Spectral curves and integrable systems. (English) Zbl 0824.14013
The author studies families of Jacobians of spectral curves over algebraic curves of any genus. Main result:
Let \(\Sigma\) be a nonsingular complete algebraic curve, \(\mathbb{K}_ \Sigma\) its canonical line bundle. Let \(L\) be a positive line bundle on \(\Sigma\) satisfying \(L \geq \mathbb{K}_ \Sigma\). Let \(M_{\text{Higgs}}\) be the moduli space of \(L\) twisted Higgs pairs (pairs consisting of a vector bundle and an \(L\)-twisted endomorphism satisfying a stability condition). Then
1. \(M_{\text{Higgs}}\) is fibered, via an invariant polynomial map \(H_ L\): \(M_{\text{Higgs}} \to B_ L\) by Jacobians of spectral curves.
2. For every nonzero \(s \in H^ 0 (\Sigma, L \otimes \mathbb{K}_ \Sigma^{- 1})\) there exists a canonical Poisson structure, \(\Omega_ S\), on \(M_{\text{Higgs}}\).
3. \(H_ L : M_{\text{Higgs}} \to B_ L\) is an integrable system.
Let \(\Sigma\) be a nonsingular complete algebraic curve, \(\mathbb{K}_ \Sigma\) its canonical line bundle. Let \(L\) be a positive line bundle on \(\Sigma\) satisfying \(L \geq \mathbb{K}_ \Sigma\). Let \(M_{\text{Higgs}}\) be the moduli space of \(L\) twisted Higgs pairs (pairs consisting of a vector bundle and an \(L\)-twisted endomorphism satisfying a stability condition). Then
1. \(M_{\text{Higgs}}\) is fibered, via an invariant polynomial map \(H_ L\): \(M_{\text{Higgs}} \to B_ L\) by Jacobians of spectral curves.
2. For every nonzero \(s \in H^ 0 (\Sigma, L \otimes \mathbb{K}_ \Sigma^{- 1})\) there exists a canonical Poisson structure, \(\Omega_ S\), on \(M_{\text{Higgs}}\).
3. \(H_ L : M_{\text{Higgs}} \to B_ L\) is an integrable system.
Reviewer: R.Salvi (Milano)
MSC:
| 14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 14J10 | Families, moduli, classification: algebraic theory |
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