The K. P. hypothesis for decomposable abelian varieties
Novikov's conjecture states that the Kadomtsev-Petviashvili (K.P.) equation characterizes Jacobians among all the indecomposable principally polarized abelian varieties. Shiota gave a geometrical proof of this conjecture. The aim of this article is to include the decomposable case in Shiota's theorem: Theorem 2. Let \((X,\left[ \Theta \right])\) be a principally polarized abelian variety and let \(\Theta \) be a generator of the vector space \( H^0(X,{\mathcal O}(\Theta)).\) Assume that there exist invariant vector fields \( D_1\neq 0,D_2,D_3\) and a constant \(d\) such that the K.P. equation \( K.P.(D_1,D_2,D_3;d)=0\) holds. Then there exists a non-zero Jacobian \( (Y,\left[ \Theta _Y\right])\) and a p.p.a.v. \((Z,\left[ \Theta _Z\right])\) such that \((X,\left[ \Theta \right])\) is isomorphic to the product of the p.p.a.v. \((Y,\left[ \Theta \right])\times (Z,\left[ \Theta _Z\right])\) and the vector fields \(D_1\) and \(D_2\) are parallel to the tangent bundle \(T(Y).\)
scientific article; zbMATH DE number 1138670