Algebraic complete integrability of an integrable system of Beauville (Q931893)

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.