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 µ(x) 2, 2 | | and show that trajectories of the negative gradient fieldx ˙ = 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 2. K¨ahler manifolds. A K¨ahler manifold (X,ω,J) satisfies ω (J xˆ ,J xˆ )= ω (ˆx , xˆ ), xˆ 2 := ω (ˆx,J xˆ) > 0 x x 1 x 2 x 1 2 | |x x x for x X,x, ˆ xˆ , xˆ T X. Hence ∈ 1 2 ∈ x xˆ , xˆ := ω (ˆx ,J xˆ ) h 1 2ix x 1 x 2 is a Riemannian metric. A function H : X R has a symplectic gradient X Vect(X) and a Riemannian gradient→ H Vect(X) characterized by H ∈ ∇ ∈ dH(x)ˆx = ω (X , xˆ)= H, xˆ . x H h∇ ix They are related by the formula H = JX . ∇ H 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 example of a K¨ahler manifold. The symplectic⊆ form is the restriction to X of the Fubini– Study form π∗ω = ∂∂K,¯ K(v)= i~ log v . FS | | 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 U and its complexification Gc GL (C). • ⊆ N ⊆ N 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.

G A sequence of generators ϕ0,...,ϕn of R(X) 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 where all 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 gives equations for Y .

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

A compact symplectic manifold (X,ω). • A Lie algebra g u of G. • ⊆ N The ad(G)-invariant inner product ξ, η = trace(ξ∗η) on g. • h i A Hamiltonian action g Vect(X): ξ X . • → 7→ ξ An equivariant moment map µ : X g for this action. • → The norm squared function f : X R defined by f(x)= 1 µ(x) 2. • → 2 | |

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

H := µ, ξ , dH = ω(X , ). ξ h i ξ ξ · That µ is equivariant means that µ(ux)= uµ(x)u−1 for x X, u G. This implies ∈ ∈ µ(x), [ξ, η] = ω(X (x), X (x)) =: H ,H h i ξ η { ξ η} for ξ, η g.1 ∈ 6. K¨ahler Actions. A Hamiltonian action g Vect(X): ξ Xξ on a closed K¨ahler manifold extends to an action → 7→

gc Vect(X): ξ + iη X + JX → 7→ ξ η by holomorphic vector fields, so the action of G on X extends to a holomorphic action 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 = 1 iξv,v , v S(V ) := S2N−1. h ig 2 h iV ∈ Warning. Distinguish G and Gc. The eigenvalues of a real element ξ g are pure imaginary and the eigenvalues of a pure imaginary iη ig are real.∈ ∈ The closure of the 1-PSG R G : t exp(tξ) is a torus, generically a maximal torus, and sometimes a circle.→ 7→

1We use the sign convention

[v,w] := ∇wv −∇vw for the Lie bracket of two vector fields so that the infinitesimal action is a homomorphism g → Vect(X) of Lie algebras (and not an anti-homomorphism). 2By Weyl’s unitarian trick G-invariance implies Gc-invariance.

3 7. Rationality Theorem. A K¨ahler G-manifold is isomorphic to a projective G-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

1 ∗ −1 −1 µ(x0, u, x¯) := x¯ ω + µ(u(t) x0),u(t) u˙(t) dt A − ZD Z 0 is integral in the sense that

(x , u, x¯) 2π~Z Aµ 0 ∈ 2πit −1 whenever x0 X, u : R/Z G, andx ¯ : D X satisfyx ¯(e ) = u(t) x0. (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.) ∈ h i ∈ 8. Moreover Where Λ := ξ g 0 exp(ξ)=1 (see 24) { ∈ \{ } | } § (i) The action integral (x, u, v) is invariant under homotopy. Aµ (ii) There is an N N such that Nα = 0 for every torsion class α H (G; Z). ∈ ∈ 1 (iii) If ω, π2(X) 2π~NZ then there is a central element τ Z(g) such that the momenth mapi⊆µ τ satisfies condition (B). ∈ − (iv) Assume (A), (B), (C), and let τ Z(g) be a central element, so µ + τ is an 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 moment map µ is replaced by µ + τ and ∈

(x , u, x¯)= (x , u, x¯)+ τ, ξ . Aµ+τ 0 Aµ 0 h i This is because τ Λ and ξ Λ so τ, ξ 2π~Z by taking η = τ/2π~ in (C). ∈ ∈ h i ∈

4 9. The simplest K¨ahler manifold. The sphere S2 is a K¨ahler manifold. The complex structure, symplectic form, and Riemannian metric are J qˆ = q q,ˆ ω (ˆq , qˆ )= q (ˆq qˆ ), qˆ , qˆ =q ˆ qˆ q × q 1 2 · 1 × 2 h 1 2iq 1 · 2 2 2 3 for q S andq, ˆ qˆ1, qˆ2 TqS . With the identification so3 (R , ) the action is Hamiltonian∈ with ∈ ≃ × X (q)= ξ q, ξ, η = ξ η, H (q)= µ(q), ξ = q ξ. ξ × h i · ξ h i · (rotation about ξ). The gradient H (q)= J X (q) = (ξ q) q = ξ (ξ q)q ∇ ξ − q ξ × × − · of Hξ generates a north pole – south pole flow with poles on the axis of rotation of ξ. ∗ 2 3 10. The action of C on S Let ξ = (0, 0, 1) R so3 generate rotation ∈ 2 ≃ about the z-axis. Then Hξ(q)= z for q = (x,y,z) S and the ODEq ˙ = Xξ(q) is ∈ x˙ = y, y˙ = x. z˙ =0. − The gradient vector field is H (q)= ξ (ξ q)q ∇ ξ − · so the equationq ˙ = H (q) is −∇ ξ x˙ = zx, y˙ = zy, z˙ = z2 1. − The solution with q(0) = q0 is 2etx 2ety z +1+(z 1)e2t x = 0 , y = 0 , z = 0 0 − . z +1 (z 1)e2t z +1 (z 1)e2t z +1 (z 1)e2t 0 − 0 − 0 − 0 − 0 − 0 − 11. Another gradient flow on S2. The moment map squared for the S1 action on S2 is 1 2 f(q)= 2 z 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 along the meridians towards the equator.

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

5 12. The archetypal example. Let SO act diagonally on X = S2 S2 3 ×···× n with moment map | {z } n µ(x) := q , x = (q ,...,q ) X. i 1 n ∈ Xi=1 The preimage µ−1(0) consists of those x with center of mass at the origin. The moment map squared is

f(x)= 1 µ(x) 2, df(x)ˆx = 1 q qˆ 2 | | 2 i · j Xi=6 j and negative gradient flow of f is

q˙ = f(x)= (q (q q )q ). i −∇ − j − j · i i Xj=6 i 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 the components as roots of a binary form of degree n. However the roots of a form give an unordered set with multiplicities whereas a point is X is an ordered sequence. 13. Critical points of f where µ = 0. In the archetypal example we can characterize the critical points using Lagrange6 multipliers. Since n f(x)= 1 µ(x) 2 = + q q , 2 | | 2 i · j Xi

qj = λiqi Xj=6 i for i =1,...,n and when this holds, λ = q q so i j=6 i j · i P  df(x)=0 µ(x) = (λ + 1)q . ⇐⇒ i i Since qi = 1, the critical points x of f where µ(x) = 0 are the points x of form x = (q| ,...,q| ) where q = p for i =1,...,n for some6 p S2. 1 n i ± ∈ Assign to x X the equivalence relation (cycle structure of a permutation) ∈ i j q = q . ≡x ⇐⇒ i j This stratifies X. Each equivalence relation on 1,...,n determines a stratum. The Gc action preserves the stratification. The gradient{ flow} f also preserves −∇ the stratification because qi = qj = ( f(x))i = ( f(x))j , i.e. f is tangent to each stratum. Each stratum is foliated⇒ ∇ by Gc oribits.∇ ∇

6 14. The moment map squared. In the K¨ahler case define f : X R by → f(x) := 1 µ(x) 2 2 | | satisfies f −1(0) = µ−1(0) so any x µ−1(0) is a critical point of f (as it is an absolute 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 | | T (Gcv)= ζv, ζ gc ,T (Gv)= ξv, ξ g . v { ∈ } v { ∈ } c The derivative of the distance isv ˆ 2 v, vˆ V . It vanishes on Tg(G V ) iff v, vˆ = v,ζv = 0 for all ζ. But for7→ ζ =h ξ +i iη, ξ, η g we have h iV h iV ∈ 1 v, iζv = µ(x), iξ η c = µ(x), η =0 2 h iV h − ig − h ig 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) = x −∇ 0 converges, i.e. the limit x∞ := lim x(t) 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 X and the map ⊆ −1 U µ (0) : x x∞ → 0 → gives an isomorphism U/Gc µ−1(0)/G. (This is like R(Y )= R(X)G.) ≃

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

ξ + iη , ξ + iη c = ξ , ξ + η , η h 1 1 2 2ig h 1 2ig h 1 2ig on the tangent space gc to Gc at the identity. This Riemannian metric is invariant 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-invariant Riemannian metric of nonpositive curvature on Gc/G. 17. The Moment Conjugacy Theorem. Fix x X and define a map c c ∈ ψx : G G x X by → ⊆ −1 ψx(g)= g x. c Then there is a function Φx : G R such that ψx intertwines the two gradient vector fields Φ Vect(Gc) and→ f Vect(X), i.e. ∇ x ∈ ∇ ∈ dψ (g) Φ (g)= f(ψ (g)). ( ) x ∇ x ∇ x ♥ In particular, f is tangent to the Gc-orbits. ∇ Definition. The function Φ : Gc R will be called the lifted Kempf–Ness x → function based at x. (It is unique if normalized by the condition Φx(1) = 0.) It is G-invariant and hence descends to a function

Φ : Gc/G R x → denoted by the same symbol and called the Kempf–Ness function.

8 c 18. Proof of the Conjugacy Theorem. Define a vector field Fx Vect(G ) c ∈ and a one form αx on H by

F (g) := giµ(g−1x), α (g)ˆg = µ(g−1x), (g−1gˆ) x − x − ℑ for g Gc,g ˆ T Gc. Then ∈ ∈ g Step 1. There is a unique Φx such that Φx(1) = 0 and

dΦx = αx.

Step 2. The map ψ intertwines the vectorfields F and f. x x ∇

dψ (g)F (g)= f(ψ (g)). x x ∇ x

Step 3. The gradient Φ of Φ is F , i.e. ∇ x x x α (g)ˆg = F (g), gˆ x h x ig where the inner product on the right is the left-invariant inner Riemannian metric 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. ∈ ∈

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

F (gu)= guiµ(u−1g−1x)= guiu−1µ(g−1x)u = giµ(g−1x)u = F (g)u x − − − x 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)ˆx, ξ = dH (x)ˆx = ω(X (x), xˆ) h i ξ ξ for x X,x ˆ T X and ξ g so taking ξ = µ(x) we get ∈ ∈ x ∈ df(x)ˆx = dµ(x)ˆx,µ(x) = ω(X (x), xˆ) h i µ(x) and hence f(x)= J(x)X (x)= Xi (x). ∇ µ(x) µ(x) Now ifg ˆ T Gc, theng ˆ = gζ T Gc where ζ gc. Hence ∈ g ∈ g ∈ d d dψ (g)ˆg = ψ (g exp(tζ)x) = exp( tζ)g−1x = X (g−1x) x dt x dt − − ζ t=0 t=0 so takingg ˆ = F (g) (so that ζ = iµ(g−1x)) we get x − −1 −1 dψ (g)F (g)= X i −1 (g x)= Xi −1 (g x)= f(ψ (g)). x x − − µ(g x) µ(g x) ∇ x

−1 ∗ −1 Because µ(ux)= uµ(x)u and u = u for u G, the one-form αx is right ∈ c G-invariant and hence descends to a one-form (also denoted by αx) on G /G. c It is easy to show that αx is closed and hence (as G /G is contractible) αx is exact on Gc/G so its pull back to Gc is also exact. The Kempf-Ness function is defined by dΦx = αx, Φx(1) = 0. By ( ) to prove ( ) we must prove Φ = F , i.e. † ♥ ∇ x x α (g)ˆg = F (g), gˆ ( ) x h x ig ‡ where the inner product on the right is the left-invariant inner Riemannian c −1 −1 metric on G . By left invariance F (g), gˆ = iµ(g x),g gˆ c so ( ) h x ig − g ‡ follows because iξ, ζ = ξ, (ζ) when ξ g and ζ gc. h i h ℑ i ∈ ∈

10 19. The Kempf–Ness function for projective manifolds. The Kempf– Ness function Φ for a projective manifold X P (V ) is x ⊆ Φ (g)= 1 log g−1v log v (#) x 2 | |V − | |V
for g Gc where x = π(v) X P (V ). ∈ ∈ ⊆ Proof: Define Φx by (#). Then Φx(1) = 0 and

g−1ggˆ −1v,g−1v dΦ (g)ˆg = V . x − 2 g−1v 2 | |V The moment map µ : X g is characterized by the formula → µ(y), η = H (y)= 1 iηw,w , y = π(w), w 2 =1, η g. h ig η 2 h iV | |V ∈ (See 3.) Let ζ = g−1gˆ = ξ + iη where ξ, η g. Then § ∈ −1 −1 ζg v,g v V −1 −1 dΦ (g)ˆg = = iµ(g x), ζ c = µ(g x), η c x − 2 g−1v 2 − g − g | |V and η = (ζ)= (g−1gˆ) so dΦ = α . ℑ ℑ x x

20. Properties of the Kempf–Ness function. (i) The Kempf–Ness function Φ : Gc/G R is Morse–Bott (usually Morse). x → (ii) The critical set of Φx is a (possibly empty) closed connected submanifold of Gc/G. It is given by

Crit(Φ )= π(g) Gc/G, µ(g−1x)=0 . x ∈  (iii) If the critical manifold of Φx is nonempty, then it consists of the absolute minima of Φx and every negative gradient flow line of Φx converges exponentially to a critical point.

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

lim Φx(γ(t)) = inf Φx. t→∞ Gc/G

(The infimum may be minus infinity.)

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

Φ (g)=Φ (g) Φ (h−1) hx x − x for h G. ∈

11 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 Xss) iff Gcx µ−1(0) = , ∈ ∩ 6 ∅ (iii) µ-polystable (x Xps) iff Gcx µ−1(0) = , ∈ ∩ 6 ∅ (iv) µ-stable (x Xs) iff x is µ-polystable and gc = 0.4 ∈ x us In the archetypal example x X more than half the qi coincide and x Xps exactly half the points∈ coincide.⇐⇒ ∈ ⇐⇒ The Moment Limit Theorem. With x0 and x∞ as in 15, us § (i) x0 X if and only if µ(x∞) = 0. ∈ ss 6 (ii) x X if and only if µ(x∞) = 0. 0 ∈ ps c (iii) x0 X if and only if µ(x∞) = 0 and x∞ G x0. ∈ s ∈ (iv) x X if and only if g ∞ = 0. 0 ∈ x Moreover, Xss and Xs 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 mass need 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 Gc is discrete. ∈ v In the archetypal example x V us more than half the roots coincide, and x V ps exactly half the∈ roots⇐⇒ coincide. ∈ ⇐⇒

12 Kempf–Ness Theorem. The two notions of stability agree for projective space in the following sense. If x X P (V ), then ∈ ⊆ (i) x Xus if and only if π−1(x) V us. ∈ ⊆ (ii) x Xss if and only if π−1(x) V ss. ∈ ⊆ (iii) x Xps if and only if π−1(x) V ps. ∈ ⊆ (iv) x Xs if and only if π−1(x) V s. ∈ ⊆ 23. The Kempf–Ness Theorem generalized. For any K¨ahler G-manifold (X,ω,J,µ) the Kempf–Ness function Φx characterizes µ-stability as follows. (i) x is µ-unstable Φ is unbounded below. ⇐⇒ x (ii) x is µ-semistable Φ is bounded below. ⇐⇒ x (iii) x is µ-polystable Φ has a critical point. ⇐⇒ x (iv) x is µ-stable Φ is bounded below and proper. ⇐⇒ x In the archetypal example take V = Cn+1. A point v V is a binary form ∈ v(x, y)= v xn + v xn−1y + + v yn. 0 1 ··· n The north pole-south pole flow is

(exp(tξ)v)(x, y)= v(etx, e−ty).

The Kempf–Ness function is

Φ (et) = log( v ent 2 + v e(n−2)t 2 + + v e−nt 2) log( v 2). x | 0 | | 1 | ··· | n | − | |

13 24. Toral elements. A nonzero element ζ gc is called toral iff it satisfies the following equivalent conditions. ∈ ζ is semi-simple and has purely imaginary eigenvalues. • The subset T := exp(tζ) t R is a torus in Gc. • ζ { | ∈ } The element ζ is conjugate to an element of g. • Denote the set of toral elements by

c := ad(Gc)(g 0 ) T \{ } 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 torus • T G in a∪{ spanning} lattice. ⊂ ⊆ The generator of any one parameter subgroup C∗ Gc is conjugate to • an element of Λ t. → ∩ 25. µ-weights. For x X and ξ g 0 the µ-weight of the pair (x, ξ) is the real number ∈ ∈ \{ } wµ(x, ξ) := lim µ(exp(itξ)x), ξ . t→∞ h i In the projective case the µ-weight is

wµ(x, ξ)= ~ max λi vi=06 for x = π(v) where λ < < λ are the eigenvalues of iξ, • 1 ··· k V V are the corresponding eigenspaces, and • i ⊆ v = k v with v V . • i=1 i i ∈ i P

14 26. The Hilbert–Mumford criterion. The µ-weight characterizes µ-stability as follows. (i) x Xus there exists a ξ Λ such that w (x, ξ) < 0. ∈ ⇐⇒ ∈ µ (ii) x Xss w (x, ξ) 0 for all ξ Λ. ∈ ⇐⇒ µ ≥ ∈ ps ps c (iii) x X x X and lim →∞ exp(itξ)x G x if w (x, ξ) = 0. ∈ ⇐⇒ ∈ t ∈ µ (iv) x Xs w (x, ξ) > 0 for all ξ Λ. ∈ ⇐⇒ µ ∈ The original Hilbert–Mumford criterion is that v V us there exists an element ξ Λ such that ∈ ⇐⇒ ∈ lim exp(itξ)v =0. t→∞ In the archetypal example take the north pole at the heavy cluster. The center of mass will lie on the line through the north pole and the center of mass of the remaining points.5 The center of mass lies on the polar axis, at the origin in the polystable case.

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

15 Notes.

2 2 27. Remark. The integral of ω0 over S = P (C ) P (V ) is 2π~. To see this let u : D S2N−1 by u(reiθ) = (reiθ, √1 r2eiθ, 0,...,⊆ 0). Then → − ∗ ∗ ∗ ¯ 1 ∗ ¯ 1 ¯ ∗ u π ω0 = u ∂∂K = 2 u d(∂ ∂)K = 2 (∂ ∂)u K. ZD ZD − ZD − − Z∂D − where we used d(∂ ∂¯) = (∂ + ∂¯)(∂ ∂¯) = 2∂∂¯. Let z = reiθ. Then u∗K = i~ log r2 = i~−log(zz¯) so on ∂D we− have −

dz¯ dz (∂ ∂¯)u∗K = i = 2 dθ −  z¯ − z  − and hence

∗ ∗ ∗ 1 ¯ ∗ ~ (π u) ω0 = u π ω0 = 2 (∂ ∂)u K = π . ZD ◦ ZD − Z∂D −

c c 28. Theorem of Mumford. (i) The map wµ : X Λ R is G -invariant, i.e. for all x X, ζ Λc, and g Gc, × → ∈ ∈ ∈ −1 wµ(gx,gζg )= wµ(x, ζ).

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

ζ ζ′ def p P (ζ) such that pζp−1 = ζ′. ∼ ⇐⇒ ∃ ∈ i.e. for all ζ Λc and p P (ζ), ∈ ∈ −1 wµ(x,pζp )= wµ(x, ζ).

Here P (ζ) is the parabolic subgroup

P (ζ) := p Gc the limit lim exp(ıtζ)p exp( ıtζ) exists in Gc n ∈ | t→∞ − o

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

(A.1) The K¨ahler manifold (X,ω,J). (A.2) A compact Lie group G acting on X by K¨ahler isometries. (A.3) The moment map µ : X g. → (A.4) The homogeneous space Gc/G (A.5) The Kempf–Ness function Φ : Gc/G R. x →

In the K¨ahler Einstein problem fix a K¨ahler manifold (M,ω0,J0). Then the following are analogous to (A.1)-(A.5).

(B.1) The space (M,ω0) of ω0-compatible integrable complex structures on a K¨ahler manifoldJ M is analogous to the K¨ahler 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 (M,ω ) C∞(M, R): J s (the scalar curvature of the J 0 → 7→ J Riemannian metric of (ω0,J)) is analogous to the moment map µ in (A.3).

∞ (B.4) The space of K¨ahler potentials 0 := ϕ C (M, R),ω0 + i∂∂ϕ¯ > 0 is analogous to the homogeneousH space G{c/G∈ in (A.4). } (B.5) The Mabuchi energy E : R (see below) is analogous to the Kempf– 0 H0 → 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 (M,ω0) and consider the complexified group orbit (there’s no complexified∈ group J here, only an orbit). It is the space

:= J (M,ω ) ψ Diff (M) such that ψ∗J = J . J0 { ∈ J 0 ∃ ∈ 0 0}

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

: J ϕ , ω + i∂∂ϕ¯ := (ψ )∗ω J0 → H0 7→ J 0 J J 0 which descends to a diffeomorphism 0/Ham ∼= 0. Now the Mabuchi energy E : R is precisely the functionJ whose gradientH is the scalar curvature, i.e. 0 H0 → the moment map, i.e. E0 is our Φx0 . If J0 does have automorphisms, they won’t preserve ω, so in this case is a little bigger than /Ham, just like Gc/G H0 J0

17 is a little bigger that Gcx /G. In fact there is then a map /Ham, 0 H0 → J0 given by Moser isotopy from ω0 to ω0 + i∂∂ϕ¯ , whose fiber consists of those K¨ahler potentials that come from diffeomorphisms, isotopic to the identity, that preserve J0. So even in the presence of automorphisms of J0, the space 0 of c H K¨ahler potentials is a perfect analogue of G /G and E0 is a perfect analogue of

Φx0 . 30. Tian discusses the Mabuchi energy (what he calls the K-energy) and metrics of constant scalar curvature on page 93 of [?]. He defines the Mabuchi energy by 1 1 n E0(ϕ) := ϕ˙ t(s(ωt) nν)ωt dt −V Z0 ZM − ∧ where (ϕ ) is any path in with ϕ = 0, ϕ = ϕ, t t H0 0 1

ωt = ω0 + t∂∂ϕ,¯ s(ωt) is the scalar curvature of ωt, and

n−1 c1(M) [ω0] ν = · n . [ωt] 31. Stable holomorphic vector bundles. An infinite dimensional analogue of geometric invariant theory is the correspondence between stable holomorphic vector bundles and hermitian Yang–Mills connections over K¨ahler manifolds due to Donaldson and Uhlenbeck–Yau. In this theory the space of hermitian connections on a hermitian vector bundle over a closed K¨ahler manifold with curvature 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 the curvature parallel to the symplectic form, and the zero set of the moment map is the space of hermitian Yang–Mills connections. For bundles over Riemann surfaces this is the picture exhibited by Atiyah–Bott [?] and they proved the analogue of the moment-weight inequality in this setting. Still in the case of bundles 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 of the identification of the Jacobian with a torus to higher rank bundles. Another proof of the Narasimhan–Seshadri theorem was given by Donaldson [?]. In di- mension four the hermitian Yang–Mills conections are anti-self-dual instantons over K¨ahler surfaces. In this setting the correspondence between stable bun- dles and ASD instantons was established by Donaldson [?] and used to prove nontriviality of the Donaldson invariants. Donaldson’s theorem was extended to higher dimensional K¨ahler manifolds by Uhlenbeck–Yau [?]. 32. Scalar curvature is the moment map (formally). Another infinite dimensional analogue of GIT emerged in the study of the K¨ahler-Einstein equa- tions. It was noted by Fujiki [?] and Donaldson [?] that the scalar curvature

18 can be interpreted as a moment map for the action of the group of Hamilto- nian symplectomorphisms of a K¨ahler manifold (X,ω ,J ) on the space of 0 0 J0 all ω0-compatible integrable complex structures. It was also noted by Donald- son [?] that in this setting the Futaki invariants [?] are the analogues of the Mumford numerical invariants, that the space of K¨ahler potentials is the ana- logue of Gc/G, that the Mabuchi functional [?, ?] is the analogue of the the log-norm function in the work of Kempf–Ness [?], and that Tians’s notion of K-stability [?] can be understodd in terms of an infinite dimensional analogue of GIT. This led to Donaldson’s conjectures relating K-stability to the exis- tence of K¨ahler–Einstein metrics [?, ?, ?, ?] and refining earlier conjectures by Tian [?] and Yau [?]. These conjectures can be viewed as the analogues of the Hilbert–Mumford criterion for stability, with the µ-weights replaced by Donaldson’s generalized Futaki invariants. The conjecture that the appropriate condition on the Donaldson–Futaki invariants implies the existence of a K¨ahler– 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 [?] and Chen [?]. In this situation the duality between the positive curvature manifold G and the negative curvature manifold Gc/G is particularly striking. The analogue of G is the group of Hamiltonian symplectomorphisms with the L2-inner product on the Lie algebra of Hamiltonian functions, and the analogue of Gc/G is the space of K¨ahler potentials, also equipped with an L2 Riemannian metric. On one hand 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 K¨ahler potentials any two points are connected by a unique geodesic of class C1,1 (a solution of the Monge–Ampere equation), that therefore the space of K¨ahler potentials is a genuine negatively curved metric space, and that constant scalar curvature metrics, when they exist, are unique in their K¨ahler class. 33. Calculations on S2 00 0 0 0  0 0 1   1  =  1  , i j = k; 01− 0 0 0 ×       0 0 1 0 0 −  00 0   0  =  1  , j k = i; 10 0 1 0 ×       0 1 0 1 0 −  1 00   0  =  1  , k i = j. 0 00 0 0 ×      

19 x xˆ 2 2 q =  y  S . qˆ =  yˆ  TqS . z ∈ zˆ ∈     0 c b 0 z y − − ξ =  c 0 a  so3, Jq =  z 0 x  b a −0 ∈ y x −0  −   −  yzˆ zyˆ − Jqqˆ = q qˆ =  zyˆ xzˆ  × xyˆ − yxˆ  −  1 2 2 2 Hξ(q)= 2 (ax + by + cz )

ωq(ˆq1, qˆ2) = q (ˆq1 qˆ2) = x(ˆ·y zˆ × zˆ yˆ )+ y(ˆz xˆ xˆ zˆ )+ z(ˆy xˆ xˆ yˆ ) 1 2 − 1 2 1 2 − 1 2 1 2 − 1 2 Check: ω (X (q), qˆ)= q ((ξ q) qˆ)= ξ qˆ = dH (q)ˆq. q ξ · × × · ξ 34. The complexified group. The map

U(n) u(n) GL(n, C) : (u, η) exp(iη)u × → 7→ is a diffeomorphism, by polar decomposition, and the image of G g under this diffeomorphism is denoted by ×

Gc := hu u G, h = exp(iη), η g . { | ∈ ∈ } The action extends to a holomorphic action

Gc X X : (g, x) gx × → 7→ This is a Lie subgroup of GL(n, C) called the complexification of G. (We always denote elements of Gc by g and elements of G by u.) It contains G as a maximal 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 from G into a complex Lie group extends uniquely to a homomorphism from Gc to that complex Lie group. An element of ζ gc has form ∈ ζ = ξ + iη, (ζ) := ξ, (ζ) := η. ℜ ℑ where ξ, η g are the real and imaginary parts of ζ. The eigenvalues of ξ and η are imaginary∈ and the eigenvalues of iξ and iη are real.

20 References

[1] Michael Atiyah & Raoul Bott, The Yang-Mills equations over Riemann surfaces. Phil. Trans. Roy. Soc. London A 308 (1982), 523–615. [2] Edward Bierstone & Pierre E. Milman, Semianalytic and subanalytic sets. Publications Math´ematiques de l’I.H.E.S.´ 67 (1988), 5–42. [3] Eugenio Calabi, The space of K¨ahler metrics. Proceedings of the International Congress of Mathematicians, Amsterdam 2 (1954), 206–207. [4] Eugenio Calabi, On K¨ahler manifolds with vanishing canonical class, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, edited by Ralph H. Fox, D.C. Spencer, A.W. Tucker, Princeton Mathematical Series 12, Princeton University Press, 1957, pp 78–89. [5] Eugenio Calabi, Extremal K¨ahler metrics. Seminar on differential geometry, edited by Yau et al, Annals of Math. Studies 102, Princeton University Press, 1982. [6] Eugenio Calabi & Xiuxiong Chen, Space of K¨ahler metrics and Calabi flow. Journal of Dif- ferential Geometry 61 (2002), 173–193. [7] Bill Casselmann, Symmetric spaces of semi-simple groups. Essays on representations of real groups, August 2012. http://www.math.ubc.ca/~cass/research/pdf/Cartan.pdf [8] Xiuxiong Chen, Space of K¨ahler metrics. Journal of Differential Geometry 56 (2000), 189– 234. [9] Xiuxiong Chen, Space of K¨ahler metrics III – On the lower bound of the Calabi energy and geodesic distance. Inventiones Mathematicae 175 (2009), 453–503. [10] Xiuxiong Chen, Space of K¨ahler metrics IV – On the lower bound of the K-energy. September 2008. http://arxiv.org/abs/0809.4081v2 [11] Xiuxiong Chen & Simon Donaldson & Song Sun, K¨ahler-Einstein metrics and stability. In- ternational Mathematics Research Notices 2013. Preprint, 28 October 2013. http://arxiv.org/abs/1210.7494. [12] Xiuxiong Chen & Simon Donaldson & Song Sun, K¨ahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities. Preprint, 19 November 2012. http://arxiv.org/abs/1211.4566 [13] Xiuxiong Chen & Simon Donaldson & Song Sun, K¨ahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2π. Preprint, 19 December 2012. http://arxiv.org/abs/1212.4714 [14] Xiuxiong Chen & Simon Donaldson & Song Sun, K¨ahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof. Preprint, 1 February 2013. http://arxiv.org/abs/1302.0282 [15] Xiuxiong Chen & Song Sun, Space of K¨ahler metrics V – K¨ahler quantization. April 2010. http://arxiv.org/abs/0902.4149v2 [16] Xiuxiong Chen & Song Sun, Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K¨ahler metrics. Preprint, April 2010, revised January 2012. http://arxiv.org/abs/1004.2012 [17] Xiuxiong Chen & Gang Tian, Geometry of K¨ahler metrics and foliations by holomorphic discs, Publications Math´ematiques de lI.H.E.S.´ 107 (2008), 1–107. http://arxiv.org/abs/math/0507148.pdf [18] Simon Donaldson, A new proof of a theorem of Narasimhan and Seshadri. Journal of Differ- ential Geometry 18 (1983), 269–277. [19] Simon Donaldson, Anti-self-dual Yang-Mills connections on complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. 50 (1985), 1–26. [20] Simon Donaldson, Moment Maps and Diffeomorphisms. Asian Journal of Mathematics 3 (1999), 1–16. [21] Simon Donaldson, Symmetric spaces, K¨ahler geometry and Hamiltonian dynamics. Northern California Symplectic Geometry Seminar, edited by Eliashberg et al, Amer. Math. Soc. Transl. Ser. 2, 196, 1999, 13–33. [22] Simon Donaldson, Scalar curvature and projective embeddings, I. Journal of Differential Geometry 59 (2001), 479–522. [23] Simon Donaldson, Conjectures in K¨ahler geometry. Strings and Geometry, Proceedings of the Clay Mathematics Institute 2002 Summer School on Strings and Geometry, Isaac New- ton Institute, Cambridge, United Kingdom, March 24–April 20, 2002. Clay Mathematical Proceedings 3, AMS 2004, pp 71–78. [24] Simon Donaldson, Scalar curvature and stability of toric varieties. Journal of Differential Geometry 62 (2002), 289–349. [25] Simon Donaldson, Lower bounds on the Calabi functional. Journal of Differential Geometry 70 (2005), 453–472.

21 [26] Simon Donaldson, Scalar curvature and projective embeddings, II. Quarterly Journal of Mathematics 56 (2005), 345–356. [27] Simon Donaldson, Lie algebra theory without algebra. In “Algebra, Arithmetic, and Geom- etry, In honour of Yu. I. Manin”, edited by Yuri Tschinkel and Yuri Zarhin, Progress in Mathematics, Birkh¨auser 269, 2009, pp 249–266. [28] Simon Donaldson & Song Sun, Gromov–Hausdorff limits of K¨ahler manifolds and algebraic geometry. Preprint, 12 June 2012. http://arxiv.org/abs/1206.2609 [29] Yasha Eliashberg & Leonid Polterovich, Bi-invariant metrics on the group of Hamiltonian diffeomorphisms. International Journal of Mathematics 4 (1993), 727–738. [30] Akira Fujiki, Moduli spaces of polarized algebraic varieties and K¨ahler metrics. Sˆugaku 42 (1990), 231–243. English Translation: Sˆugaku Expositions 5 (1992), 173–191. [31] Akito Futaki, An obstruction to the existence of Einstein-K¨ahler metrics. Inventiones Math- ematicae 73 (1983), 437–443. [32] Victor Guillemin & Shlomo Sternberg, Geometric quantization and multiplicities of group represnetations. Inventiones Mathamticae 67 (1982), 515–538. [33] Alain Haraux, Some applications of the Lojasiewicz gradient inequality. Communications on Pure and Applied Analysis 6 (2012), 2417–2427. [34] George Kempf, Instability in invariant theory. Annals of Mathematics 108 (1978), 299–317. [35] George Kempf & Linda Ness, The length of vectors in representation spaces. Springer Lecture Notes 732, Algebraic Geometry, Proceedings, Copenhagen, 1978, pp 233–244. [36] Frances Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry. Princeton University Press, 1984. [37] Eugene Lerman, Gradient flow of the norm squared of the moment map. L’Enseignement Math´emathique 51 (2005), 117–127. [38] Toshiki Mabuchi, K-energy maps integrating Futaki invariants, Tohoku Math. Journal 38 (1986), 575–593. [39] Toshiki Mabuchi, Einstein–K¨ahler forms, Futaki invariants and convex geometry on toric Fano varieties. Osaka Journal of Mathematics 24 (1987), 705–737. [40] Dusa McDuff & Dietmar Salamon, J-holomorphic Curves and Symplectic Topology, Second Edition. AMS Colloquium Publication 52, 2012. [41] David Mumford & John Fogarty & Frances Kirwan, Geometric Invariant Theory, Third Enlarged Edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer Verlag, New York, 1994. [42] Mudumbai S. Narasimhan & Conjeevaram S. Seshadri, Stable and unitary vector bundles on compact Riemann surfaces. Annals of Mathematics 82 (1965), 540–567. [43] Linda Ness, Mumford’s numerical function and stable projective hypersurfaces. Springer Lec- ture Notes 732, Algebraic Geometry, Proceedings, Copenhagen, 1978, pp 417–454. [44] Linda Ness, A stratification of the null cone by the moment map. American Journal of Mathematics 106 (1984), 1281–1329. [45] Sean Paul, Geometric analysis of Chow Mumford stability, Advances in Mathematics, 182 (2004) no. 2, 333-356. [46] Sean Paul, Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics, Annals of Mathematics, 175 (2012), 255-296. [47] Joel Robbin & Dietmar Salamon, Introduction to Differential Geometry, 2013. http://www.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf [48] Gabor Szekelyhidi, Extremal metrics and K-stability. PhD thesis, Imperial College, London, 2006. http://arxiv.org/pdf/math.DG/0611002.pdf [49] Richard P. Thomas, Notes on GIT and symplectic reduction for bundles and varieties. Surveys in Differential Geometry 10, A tribute to S.-S. Chern, edited by Shing-Tung Yau, Intern. Press, 2006. http://arxiv.org/abs/math.AG/0512411 [50] Gang Tian, Extremal metrics and geometric stability, Houston Journal of Mathematics 28 (2002) no.2, 411-432. [51] Gang Tian, K¨ahler-Einstein metrics with positive scalar curvature. Inventiones Mathematicae 130 (1997), 1–37. [52] Gang Tian: Canonical Metrics in K¨ahler Geometry, (Notes by Meike Akveld) Lectures in Mathematics ETH Z¨urich, Birk¨auser, 2000. [53] Karen Uhlenbeck & Shing-Tung Yau, On the existence of Hermitian Yang-Mills connections in stable vector bundles. Communications on Pure and Applied Mathematics 39 (1986), 257–293. [54] Lijing Wang, Hessians of the Calabi Functional and the Norm Function. Annals of Global Analysis and Geometry 29 (2006), 187–196. [55] Christopher T. Woodward, Moment maps and geometric invariant theory. June 2011. http://arxiv.org/abs/0912.1132v6

22 [56] Shing-Tung Yau, Calabi’s conjecture and some new results in algebraic geometry. Proceedings of the National Academy of Sciences of the United States of America 74 (1977), 1798–1799. [57] Shing-Tung Yau, On the Ricci curvature of a compact K¨ahler manifold and the complex Monge- Amp`ere equation I. Communications in Pure and Applied Mathematics 31 (1978), 339–411. [58] Shing-Tung Yau, Open problems in geometry. L’Enseignement Math´emathique 33 (1987), 109–158.

23