Characterization of Jacobian varieties in terms of soliton equations

Pour Ω dans le demi-espace superieur de Siegel h g , soit X=C g /(Z g +ΩZ g ) la variete abelienne principalement polarisee correspondante et soit θ(Z)=θ(Z,Ω)=∑ m e Z gexp(2Πi t mz+πi t mΩm) la fonction theta de Riemann pour X. Les deux conditions (A) et (B) pour X sont equivalentes: A) 1) il existe des vecteurs a 1 ,a 2 ,a 3 ∈C g , a 1 ¬=0, et une forme quadratique Q(t)=∑ i/j=1 3 Q ij t i t j ,Q ij ∈C, tel que pour ζ∈C g τ(t)=τ(t,ξ)=exp(Q(t))θ(t 1 a 1 +t 2 a 2 +t 3 a 3 +ξ) donne une solution a l'equation de Kadomtsev-Petviashvili sous la forme d'Hirota: (D 1 4 +3D 2 2 -4D 1 D 3 )τ•τ=0; 2) le diviseur Θ de X est irreductible. B) X est isomorphe a la variete jacobienne d'une courbe lisse complete de genre g sur C

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