1. GIT and Μ-GIT

1. GIT and µ-GIT Valentina Georgoulas Joel W. Robbin Dietmar A. Salamon ETH Zürich UW Madison ETH Zürich 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 symplectic 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 Kähler 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 Kähler 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 Kähler 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. Kähler manifolds. A Kähler 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 Kähler 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 manifold, 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. Kähler Actions. A Hamiltonian action g Vect(X): ξ Xξ on a closed Kähler 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 Kähler 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 Kähler manifold. The sphere S2 is a Kähler 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.

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