ urich Dietmar did the heavy lifting; Valentina and I made him explain it to us. A key idea comes from Xiuxiong Chen and Song Sun; they were doing an analogous infinite dimensional problem. I learned a lot from Sean. The 1994 edition of Mumford’s book lists 926 items in the bibliography; I have read fewer than 900 of them. Dietmar will talk in the Geometric Analysis Seminar next Monday. Follow the talk on your cell phone. Calc II example of GIT: conics, eccentricity, major axis. Many important problems in geometry can be reduced to a partial differential equation of the form μ(x)=0, where x ranges over a complexified group orbit in an infinite dimensional sym- plectic manifold X and μ : X g is an associated moment map. Here we study the finite dimensional version. Because we want to gain intuition for the infinite dimensional problems, our treatment avoids the structure theory of compact groups. We also generalize from projective manifolds (GIT) to K¨ ahler manifolds (μ-GIT). In GIT you start with (X,J,G) and try to find Y with R(Y ) R(X ) G . In μ-GIT you start with (X,ω,G) and try to solve μ(x) = 0. GIT = μ-GIT + rationality. The idea is to find analogs of the GIT definitions for K¨ ahler manifolds, show that the μ-GIT definitions and the GIT definitions agree for projective manifolds, and prove the analogs of the GIT theorems in the K¨ ahler case. To solve μ(x)=0 we form the function f (x)= 1 2 |μ(x)| 2 , and show that trajectories of the negative gradient field ˙ x = −∇f (x) converge to limits in f 1 (0) = μ 1 (0). Just like in ordinary Morse theory not every critical point is a minimum. The points which flow to a minimum are called (poly)stable. Just like in Morse theory the stable points are open dense. I’ll give an “archetypal example” and illustrate the definitions and theorems by applying them to this example. 1

Transcript of GITand µ-GITrobbin/GIT_notes.pdf1. GITandµ-GIT Valentina Georgoulas Joel W. Robbin Dietmar A....

Page 1: GITand µ-GITrobbin/GIT_notes.pdf1. GITandµ-GIT Valentina Georgoulas Joel W. Robbin Dietmar A. Salamon ETH Zu¨rich UW Madison ETH Zu¨rich • Dietmar did the heavy lifting; Valentina

1. GIT and µ-GIT

Valentina Georgoulas Joel W. Robbin Dietmar A. SalamonETH Zurich UW Madison ETH Zurich

• Dietmar did the heavy lifting; Valentina and I made him explain it to us.

• A key idea comes from Xiuxiong Chen and Song Sun; they were doing ananalogous infinite dimensional problem.

• I learned a lot from Sean.

• The 1994 edition of Mumford’s book lists 926 items in the bibliography; Ihave read fewer than 900 of them.

• Dietmar will talk in the Geometric Analysis Seminar next Monday.

• Follow the talk on your cell phone.

• Calc II example of GIT: conics, eccentricity, major axis.

Many important problems in geometry can be reduced to a partial differentialequation of the form

µ(x) = 0,

where x ranges over a complexified group orbit in an infinite dimensional sym-plectic manifold X and µ : X → g is an associated moment map. Here we studythe finite dimensional version.

Because we want to gain intuition for the infinite dimensional problems, ourtreatment avoids the structure theory of compact groups. We also generalizefrom projective manifolds (GIT) to Kahler manifolds (µ-GIT).

• In GIT you start with (X, J,G) and try to find Y with R(Y ) ≃ R(X)G.

• In µ-GIT you start with (X,ω,G) and try to solve µ(x) = 0.

• GIT = µ-GIT + rationality.

The idea is to find analogs of the GIT definitions for Kahler manifolds, show thatthe µ-GIT definitions and the GIT definitions agree for projective manifolds,and prove the analogs of the GIT theorems in the Kahler case. To solve µ(x) = 0we form the function

f(x) = 12 |µ(x)|2,

and show that trajectories of the negative gradient field x = −∇f(x) convergeto limits in f−1(0) = µ−1(0). Just like in ordinary Morse theory not everycritical point is a minimum. The points which flow to a minimum are called(poly)stable. Just like in Morse theory the stable points are open dense.

I’ll give an “archetypal example” and illustrate the definitions and theorems byapplying them to this example.

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Page 2: GITand µ-GITrobbin/GIT_notes.pdf1. GITandµ-GIT Valentina Georgoulas Joel W. Robbin Dietmar A. Salamon ETH Zu¨rich UW Madison ETH Zu¨rich • Dietmar did the heavy lifting; Valentina

2. Kahler manifolds. A Kahler manifold (X,ω, J) satisfies

ωx(Jxx1, Jxx2) = ωx(x1, x2), |x|2x := ωx(x, Jxx) > 0

for x ∈ X , x, x1, x2 ∈ TxX . Hence

〈x1, x2〉x := ωx(x1, Jxx2)

is a Riemannian metric. A function H : X → R has a symplectic gradientXH ∈ Vect(X) and a Riemannian gradient ∇H ∈ Vect(X) characterized by

dH(x)x = ωx(XH , x) = 〈∇H, x〉x .They are related by the formula

∇H = JXH .

The symplectic gradient is also called the Hamiltonian vector field.

3. Projective manifolds.Ingredients:

• The complex vector space V = CN and the projective space P (V ).

• The (frame bundle of the) tautological line bundle π : V \ 0 → P (V ).

Any complex submanifold X ⊆ P (V ), V = CN of projective space is an exampleof a Kahler manifold. The symplectic form is the restriction to X of the Fubini–Study form

π∗ωFS = ∂∂K, K(v) = i~ log |v|.By Chow, a complex submanifold of P (V ) is the same as a projective mani-fold, i.e. a smooth projective variety.

4. GIT. In GIT the ingredients are

• A compact subgroup G ⊆ UN and its complexification Gc ⊆ GLN (C).

• A Gc invariant closed complex submanifold X ⊆ P (V ).

• The homogenous coordinate ring R(X).

• The ring R(X)G of invariants.

Hilbert showed how to construct a projective variety Y with

R(Y ) = R(X)G.

A sequence of generators ϕ0, . . . , ϕn of R(X)G gives an embedding

ϕ : P (U) → P (Cn).

The set U ⊆ V is the complement of the null cone, the set of points in V whereall the invariants vanish. (The embedding ϕ is not defined on this set. If neces-sary raise the components to suitable powers so that the map is homogeneous.)The closure Y of the image of ϕ is an algebraic variety. A syzygy for ϕ givesequations for Y .

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5. µ-GIT. In µ-GIT the ingredients are

• A compact symplectic manifold (X,ω).

• A Lie algebra g ⊆ uN of G.

• The ad(G)-invariant inner product 〈ξ, η〉 = trace(ξ∗η) on g.

• A Hamiltonian action g → Vect(X) : ξ 7→ Xξ.

• An equivariant moment map µ : X → g for this action.

• The norm squared function f : X → R defined by f(x) = 12 |µ(x)|2.

That µ is a moment map means that the Hamiltonian for Xξ is

Hξ := 〈µ, ξ〉 , dHξ = ω(Xξ, ·).

That µ is equivariant means that µ(ux) = uµ(x)u−1 for x ∈ X , u ∈ G. Thisimplies

〈µ(x), [ξ, η]〉 = ω(Xξ(x),Xη(x)) =: {Hξ, Hη}for ξ, η ∈ g.1

6. Kahler Actions. A Hamiltonian action g → Vect(X) : ξ 7→ Xξ on a closedKahler manifold extends to an action

gc → Vect(X) : ξ + iη 7→ Xξ + JXη

by holomorphic vector fields, so the action of G on X extends to a holomorphicaction of Gc on X .

Any subgroup of GL(V ) = GLN induces an action on P (V ). When G ⊆ UN

and X ⊆ P (V ) is a complex G-invariant submanifold,2 the moment map is

〈µ(π(v)), ξ〉g= 1

2 〈iξv, v〉V , v ∈ S(V ) := S2N−1.

Warning. Distinguish G and Gc. The eigenvalues of a real element ξ ∈ g arepure imaginary and the eigenvalues of a pure imaginary iη ∈ ig are real.

The closure of the 1-PSG R → G : t 7→ exp(tξ) is a torus, generically amaximal torus, and sometimes a circle.

1We use the sign convention

[v, w] := ∇wv −∇vw

for the Lie bracket of two vector fields so that the infinitesimal action is a homomorphismg → Vect(X) of Lie algebras (and not an anti-homomorphism).

2By Weyl’s unitarian trick G-invariance implies Gc-invariance.

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7. Rationality Theorem. A Kahler G-manifold is isomorphic to a projectiveG-manifold if and only if

(A) Integrality of ω. The cohomology class of ω lies in H2(X ; 2π~Z).

(B) Integrality of µ. The action integral

Aµ(x0, u, x) := −∫

D

x∗ω +

∫ 1

0

⟨µ(u(t)−1x0), u(t)

−1u(t)⟩dt

is integral in the sense that

Aµ(x0, u, x) ∈ 2π~Z

whenever x0 ∈ X , u : R/Z → G, and x : D → X satisfy x(e2πit) = u(t)−1x0.(Here D := {z ∈ C | |z| ≤ 1} denotes the closed unit disc in the complex plane.)

(The inner product on g satisfies

(C) Integrality of g. If ξ, η ∈ g, exp(ξ) = exp(η) = 1, and [ξ, η] = 0, then〈ξ, η〉 ∈ Z.)

8. Moreover Where Λ := {ξ ∈ g \ {0} | exp(ξ) = 1} (see §24)(i) The action integral Aµ(x, u, v) is invariant under homotopy.

(ii) There is an N ∈ N such that Nα = 0 for every torsion class α ∈ H1(G;Z).

(iii) If 〈ω, π2(X)〉 ⊆ 2π~NZ then there is a central element τ ∈ Z(g) such thatthe moment map µ− τ satisfies condition (B).

(iv) Assume (A), (B), (C), and let τ ∈ Z(g) be a central element, so µ + τ isan equivariant moment map. Then µ+ τ satisfies (B) if and only if τ ∈ 2π~Λ.

Note that if ξ in (iii) is replaced by ξ + τ where τ ∈ Z(g) then the momentmap µ is replaced by µ+ τ and

Aµ+τ (x0, u, x) = Aµ(x0, u, x) + 〈τ, ξ〉 .

This is because τ ∈ Λ and ξ ∈ Λ so 〈τ, ξ〉 ∈ 2π~Z by taking η = τ/2π~ in (C).

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9. The simplest Kahler manifold. The sphere S2 is a Kahler manifold. Thecomplex structure, symplectic form, and Riemannian metric are

Jq q = q × q, ωq(q1, q2) = q · (q1 × q2), 〈q1, q2〉q = q1 · q2

for q ∈ S2 and q, q1, q2 ∈ TqS2. With the identification so3 ≃ (R3,×) the action

is Hamiltonian with

Xξ(q) = ξ × q, 〈ξ, η〉 = ξ · η, Hξ(q) = 〈µ(q), ξ〉 = q · ξ.

(rotation about ξ). The gradient

∇Hξ(q) = −JqXξ(q) = (ξ × q)× q = ξ − (ξ · q)q

of Hξ generates a north pole – south pole flow with poles on the axis of rotationof ξ.

10. The action of C∗ on S2 Let ξ = (0, 0, 1) ∈ R3 ≃ so3 generate rotationabout the z-axis. Then Hξ(q) = z for q = (x, y, z) ∈ S2 and the ODE q = Xξ(q)is

x = −y, y = x. z = 0.

The gradient vector field is

∇Hξ(q) = ξ − (ξ · q)q

so the equation q = −∇Hξ(q) is

x = zx, y = zy, z = z2 − 1.

The solution with q(0) = q0 is

x =2etx0

z0 + 1− (z0 − 1)e2t, y =

2ety0z0 + 1− (z0 − 1)e2t

, z =z0 + 1 + (z0 − 1)e2t

z0 + 1− (z0 − 1)e2t.

11. Another gradient flow on S2. The moment map squared for the S1

action on S2 isf(q) = 1

2z2

for q = (x, y, z) ∈ S2. The gradient is

∇f(q) = ξ − (ξ · q)q

where ξ = (0, 0, z) so the ODE

q = −∇f(q)

is3

x = z2x, y = z2y, z = z3 − z.

The equator consists of rest points and the orbits run away from the poles alongthe meridians towards the equator.

3Check: xx+ yy + zz = z2x2 + z2y2 + z4 − z2 = z2(x2 + y2 + z2 − 1) = 0.

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12. The archetypal example. Let SO3 act diagonally on X = S2 × · · · × S2

︸ ︷︷ ︸

n

with moment map

µ(x) :=

n∑

i=1

qi, x = (q1, . . . , qn) ∈ X.

The preimage µ−1(0) consists of those x with center of mass at the origin. Themoment map squared is

f(x) = 12 |µ(x)|2, df(x)x = 1

2

i6=j

qi · qj

and negative gradient flow of f is

qi = −∇f(x) = −∑

j 6=i

(qj − (qj · qi)qi).

This example is closely related to the space of binary forms of degree n.

The relation to binary forms is via the identification S2 = P (C2). Treat thecomponents as roots of a binary form of degree n. However the roots of a formgive an unordered set with multiplicities whereas a point is X is an orderedsequence.

13. Critical points of f where µ 6= 0. In the archetypal example we cancharacterize the critical points using Lagrange multipliers. Since

f(x) = 12 |µ(x)|2 =

n

2+∑

i<j

qi · qj ,

we have that df(x) = 0 if and only if there exist λ1, . . . , λn such that∑

j 6=i

qj = λiqi

for i = 1, . . . , n and when this holds, λi =(∑

j 6=i qj

)

· qi so

df(x) = 0 ⇐⇒ µ(x) = (λi + 1)qi.

Since |qi| = 1, the critical points x of f where µ(x) 6= 0 are the points x of formx = (q1, . . . , qn) where qi = ±p for i = 1, . . . , n for some p ∈ S2.

Assign to x ∈ X the equivalence relation (cycle structure of a permutation)

i ≡x j ⇐⇒ qi = qj .

This stratifies X . Each equivalence relation on {1, . . . , n} determines a stratum.The Gc action preserves the stratification. The gradient flow−∇f also preservesthe stratification because qi = qj =⇒ (∇f(x))i = (∇f(x))j , i.e. ∇f is tangentto each stratum. Each stratum is foliated by Gc oribits.

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14. The moment map squared. In the Kahler case define f : X → R by

f(x) := 12 |µ(x)|2

satisfies f−1(0) = µ−1(0) so any x ∈ µ−1(0) is a critical point of f (as it is anabsolute minimum).

In the projective case for x = π(v), v ∈ V \ 0 the following are equivalent:

• v minimizes the distance from the orbit Gcv to the origin.

• x ∈ µ−1(0).

Proof: Assume w.l.o.g. that |v| = 1. The tangent space to the orbit is

Tv(Gcv) = {ζv, ζ ∈ gc}, Tv(Gv) = {ξv, ξ ∈ g}.

The derivative of the distance is v 7→ 2 〈v, v〉V . It vanishes on Tg(GcV ) iff

〈v, v〉V = 〈v, ζv〉V = 0 for all ζ. But for ζ = ξ + iη, ξ, η ∈ g we have

12 〈v, iζv〉V = 〈µ(x), iξ − η〉

gc = −〈µ(x), η〉g= 0

if x ∈ µ−1(0). For the converse see §20 items (ii) and (iii).

15. Convergence Theorem. Every solution of the negative gradient equation

x = −∇f(x), x(0) = x0

converges, i.e. the limitx∞ := lim

t→∞x(t)

exists in X . (Proof: Lojasiewicz.)

The (not quite right) idea is to view f as a G-invariant Morse–Bott function.The stable manifold of the G-invariant set µ−1(0) is an open dense set in U ⊆ Xand the map

U → µ−1(0) : x0 → x∞

gives an isomorphism U/Gc ≃ µ−1(0)/G. (This is like R(Y ) = R(X)G.)

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16. The homogeneous space Gc/G. Equip Gc with the unique left invariantRiemannian metric which agrees with the inner product

〈ξ1 + iη1, ξ2 + iη2〉gc = 〈ξ1, ξ2〉g + 〈η1, η2〉gon the tangent space gc to Gc at the identity. This Riemannian metric isinvariant under the right G-action. Let

π : Gc → Gc/G

be the projection onto the right cosets of G. This is a principal G-bundle. The(orthogonal) splitting

gc = g⊕ ig

extends to a left invariant principal connection on π. Projection from the hor-izontal bundle (i.e. the summand corresponding to ig) defines a Gc-invariantRiemannian metric of nonpositive curvature on Gc/G.

17. The Moment Conjugacy Theorem. Fix x ∈ X and define a mapψx : Gc → Gcx ⊆ X by

ψx(g) = g−1x.

Then there is a function Φx : Gc → R such that ψx intertwines the two gradientvector fields ∇Φx ∈ Vect(Gc) and ∇f ∈ Vect(X), i.e.

dψx(g)∇Φx(g) = ∇f(ψx(g)). (♥)

In particular, ∇f is tangent to the Gc-orbits.

Definition. The function Φx : Gc → R will be called the lifted Kempf–Nessfunction based at x. (It is unique if normalized by the condition Φx(1) = 0.)It is G-invariant and hence descends to a function

Φx : Gc/G→ R

denoted by the same symbol and called the Kempf–Ness function.

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Page 9: GITand µ-GITrobbin/GIT_notes.pdf1. GITandµ-GIT Valentina Georgoulas Joel W. Robbin Dietmar A. Salamon ETH Zu¨rich UW Madison ETH Zu¨rich • Dietmar did the heavy lifting; Valentina

18. Proof of the Conjugacy Theorem. Define a vector field Fx ∈ Vect(Gc)and a one form αx on Hc by

Fx(g) := −giµ(g−1x), αx(g)g = −⟨µ(g−1x),ℑ(g−1g)

for g ∈ Gc, g ∈ TgGc. Then

Step 1. There is a unique Φx such that Φx(1) = 0 and

dΦx = αx.

Step 2. The map ψx intertwines the vectorfields Fx and ∇f .

dψx(g)Fx(g) = ∇f(ψx(g)).

Step 3. The gradient ∇Φx of Φx is Fx, i.e.

αx(g)g = 〈Fx(g), g〉gwhere the inner product on the right is the left-invariant inner Riemannianmetric on Gc

The vector field F is is horizontal since µ(x) ∈ g and is right G-equivariant,i.e.

Fx(gu) = Fx(g)u

for g ∈ Gc and u ∈ G.

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The vector field F is is horizontal since µ(x) ∈ g and is right G-equivariantsince

Fx(gu) = −guiµ(u−1g−1x) = −guiu−1µ(g−1x)u = −giµ(g−1x)u = Fx(g)u

for u ∈ G. In the second step we used the G-equivariance of µ, i.e.

µ(u−1y) = u−1µ(y)u).

By definition

〈dµ(x)x, ξ〉 = dHξ(x)x = ω(Xξ(x), x)

for x ∈ X , x ∈ TxX and ξ ∈ g so taking ξ = µ(x) we get

df(x)x = 〈dµ(x)x, µ(x)〉 = ω(Xµ(x)(x), x)

and hence∇f(x) = J(x)Xµ(x)(x) = Xiµ(x)(x).

Now if g ∈ TgGc, then g = gζ ∈ TgG

c where ζ ∈ gc. Hence

dψx(g)g =d

dtψx(g exp(tζ)x)

∣∣∣∣t=0

=d

dtexp(−tζ)g−1x

∣∣∣∣t=0

= −Xζ(g−1x)

so taking g = Fx(g) (so that ζ = −iµ(g−1x)) we get

dψx(g)Fx(g) = −X−iµ(g−1x)(g−1x) = Xiµ(g−1x)(g

−1x) = ∇f(ψx(g)).

Because µ(ux) = uµ(x)u−1 and u∗ = u−1 for u ∈ G, the one-form αx is rightG-invariant and hence descends to a one-form (also denoted by αx) on Gc/G.It is easy to show that αx is closed and hence (as Gc/G is contractible) αx isexact on Gc/G so its pull back to Gc is also exact. The Kempf-Ness function isdefined by

dΦx = αx, Φx(1) = 0.

By (†) to prove (♥) we must prove ∇Φx = Fx, i.e.

αx(g)g = 〈Fx(g), g〉g (‡)

where the inner product on the right is the left-invariant inner Riemannianmetric on Gc. By left invariance 〈Fx(g), g〉g = −

⟨iµ(g−1x), g−1g

gcso (‡)

follows because 〈iξ, ζ〉 = 〈ξ,ℑ(ζ)〉 when ξ ∈ g and ζ ∈ gc.

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19. The Kempf–Ness function for projective manifolds. The Kempf–Ness function Φx for a projective manifold X ⊆ P (V ) is

Φx(g) =12

(log |g−1v|V − log |v|V

)(#)

for g ∈ Gc where x = π(v) ∈ X ⊆ P (V ).

Proof: Define Φx by (#). Then Φx(1) = 0 and

dΦx(g)g = −⟨g−1gg−1v, g−1v

V

2|g−1v|2V.

The moment map µ : X → g is characterized by the formula

〈µ(y), η〉g= Hη(y) =

12 〈iηw,w〉V , y = π(w), |w|2V = 1, η ∈ g.

(See §3.) Let ζ = g−1g = ξ + iη where ξ, η ∈ g. Then

dΦx(g)g = −⟨ζg−1v, g−1v

V

2|g−1v|2V= −

⟨iµ(g−1x), ζ

gc= −

⟨µ(g−1x), η

gc

and η = ℑ(ζ) = ℑ(g−1g) so dΦx = αx.

20. Properties of the Kempf–Ness function.

(i) The Kempf–Ness function Φx : Gc/G→ R is Morse–Bott (usually Morse).

(ii) The critical set of Φx is a (possibly empty) closed connected submanifoldof Gc/G. It is given by

Crit(Φx) ={π(g) ∈ Gc/G, µ(g−1x) = 0

}.

(iii) If the critical manifold of Φx is nonempty, then it consists of the absoluteminima of Φx and every negative gradient flow line of Φx converges exponentiallyto a critical point.

(iv) Even if the critical manifold of Φx is empty, every negative gradient flowline γ : R → Gc/G of Φx satisfies

limt→∞

Φx(γ(t)) = infGc/G

Φx.

(The infimum may be minus infinity.)

The function Φx is uniquely determined by the normalization Φx(1) = 0 sothe dependence on the base point x is given by

Φhx(g) = Φx(g)− Φx(h−1)

for h ∈ G.

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21. Stability in symplectic geometry. A point x ∈ X is called

(i) µ-unstable (x ∈ Xus) iff Gcx ∩ µ−1(0) = ∅,(ii) µ-semistable (x ∈ X ss) iff Gcx ∩ µ−1(0) 6= ∅,(iii) µ-polystable (x ∈ Xps) iff Gcx ∩ µ−1(0) 6= ∅,(iv) µ-stable (x ∈ X s) iff x is µ-polystable and gcx = 0.4

In the archetypal example x ∈ Xus ⇐⇒ more than half the qi coincide andx ∈ Xps ⇐⇒ exactly half the points coincide.

The Moment Limit Theorem. With x0 and x∞ as in §15,(i) x0 ∈ Xus if and only if µ(x∞) 6= 0.

(ii) x0 ∈ X ss if and only if µ(x∞) = 0.

(iii) x0 ∈ Xps if and only if µ(x∞) = 0 and x∞ ∈ Gcx0.

(iv) x0 ∈ X s if and only if gx∞= 0.

Moreover, X ss and X s are open subsets of X .

In both cases the north-pole south-pole limits to a pair of antipodal clusters.Even when there are only two clusters and they are antipodal the center of massneed not be at the origin.

22. Stability in algebraic geometry. A vector v ∈ V \ 0 is called

(i) unstable (v ∈ V us) iff 0 ∈ Gcv,

(ii) semistable (v ∈ V ss) iff 0 /∈ Gcv,

(iii) polystable (v ∈ V ps) iff Gcv = Gcv,

(iv) stable (v ∈ V s) iff Gcv = Gcv and Gcv is discrete.

In the archetypal example x ∈ V us ⇐⇒ more than half the roots coincide,and x ∈ V ps ⇐⇒ exactly half the roots coincide.

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Kempf–Ness Theorem. The two notions of stability agree for projectivespace in the following sense. If x ∈ X ⊆ P (V ), then

(i) x ∈ Xus if and only if π−1(x) ⊆ V us.

(ii) x ∈ X ss if and only if π−1(x) ⊆ V ss.

(iii) x ∈ Xps if and only if π−1(x) ⊆ V ps.

(iv) x ∈ X s if and only if π−1(x) ⊆ V s.

23. The Kempf–Ness Theorem generalized. For any Kahler G-manifold(X,ω, J, µ) the Kempf–Ness function Φx characterizes µ-stability as follows.

(i) x is µ-unstable ⇐⇒ Φx is unbounded below.

(ii) x is µ-semistable ⇐⇒ Φx is bounded below.

(iii) x is µ-polystable ⇐⇒ Φx has a critical point.

(iv) x is µ-stable ⇐⇒ Φx is bounded below and proper.

In the archetypal example take V = Cn+1. A point v ∈ V is a binary form

v(x, y) = v0xn + v1x

n−1y + · · ·+ vnyn.

The north pole-south pole flow is

(exp(tξ)v)(x, y) = v(etx, e−ty).

The Kempf–Ness function is

Φx(et) = log(|v0ent|2 + |v1e(n−2)t|2 + · · ·+ |vne−nt|2)− log(|v|2).

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24. Toral elements. A nonzero element ζ ∈ gc is called toral iff it satisfiesthe following equivalent conditions.

• ζ is semi-simple and has purely imaginary eigenvalues.

• The subset Tζ := {exp(tζ) | t ∈ R} is a torus in Gc.

• The element ζ is conjugate to an element of g.

Denote the set of toral elements by

T c := ad(Gc)(g \ {0})

and also use the abbreviations

Λ := {ξ ∈ g \ {0} | exp(ξ) = 1} , Λc := {ζ ∈ gc \ {0} | exp(ζ) = 1} .

• The set Λ ∪ {0} intersects the Lie algebra t ⊂ g of any maximal torusT ⊆ G in a spanning lattice.

• The generator of any one parameter subgroup C∗ → Gc is conjugate toan element of Λ ∩ t.

25. µ-weights. For x ∈ X and ξ ∈ g \ {0} the µ-weight of the pair (x, ξ) isthe real number

wµ(x, ξ) := limt→∞

〈µ(exp(itξ)x), ξ〉 .

In the projective case the µ-weight is

wµ(x, ξ) = ~maxvi 6=0

λi

for x = π(v) where

• λ1 < · · · < λk are the eigenvalues of iξ,

• Vi ⊆ V are the corresponding eigenspaces, and

• v =∑k

i=1 vi with vi ∈ Vi.

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26. The Hilbert–Mumford criterion. The µ-weight characterizes µ-stabilityas follows.

(i) x ∈ Xus ⇐⇒ there exists a ξ ∈ Λ such that wµ(x, ξ) < 0.

(ii) x ∈ X ss ⇐⇒ wµ(x, ξ) ≥ 0 for all ξ ∈ Λ.

(iii) x ∈ Xps ⇐⇒ x ∈ Xps and limt→∞ exp(itξ)x ∈ Gcx if wµ(x, ξ) = 0.

(iv) x ∈ X s ⇐⇒ wµ(x, ξ) > 0 for all ξ ∈ Λ.

The original Hilbert–Mumford criterion is that v ∈ V us ⇐⇒ there exists anelement ξ ∈ Λ such that

limt→∞

exp(itξ)v = 0.

In the archetypal example take the north pole at the heavy cluster. The centerof mass will lie on the line through the north pole and the center of mass of theremaining points.5 The center of mass lies on the polar axis, at the origin inthe polystable case.

5Slogan: The center of mass of the centers of mass is the center of mass.

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Notes.

27. Remark. The integral of ω0 over S2 = P (C2) ⊆ P (V ) is 2π~. To see thislet u : D → S2N−1 by u(reiθ) = (reiθ,

√1− r2eiθ, 0, . . . , 0). Then

D

u∗π∗ω0 =

D

u∗∂∂K = − 12

D

u∗d(∂ − ∂)K = − 12

∂D

(∂ − ∂)u∗K.

where we used d(∂ − ∂) = (∂ + ∂)(∂ − ∂) = −2∂∂. Let z = reiθ. Thenu∗K = i~ log r2 = i~ log(zz) so on ∂D we have

(∂ − ∂)u∗K = i

(dz

z− dz

z

)

= −2 dθ

and hence∫

D

(π ◦ u)∗ω0 =

D

u∗π∗ω0 = − 12

∂D

(∂ − ∂)u∗K = π~.

28. Theorem of Mumford. (i) The map wµ : X × Λc → R is Gc-invariant,i.e. for all x ∈ X , ζ ∈ Λc, and g ∈ Gc,

wµ(gx, gζg−1) = wµ(x, ζ).

(ii) For every x ∈ X the map Λc → R : ζ 7→ wµ(x, ζ) is constant on theequivalence classes

ζ ∼ ζ′def⇐⇒ ∃ p ∈ P (ζ) such that pζp−1 = ζ′.

i.e. for all ζ ∈ Λc and p ∈ P (ζ),

wµ(x, pζp−1) = wµ(x, ζ).

Here P (ζ) is the parabolic subgroup

P (ζ) :={

p ∈ Gc | the limit limt→∞

exp(ıtζ)p exp(−ıtζ) exists in Gc}

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29. Kahler–Einstein. There is an analogy between µ-GIT and the Kahler–Einstein problem explained in Donaldson’s first paper [?] on this subject. Inµ-GIT the ingredients are

(A.1) The Kahler manifold (X,ω, J).

(A.2) A compact Lie group G acting on X by Kahler isometries.

(A.3) The moment map µ : X → g.

(A.4) The homogeneous space Gc/G

(A.5) The Kempf–Ness function Φx : Gc/G→ R.

In the Kahler Einstein problem fix a Kahler manifold (M,ω0, J0). Then thefollowing are analogous to (A.1)-(A.5).

(B.1) The space J (M,ω0) of ω0-compatible integrable complex structures on aKahler manifoldM is analogous to the Kahler manifold (X,ω, J) in (A.1).

(B.2) The group Ham(M,ω0) of Hamiltonian symplectomorphisms is analo-gous to the compact group G in (A.2).

(B.3) The map J (M,ω0) → C∞(M,R) : J 7→ sJ (the scalar curvature of theRiemannian metric of (ω0, J)) is analogous to the moment map µ in (A.3).

(B.4) The space of Kahler potentials H0 := {ϕ ∈ C∞(M,R), ω0 + i∂∂ϕ > 0}is analogous to the homogeneous space Gc/G in (A.4).

(B.5) TheMabuchi energyE0 : H0 → R (see below) is analogous to the Kempf–Ness function Φx in (A.5).

Here is the explanation of the analogy under the assumption π1(M) = 1,so Ham = Symp0. Fix a complex structure J0 ∈ J (M,ω0) and consider thecomplexified group orbit (there’s no complexified group here, only an orbit). Itis the space

J0 := {J ∈ J (M,ω0) ∃ψ ∈ Diff0(M) such that ψ∗J = J0} .

Assume first that J0 has no automorphisms. Then the diffeomorphism ψ = ψJ

is uniquely determined by J . Since the pair (ω0, J) is compatible, so is the pair((ψJ )

∗ω0, (ψJ )∗J) = ((ψJ )

∗ω0, J0). Thus there is a map

J0 → H0 : J 7→ ϕJ , ω0 + i∂∂ϕJ := (ψJ )∗ω0

which descends to a diffeomorphism J0/Ham ∼= H0. Now the Mabuchi energyE0 : H0 → R is precisely the function whose gradient is the scalar curvature, i.e.the moment map, i.e. E0 is our Φx0

. If J0 does have automorphisms, they won’tpreserve ω, so in this case H0 is a little bigger than J0/Ham, just like Gc/G

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is a little bigger that Gcx0/G. In fact there is then a map H0 → J0/Ham,given by Moser isotopy from ω0 to ω0 + i∂∂ϕ, whose fiber consists of thoseKahler potentials that come from diffeomorphisms, isotopic to the identity, thatpreserve J0. So even in the presence of automorphisms of J0, the space H0 ofKahler potentials is a perfect analogue of Gc/G and E0 is a perfect analogue ofΦx0

.

30. Tian discusses the Mabuchi energy (what he calls the K-energy) and metricsof constant scalar curvature on page 93 of [?]. He defines the Mabuchi energyby

E0(ϕ) := − 1

V

∫ 1

0

M

ϕt(s(ωt)− nν)ωnt ∧ dt

where (ϕt)t is any path in H0 with ϕ0 = 0, ϕ1 = ϕ,

ωt = ω0 + t∂∂ϕ,

s(ωt) is the scalar curvature of ωt, and

ν =c1(M) · [ω0]

n−1

[ωt]n.

31. Stable holomorphic vector bundles. An infinite dimensional analogueof geometric invariant theory is the correspondence between stable holomorphicvector bundles and hermitian Yang–Mills connections over Kahler manifoldsdue to Donaldson and Uhlenbeck–Yau. In this theory the space of hermitianconnections on a hermitian vector bundle over a closed Kahler manifold withcurvature of type (1, 1) is viewed as an infinite dimensional symplectic manifold,the group of unitary gauge transformations acts on it by Hamiltonian symplec-tomorphisms, the moment map assigns to a connection the component of thecurvature parallel to the symplectic form, and the zero set of the moment mapis the space of hermitian Yang–Mills connections. For bundles over Riemannsurfaces this is the picture exhibited by Atiyah–Bott [?] and they proved theanalogue of the moment-weight inequality in this setting. Still in the case ofbundles over Riemann surfaces the analogue of the Hilbert–Mumford numeri-cal criterion is the correspondence between stable bundles and flat connections,established by Narasimhan–Seshadri [?]. It can be viewed as an extension ofthe identification of the Jacobian with a torus to higher rank bundles. Anotherproof of the Narasimhan–Seshadri theorem was given by Donaldson [?]. In di-mension four the hermitian Yang–Mills conections are anti-self-dual instantonsover Kahler surfaces. In this setting the correspondence between stable bun-dles and ASD instantons was established by Donaldson [?] and used to provenontriviality of the Donaldson invariants. Donaldson’s theorem was extendedto higher dimensional Kahler manifolds by Uhlenbeck–Yau [?].

32. Scalar curvature is the moment map (formally). Another infinitedimensional analogue of GIT emerged in the study of the Kahler-Einstein equa-tions. It was noted by Fujiki [?] and Donaldson [?] that the scalar curvature

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can be interpreted as a moment map for the action of the group of Hamilto-nian symplectomorphisms of a Kahler manifold (X,ω0, J0) on the space J0 ofall ω0-compatible integrable complex structures. It was also noted by Donald-son [?] that in this setting the Futaki invariants [?] are the analogues of theMumford numerical invariants, that the space of Kahler potentials is the ana-logue of Gc/G, that the Mabuchi functional [?, ?] is the analogue of the thelog-norm function in the work of Kempf–Ness [?], and that Tians’s notion ofK-stability [?] can be understodd in terms of an infinite dimensional analogueof GIT. This led to Donaldson’s conjectures relating K-stability to the exis-tence of Kahler–Einstein metrics [?, ?, ?, ?] and refining earlier conjecturesby Tian [?] and Yau [?]. These conjectures can be viewed as the analoguesof the Hilbert–Mumford criterion for stability, with the µ-weights replaced byDonaldson’s generalized Futaki invariants. The conjecture that the appropriatecondition on the Donaldson–Futaki invariants implies the existence of a Kahler–Einstein metric in the Fano case (i.e. the analogue of part (iii) in Theorem ??)was confirmed by Chen–Donaldson–Sun [?, ?, ?] (and then also by Tian [?]with the same methods). The more general conjecture, in the nonFano case,that the same conditions on the Donaldson–Futaki invariants imply the exis-tence of a constant scalar curvature metric, is still open at the time of writing.The moment-weight inequality in this setting was proved by Donaldson [?] andChen [?].

In this situation the duality between the positive curvature manifold G andthe negative curvature manifold Gc/G is particularly striking. The analogue ofG is the group of Hamiltonian symplectomorphisms with the L2-inner producton the Lie algebra of Hamiltonian functions, and the analogue of Gc/G is thespace of Kahler potentials, also equipped with an L2 Riemannian metric. On onehand the distance function associated to the L2 metric on the group of Hamil-tonian symplectomorphisms is trivial by a result of Eliashberg–Polterovich [?].On the other hand Chen proved in [?] that in the space of Kahler potentialsany two points are connected by a unique geodesic of class C1,1 (a solution ofthe Monge–Ampere equation), that therefore the space of Kahler potentials isa genuine negatively curved metric space, and that constant scalar curvaturemetrics, when they exist, are unique in their Kahler class.

33. Calculations on S2

0 0 00 0 −10 1 0

010

=

010

, i× j = k;

0 0 −10 0 01 0 0

001

=

010

, j× k = i;

0 −1 01 0 00 0 0

100

=

010

, k× i = j.

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q =

xyz

∈ S2. q =

xyz

∈ TqS2.

ξ =

0 −c bc 0 −a

−b a 0

∈ so3, Jq =

0 −z yz 0 −x

−y x 0

Jq q = q × q =

yz − zyzy − xzxy − yx

Hξ(q) =12 (ax

2 + by2 + cz2)

ωq(q1, q2) = q · (q1 × q2)= x(y1z2 − z1y2) + y(z1x2 − x1z2) + z(y1x2 − x1y2)

Check:ωq(Xξ(q), q) = q · ((ξ × q)× q) = ξ · q = dHξ(q)q.

34. The complexified group. The map

U(n)× u(n) → GL(n,C) : (u, η) 7→ exp(iη)u

is a diffeomorphism, by polar decomposition, and the image of G× g under thisdiffeomorphism is denoted by

Gc := {hu |u ∈ G, h = exp(iη), η ∈ g} .

The action extends to a holomorphic action

Gc ×X → X : (g, x) 7→ gx

This is a Lie subgroup of GL(n,C) called the complexification of G. (Wealways denote elements of Gc by g and elements of G by u.) It contains G as amaximal compact subgroup and the Lie algebra of Gc is the complexification

gc := g⊕ ig

of the Lie algebra g of G. Equivalently, every Lie group homomorphism fromG into a complex Lie group extends uniquely to a homomorphism from Gc tothat complex Lie group. An element of ζ ∈ gc has form

ζ = ξ + iη, ℜ(ζ) := ξ, ℑ(ζ) := η.

where ξ, η ∈ g are the real and imaginary parts of ζ. The eigenvalues of ξ andη are imaginary and the eigenvalues of iξ and iη are real.

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