An Extended Abel-Jacobi Map

TL;DR

This paper addresses the inversion of an extended Abel-Jacobi map involving meromorphic differentials with common poles, with applications in algebraic geometry, monopoles, and integrable systems.

Contribution

It provides a solution to a new class of Abel-Jacobi inversion problems involving differentials with shared poles, extending previous work.

Findings

Solved the inversion problem involving differentials with common poles

Applied results to algebraic geometry and integrable systems

Enhanced understanding of monopole descriptions

Abstract

We solve the problem of inversion of an extended Abel-Jacobi map $$ \int_{P_{0}}^{P_{1}}\omega +...+\int_{P_{0}}^{P_{g+n-1}}\omega ={\bf z}, \qquad \int_{P_{0}}^{P_{1}}\Omega_{j1}+... +\int_{P_{0}}^{P_{g+n-1}}\Omega_{j1} =Z_{j},\quad j=2,...,n, $$ 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.