POISSON BRACKETS AND COMPLEX TORI

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12 A. P. VESELOV AND S. P. NOVIKOVto Liouville, but also as integrals over the elements a j of the group H 1 (Γ\P, Z)where P is the set of poles of the form Q or the forms ∇ τj Q for its derivativesalong M A ; this proves the proposition.Lemma. Compact real tori T k are possible only whenk 2g + κ, (16)where κ is the number of essential residues of the poles of the form Q modulothe annihilator A.Proof. As indicated in §1, the number 2g + κ coincides for “general” bracketsgiven by the pair (A, Q) with the rank of the lattice D in the space C k of thevariables (8). Since the real anti-involution τ is an automorphism of the realgroup J Q (Γ) and the dimension of the torus over R is equal to k, inequality(16) means simply that the dimension of the torus cannot be greater than therank of the lattice. The lemma is proved.We shall call δ = 2k − 2g − κ (the corank of the lattice) the number ofvariables of “phase type”, if the real parts are compact tori.The varieties J Q (Γ) are fibered in the formJ Q (Γ) Cs /Z δ−−−→k−s ˜JQ(Γ), s δ,where ˜J Q (Γ) is a compact complex torus. Under the conditions of the lemmawe have s = δ.3 Poisson brackets coordinated with KdV theory.The most important examplesDefinition. A Poisson bracket on the phase space (4), where Γ are hyperellipticcurves of the form y 2 = ∏ 2gj=0 (λ − λ j) = R(λ) is said to be coordinated withKdV theory if all the higher KdV are Hamiltonian in this bracket. Here k = g.In [14] a number of examples were given of brackets coordinated with KdVtheory (see below). For such brackets J Q (Γ) = J(Γ) always, i.e., the usual Abeltransformation defined by means of the collection of basis holomorphic differentialforms linearizes the dynamics of virtue of Hamiltonians of the form H(Γ).
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.
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