## Stable bundles and integrable systems.(English)Zbl 0627.14024

The author shows that cotangent bundles of moduli spaces of vector bundles over a Riemann surface are algebraically completely integrable Hamiltonian systems. More precisely, let G be a complex semisimple Lie group, let N be the moduli space of stable G-bundles with prescribed topological invariants on a compact Riemann surface and let n be the dimension of N. The cotangent space to N at the point represented by a G- bundle P is $$H^ 0(M;ad(P\otimes K))$$ where ad(P) is the bundle associated to P via the adjoint representation of G on its Lie algebra g. Thus a choice of basis $$p_ 1,...,p_ k$$ for the ring of invariant polynomials on g induces a holomorphic map $$\phi: T*N\to \oplus H^ 0(M;K^{d_ i})$$ where $$d_ i$$ is the degree of $$p_ i$$. The components of $$\phi$$ are n functionally independent Poisson-commuting functions on T*N, and when G is a classical group the generic fibre of $$\phi$$ is an open set in an abelian variety on which the Hamiltonian vector fields defined by the components of $$\phi$$ are linear. This is what it means to say that T*N is an algebraically completely integrable Hamiltonian system. The abelian varieties occurring are either Jacobian or Prym varieties of curves covering M.
Reviewer: F.Kirwan

### MSC:

 14H10 Families, moduli of curves (algebraic) 14D20 Algebraic moduli problems, moduli of vector bundles 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14H40 Jacobians, Prym varieties 14K10 Algebraic moduli of abelian varieties, classification
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### References:

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