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Symplectic geometry on moduli spaces of stable pairs. (English) Zbl 0864.14004

The cotangent bundle of the moduli space of stable bundles of fixed rank and degree on a smooth projective curve of genus \(g\geq 2\) can be described as the set of isomorphism classes of Higgs pairs \((E,\varphi)\); here \(E\) is a stable bundle with the prescribed rank and degree and \(\varphi\) is a \(K\)-valued endomorphism of \(E\). Hitchin studied the symplectic structure of this cotangent bundle by means of the so-called Hitchin map, a map from the cotangent bundle to the direct sum of the sections of the powers of \(K\), which associates to every Higgs pair \((E,\varphi)\) the coefficients of the characteristic polynomial of \(\varphi\). The Hitchin map is an algebraically completely integrable Hamiltonian system [N. Hitchin, Duke Math. J. 54, 90-114 (1987; Zbl 0627.14024)].
This very interesting paper generalizes these results when the canonical bundle \(K\) is replaced by any line bundle \(L\) such that \(K^{-1} \otimes L\) has a non-zero section. In this case one has to consider the moduli space \({\mathcal M}' (r,d,L)\) of stable pairs \((E,\varphi)\), where \(\varphi:E \to E\otimes L\) is a morphism of \({\mathcal O}\)-modules. This moduli space was constructed by N. Nitsure [Proc. Lond. Math. Soc., III. Ser. 62, No. 2, 275-300 (1991; Zbl 0733.14005)] and in the case \(L=K\) by C. T. Simpson [Publ. Math., Inst. Hautes Étud. Sci. 75, 5-95 (1992; Zbl 0814.32003)]. It happens that the connected component \({\mathcal M}_0'(r,d,L)\) containing the stable pairs \((E,\varphi)\) for which \(E\) is a stable bundle, admits a Poisson structure. Then there is a Hitchin map \(H\) from \({\mathcal M}'(r,d,L)\) to the direct sum of the sections of the tensor powers of \(L\) associating as above to a stable pair \((E,\varphi)\) the coefficients of the characteristic polynomial of \(\varphi\), and the author proves that \(H\) can be considered as a family of completely integrable Hamiltonian systems on the symplectic leaves of the Poisson variety \({\mathcal M}_0'(r,d,L)\). This construction generalizes earlier results of A. Beauville in the case of genus 0 [Acta Math. 164, No. 3/4, 211-235 (1990; Zbl 0712.58031)]. Similar results are also obtained for moduli of stable pairs with fixed determinant. In the last section, the paper deals with the moduli space of stable parabolic bundles with fixed degree, weight and sequence of multiplicities. In this case, explicit descriptions of the canonical symplectic structure of the cotangent bundles to these moduli spaces are given and some symplectic subvarieties of the tangent bundle of the moduli space of stable parabolic bundle are identified, as symplectic manifolds, with certain special symplectic subvarieties of \({\mathcal M}_0'\).

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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References:

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