## 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.
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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### References:

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