an:04019221
Zbl 0627.14024
Hitchin, Nigel
Stable bundles and integrable systems
EN
Duke Math. J. 54, 91-114 (1987).
00153597
1987
j
14H10 14D20 37J99 14F05 14H40 14K10
cotangent bundles of moduli spaces of vector bundles over a Riemann surface; completely integrable Hamiltonian systems; adjoint representation; Jacobian; Prym varieties
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.
F.Kirwan