Polynomial integrable Hamiltonian systems on symmetric powers of plane curves

Alexander Mikhailov, University of Leeds. IS Seminar series.

We have found quite general construction of $k$ commuting vector fields on $k$--th symmetric power of $\mathbb{C}^{m}$ and also of $k$ commuting tangent vector fields to the $k$--th symmetric power of an affine variety $V\subset  \mathbb{C}^{m}$.

Application of this construction to $k$--th symmetric power of a plane algebraic curve $V_g$ of genus $g$ leads to $k$ integrable Hamiltonian systems on $\mathbb{C}^{2k}$ (or on $\mathbb{R}^{2k}$, if the base field is $\mathbb{R}$).

In the case of a hyperelliptic curve  $V_g$ of genus $g$ and  $k=g$ our system is equivalent to the well known Dubrovin system which has been derived and studied in the theory of finite gap solutions (algebra-geometric integration) of the Korteweg-de-Vrise equation.

We have found the coordinates in which the systems obtained and their Hamiltonians are polynomial. For $k=2,3$ and $g=1,2,3$ we present these systems explicitly as well as we discuss the problem of their integration.