Singularities of Integrable Systems and Algebraic Curves

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Abstract

We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework for various multidimensional spinning tops, as well as Beauville systems, we prove that if the spectral curve is nodal, then all singularities on the corresponding fiber of the system are non-degenerate. We also show that the type of a non-degenerate singularity can be read off from the behavior of double points on the spectral curve under an appropriately defined antiholomorphic involution. Our analysis is based on linearization of integrable flows on generalized Jacobian varieties, as well as on the possibility to express the eigenvalues of an integrable vector field linearized at a singular point in terms of residues of certain meromorphic differentials.

Original languageEnglish (US)
Pages (from-to)5475-5524
Number of pages50
JournalInternational Mathematics Research Notices
Volume2017
Issue number18
DOIs
StatePublished - Jan 1 2017
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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