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An Extended Abel-Jacobi Map

H. W. Braden·Yu. N. Fedorov
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Abstract

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

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H. W. Braden, Yu. N. Fedorov. 2007-01-15. An Extended Abel-Jacobi Map. https://doi.org/10.1016/j.geomphys.2008.05.009

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