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


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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