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Algebraic complete integrability of an integrable system of Beauville. (English) Zbl 1144.14037
The authors consider the following integrable system constructed by Beauville. Given a $$K3$$ surface $$S$$ of degree $$6$$ in $${\mathbb P}^4$$, let us consider the moduli space $$M^v_S$$ of stable bundles with rank $$2$$, $$c_1=0$$ and $$c_2=4$$ over $$S$$. This space in endowed by the Mukai symplectic structure. Let a smooth cubic threefold $$X \subset {\mathbb P}^4$$ contains $$S$$ and $$M^v_X$$ be the moduli space of stable bundles with rank $$2$$, $$c_1=0$$ and $$c_2=2$$ over $$X$$. The restriction of bundles over $$X$$ to $$S$$ is an embedding $$M^v_X \to M^v_S$$ whose fibers are Lagrangian submanifolds. Varying $$X$$ we obtain a fibration of an open dense subset of $$M^v_S$$ by Lagrangian submanifolds which are invariant sets of some integrable Hamiltonian system, the Beauville system.
The authors prove the following theorem: Let $$\bar{M}_S$$ be the moduli space of semi-stable sheaves which compatifies $$M^v_S$$ and $$\tilde{M}_S$$ be the O’Grady’s resolution of $$M_S$$. Then there exists an open subset $$U\subset \tilde{M}_S$$ such that the Beauville system extends to $$U$$ and any fiber is the complement of a subvariety of codimension $$2$$ in the intermediate Jacobian $$J(X)$$. In particular, this implies that for any threefold $$X$$ the Hamiltonian vector fields on $$M^v_X$$ are extended to $$J(X)$$, i.e. the system is algebraically completely integrable.
##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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##### References:
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