Lecture 35 Let G be a compact connected Lie g roup and n = dim G, with Lie algebra g. We have a group lattice ZG ⊂ g, and the dual Z∗G ⊂ g∗ the weight lattice. Then G = g/ZG. We can define exp : g → g/ZG. Take elements αi ∈ Z∗ G, i = 1, . . . , d then we get a representation τ : G GL(d, C) given by →τ(exp v)z = (e √−1α1(v)z1, . . . , e √−1αd zd). We can think of τ as an action. As such it preserves the Kaehler form ω = √−1� dzi ∧ dz¯i In fact, τ is Hamiltonian with momen t map Φ : Cd → g∗, Φ(z) = � |zi2αi|Note that α1, . . . , αd are polarized if and only if there exists a v ∈ g such that αi(v) > 0 for all i. Theorem. αis are polarized if and only if Φ, the moment map, is proper. What are the regular values of Φ? Let Rd = {(t1, . . . , td) ∈ Rd, ti ≥ 0} + and take I ⊆ {1, . . . , d}. Notation. Rd = t ∈ Rd, ti = 0I { � ⇔ i ∈ I} Consider the following maps: L : Rd → g∗ given by t �→� tiαi and γ : Cd Rd + given by →2, . . . , zn).z �→ (|z1| 2 | | Then for any a ∈ g∗, let Δa = L−1(a) ∩ Rd +. Then Φ = L γ, so z ∈ Φ−1(a) if and only if γ(z) ∈ Δa.◦Suppo se that the αis a re polarized. Then Δa is a compact convex set, and in fact it is a convex polytope Definition. The index set of a polytope is defined to be = = Δa I ∩ ΔaI {I | Rd � 0} The faces of the polytope Δa are the sets ΔI = Δa ∩ Rd I , ΔI ∈ ITheorem (1). Let a ∈ g∗. Then (a) a is a regular value of Φ if and only if for every I ∈ IΔa spanR{ai, i ∈ I} = g∗ (b) G acts freely on Φ−1(a) if and only if for all I ∈ IΔa spanZ{ai, i ∈ I} = Z∗ G Δ is partially order by inclusion, i.e. I1 < I2 if I1 ⊆ I2. Δ is minimal iff the corresponding face ΔI is a vertex of Δa, i.e. ΔI = vI } where vI is a vertex of Δa. I ∈ I{Theorem (2). (a) a is a regular value of the moment map Φ if and only if for every vertex vI of Δa, αi, i ∈ I are a basis of g∗. I(b) G acts freely on Φ−1(a) if andonly if for every vertex vI of Δa, αi, i ∈ I are a lattice basis for Z∗G. Proof. In Theorem 1 it suffices to check a) and b) for the minimal elements I of IΔ. Check that a) of Thm. 1 implies b) of Thm 2. So we just have to check a) of Thm. 2. Let ΔI = {vI }, where I is a minimal element of IΔ. By Thm 1., span{αi, i ∈ I} = g∗. Suppose αis are not a basis, then there exist ci so that � ciαi = 0 i∈I Now, vI = (t1, . . . , td), ti > 0 for i ∈ I and ti = 0 for i /∈ I. Define (s1, . . . , sd) ∈ Δa by �ti + ǫci i ∈ I si = 0 i /∈ I Then L(s) = a, s ∈ ΔI , so this contradicts that ΔI is a sing ular point. Notation. Δ ∈ Rd a convex polytope, v, v′ ∈ V ert(Δ). Then v and v′ are adjacent if they lie on a commo n edge of Δ. Definition. An m-dimensional p olytope Δ is simple if for every vertex v there are exactly m vertices adjacent to it. [Next time we’ll show that a is a regular value of Φ iff Δa is simple] Example. A tetrahedron or a cube in R3 . A pyramid is not simple. Φ : Cd → g∗, and a a re gular value. G acts freely on Za = Φ−1(a). Then we can form the symplectic quotient Ma = Φ−1(a)/G, which is a compact Kaehler manifold. We want to compute the de Rham and Dolbeault cohomology groups, H∗a), H∗a). To compute the de Rham cohomology we’re going to DR(MDo(Muse Morse Theory. A Digression on Morse Theory Let Mm be a compact C∞ manifold and let f : M R be a smooth function. →p ∈ Crit(f) if and only if dfp = 0 (by definition). For any p ∈ Crit(f) we have the Hessian d2fp a quadratic form o n Tp. Let (U, x1, . . . , xn) be a coordinate patch centered at p. Then 3 3f(x) = c + � aij xixj + O(x ) = d2fp + O(x ) and p is called non-degenerate if d2fp is non-degenerate. If p is a non-degenerate critical point, then p is isolated. Definition. f is Morse if all p ∈ Crit(f ) are non-degenerate, which implies that #Crit(f) < ∞ Definition. p ∈ Crit(f) then indp = ind d2fp, i.e. if 2 2 2d2fp = −(x1 + + xk) + x 2 + xk+1 + m··· ···then ind d2fp = k. Theorem. Let f : M → R be a Morse function with the property tat ind p is even for all p ∈ Crit(f). Then H2k+1(M) = 0 H2k(M) = {p ∈ Crit(f), ind p = 2k}��� �� � �Back to Symplectic Reduction Again, we’re talking about the moment map Φ : Cd g∗, with a a regular value of Φ. G acts freely on Za→and let Ma = Za/G. Then we have the following diagram: i ��CnZa π Ma and the mapping γ : Cd Rd , . . . , zd|2). γ is G-invariant. This implies that there exists ψ : Ma → +, z �→ (|z1|2 Rd |with the property that ψ π = γ i. Moreover γ : Za Δa.→ + ◦ ◦ →So ψ : Ma Δa, Δa is called the moment polytope.→Now take ξ ∈ Rd and let f : Ma R be f(p) = ψ(p), ξ , i.e. π∗f = i∗f0 where→ � �2f0(z) = � ξi|zi| Theorem (Mai n Theorem). Assume for v, v′ ∈ V ert(Δa), v, v′ adjacent that v − v ′ , ξ = 0 then (a) f : Ma R is Morse→ (b) ψ : Ma Δa maps Crit(f) bijectively onto V ert(Δa).→ (c) For p ∈ Crit(f) and v the corresponding vertex let v1, . . . , vm be the vertices adjacent to v. Then indp = #{vi vi − v, ξ < 0} := indv ξ 2 | � �Corollary. H2k+1(Ma) = 0 then bk = H2k (Ma) = #{v ∈ V ert(Δa), indv ξ = k} that is, bk is independent of
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