POISSON BRACKETS AND COMPLEX TORI

14 A. P. VESELOV AND S. P. NOVIKOVThe forms α s are holomorphic and have at λ = ∞ the expansion (z = λ −1/2 )[ ( z) 2sα s = 2 + O(z )]2g dz (s = 0, . . . , g −1 ).2The transition matrix ω s = A sj α j is formed from the coefficients of the expansionof the forms ω s , namely∞∑ω s = c sj z 2j dz,at the point λ = ∞, so thatj=0A sj = 2 1−2j c sj .For Abelian differentials of second kind Ω j with zero a-periods and having asingle pole at λ = ∞ with asymptotics Ω j = −d(z 2j−1 ) + O(1) the followingequality holds (see [19], §10–3):∮b sΩ j = c sj = 2 2j−1 A sj .From this it follows that the columns of the matrix A ij coincide with the vectorswhich on the Jacobians J(Γ) give the direction of the flows of the higher KdVwith the corresponding indices j = 0, 1, . . . , g − 1. On the other hand, thesecolumns, by definition, are the direction vectors of Hamiltonian flows withHamiltonians H 0 , . . . , H g−1 . Thus, for the first g flows coordination has beenproved. For the remaining flows it is automatically satisfied, since they can beexpressed in terms of the first g on our phase space.The proof of the theorem is complete.We now list the most important examples.Example 1. The standard bracket [20], [21]. From the results of [22] we derive{∫1 T}Q = 2ip(λ) dλ, A = T 1 , . . . , T g , ū = lim u dx ,T →∞ Twhere the T i are the periods of the quasiperiodic potential u(x), p(λ) is thequasimomentum, and dp(λ) is a differential of second kind with a single poleat λ = ∞,dp(λ) = d(z −1 ) + O(1),∮z = λ −1/2 ,a jdp(λ) = 0, j = 1, . . . g.0