POISSON BRACKETS AND COMPLEX TORI

POISSON BRACKETS AND COMPLEX TORI 13Theorem 2. a) An analytic Poisson bracket is coordinated with KdV theoryif and only if all the forms ∇ τ Q—the derivatives in directions τtangent to a level surface of functions from the annihilator A—are holomorphicand generate the group H 1,0 (Γ).b) If the bracket is coordinated with KdV, then the coefficients of the expansion∞∑ ( z) kQ(Γ, λ) =qk (Γ), z = λ −1/2 ,2k=−Nare such that q 2l+3 (Γ) = h l (Γ) are Hamiltonians of the higher KdV withindex l 0, while the remaining coefficients q k belong to the annihilatorA.Remark 1. Part b) was established by the authors in [14].Proof. The usual Abel transformation linearizes the dynamics of all higherKdV; the necessity of the condition is therefore obvious. Suppose now thatthere is given a bracket (A, Q) such that ∇ τ Q = ω τ are holomorphic differentialsgenerating the entire space H 1,0 (Γ). Since z = λ −1/2 , from the holomorphicityof the derivatives ∇ τ Q along tangent directions to M A it follows thatthe expansion for large λ has the formQ(Γ, λ) =∞∑ ( z) 2k+3Hk (Γ) + Ann,2k=0where Ann is a series with coefficients in the annihilator. We shall prove thatH k (Γ) for k = 0, 1, . . . , g −1 are the Hamiltonians of precisely the first g higherKdV. We consider the standard set of holomorphic forms ω 1 , . . . , ω g normalizedby the conditions∮a jω s = 2πiδ js ,where a j , b j is the basis of cycles on Γ such thata i ◦ a j = b i ◦ b j = 0, a i ◦ b j = δ ij .We define two collections of variables ϕ i (i = 1, . . . , g) and ψ j (j = 0, . . . , g−1):ϕ s =g∑j=1∫ (γj ,ε j )P 0ω s , ψ s =g∑j=1∫ (γj ,ε j )P 0α s , α s = ∂Q∂H s∣ ∣∣∣A.