Poisson and Foliated Complex Structures - Paolo Caressa

12 Paolo Caressa and Adriano TomassiniTheorem 4.7 If there exists a Poisson structure π on M with Hamiltonianfoliation given by F which is almost Poisson–Kähler (w.r.t. J) then the followinghold true[ ∣ ∣ ] ∣ ∣∂ϕ ∣∣∣∣0 ∣∣∣∣0 ∣∣∣∣011− ∂ϕT 11,J(0) = ∂JT 11− ∂J ∣∣∣∣011∂x i ∂x i ∂x i ∂x i(i = 1,...,2n+m).Proof: Let us consider the first equation of (1):ω 0 ((ϕ ∗ Jϕ −1∗ )(X),(ϕ ∗Jϕ −1∗ )(Y)) = ω 0(X,Y)Now looking at the first order jets in both hands of this equation, we get that(being (e 1 ,...,e 2n ) the canonical symplectic basis on the leaves)if and only if( ([∂ϕ∗ ],J∂x ij 1 0 (ω 0((ϕ ∗ Jϕ −1∗ )(e r),(ϕ ∗ Jϕ −1∗ )(e s))) = j 1 0 (ω 0(e r ,e s ))+ ∂J ) ∣ ) T ([ ]∣∣∣0e r π 0 J(0)e s +(J(0)e r ) T ∂ϕ∗π 0 ,J + ∂J ) ∣ ∣∣∣0e s = 0∂x i ∂x i ∂x iwhich is equivalent to say that the following matrix[∂ϕ11∂x i,J 0] ∣ ∣ ∣∣∣0+ ∂J∂x i∣ ∣∣∣∣0is symmetric.Needless to say, a similar result holds for Dirac manifolds.qedExample 4.8 A foliated almost complex structures which admits no compatiblePoisson structures: take R 7 with coordinates (x 1 ,...,x 7 ) endowed withthe trivial foliation of codimension one, just given byx 7 = const.Consider the foliated ( almost)complex structure whose matrix, in the givenJ(x) H(x)coordinates, is , being J(x)0 ψ(x)2 = −Id and ψ(x) a non vanishing