Symplectic Topology Example Sheet 7 Dietmar Salamon ETH Z¨urich 11 April 2013 Exercise 7.1. Let (V; !) be a symplectic vector space and consider the space 8 2 9 J = −1l; < = J (V; !) := J 2 End(V ) !(·;J·) + !(J·; ·) = 0; : !(v; Jv) > 0 8 v 2 V n f0g ; of !-compatible linear complex structure on V . (i) Prove that the formula 1 Jb1; Jb2 := 2 trace(Jb1Jb2) (1) for Jbi 2 TJ J (V; !) defines a Riemannian metric on J (V; !) and the formula 1 ΩJ (Jb1; Jb2) := 2 trace(Jb1JJb2) (2) defines a nondegenerate 2-form that is compatible with the Riemannian ma- tric (1) and the almost complex structure Jb 7! −JJb. (ii) Prove that the formulas (1) and (2) define a K¨ahlerstructure on J (V; !). Hint: Show that the Levi-Civita connection of the metric (1) is given by 1 rtJb = @tJb− 2 J (@tJ)Jb+ Jb(@tJ) (3) for every smooth curve R !J (M; !): t 7! J(t) and every smooth vector field Jb(t) 2 TJ(t)J (V; !) along this curve. Show that it preserves the almost complex structure Jb 7! −JJb. 1 2n Exercise 7.2. Prove that the diffeomorphism Sn !J (R ;!0): Z 7! J(Z) in Exercise 2.11 is a K¨ahlerisometry, where Sn denotes Siegel upper half space with its standard complex structure and the Riemannian metric on Sn is determined by the norm s 2 2 −1 −1 jZbjZ := trace XYb + YYb : Exercise 7.3. (i) Prove that the Riemann curvature tensor of the Rieman- nian metric (1) on J (V; !) is given by 1 R(Jb1; Jb2)Jb3 = − Jb1; Jb2 ; Jb3 : (4) 2 Deduce that J (V; !) has nonpositive sectional curvature. (ii) Prove that the geodesics on J (M; !) are given by t 7! J exp(−JJtb ) for Jb 2 TJ J (V; !) and t 2 R. Deduce that J (V; !) is geodesically complete. (iii) Prove that the exponential map TJ J (V; !) !J (V; !): Jb 7! J exp(−JJb) (5) is a diffeomorphism for every J 2 J (V; !). Exercise 7.4. Let (M; !) be a compact symplectic manifold and denote by J (M; !) the space of !-compatible almost complex structures on M. Fix an element J0 2 J (M; !) and define the distance function d : J (M; !) × J (M; !) by 1 X −k kJb1 − Jb2kCk d(J1;J2) := 2 ;Ji =: J0 exp(−J0Jbi): k=0 1 + kJb1 − Jb2kCk (Define what is meant by the Ck-norm of a section of a vector bundle.) Prove that (J (M; !); d) is a complete metric space. Prove that a sequence Jν 2 J (M; !) converges to J 2 J (M; !) with respect to the metric d if and only if it converges to J with respect to the C1 topology. Deduce that every countable intersection of open and dense (both with respect to the C1 topology) subsets of J (M; !) is dense. 2 Exercise 7.5. Let X, Y , Z be Banach spaces and let D : X ! Y and Φ: Z ! Y be bounded linear operators such that D has a closed image and the operator D ⊕ Φ: X × Z ! Y; (D ⊕ Φ)(x; z) := Dx + Φz is surjective. Consider the bounded linear operator Π : W ! Z defined by W := ker(D ⊕ Φ) = f(x; z) 2 X × Z j Dx + Φz = 0g ; Π(x; z) := z: Prove that Π has a closed image and that the linear operators Z Y ker D ! ker Π : x 7! (x; 0); ! :[z] 7! [Φz] im Π im D are (well defined) Banach space isomorphisms. Exercise 7.6. Let (M; J) be an almost complex manifold, (Σ; j) be a Rie- mann surface, and u :Σ ! M be a J-holomorphic curve. Denote by 0 ∗ 0;1 ∗ Du :Ω (Σ; u TM) ! ΩJ (Σ; u TM) the linearized operator defined by 1 D u := 1 (ru + J(u)ru ◦ j) − J(u)(r J(u)) @ (u) ub 2 b b 2 ub J 0 ∗ for ub 2 Ω (Σ; u TM). (See Exercises 6.6 and 6.8.) Prove that ¯ Du(du · ξ) = du · @jξ 0 ¯ 0 0;1 for ξ 2 Vect(Σ) = Ω (Σ;T Σ). Here @j :Ω (Σ;T Σ) ! Ωj (Σ;T Σ) is the Cauchy{Riemann on the tangent bundle of Σ, determined by the complex structrure j. Exercise 7.7. Let (Σ; j) be closed connected Riemann surfaces of positive 1 genus and define M := Σ × CP with the product complex structure J. 1 (i) Let u : CP ! M be a J-holomorphic curve of the form u(z) := (p; φ(z)) 1 ∼ 1 1 for z 2 CP = C [ f1g, where p 2 Σ and φ : CP ! CP is a rational map of degree d. Compute the Fredholm index of Du and show that Du is surjective. (ii) Let u :Σ ! M be the J-holomorphic curve u(z) := (z; q) for z 2 Σ, 1 where q 2 CP . Compute the Fredholm index of Du and show that Du is not surjective. 3.