An Extended Abel-Jacobi Map

Abstract

We solve the problem of inversion of an extended Abel-Jacobi map $${\int }_{P_0}^{P_1}\omega +...+{\int }_{P_0}^{P_{g+n-1}}\omega =\mathbf{z},{\int }_{P_0}^{P_1}{\Omega }_{j1}+...+{\int }_{P_0}^{P_{g+n-1}}{\Omega }_{j1}=Z_j,j=2,...,n,$$ where ${\Omega }_{j1}$ are (normalised) abelian differentials of the third kind. In contrast to the extensions already studied, this one contains meromorphic differentials having a common pole $Q_1$. This inversion problem arises in algebraic geometric description of monopoles, as well as in the linearization of integrable systems on finite-dimensional unreduced coadjoint orbits on loop algebras.

11 pages

References in corpus (1)

• Remarks on the complex geometry of the 3-monopole