A note on automorphisms and birational transformations of holomorphic symplectic manifolds (Q2846830)

Let \(X\) be a holomorphic symplectic manifold, that is a compact Kähler manifold admitting a holomorphic two-form that is non-degenerate in every point. Holomorphic symplectic manifolds are a natural higher-dimensional analogue of \(K3\) surfaces and have been the object of an intensive study over the last ten years. An interesting and natural problem is to what degree an automorphism of \(X\) is determined by its action on the cohomology group \(H^2(X, \mathbb Z)\). In the case of \(K3\) surfaces the Torelli theorem gives a complete answer: the automorphism group injects into the orthogonal group \(O(H^2(X, \mathbb Z), q_X)\) where \(q_X\) is the Beauville form. Moreover every isometry \(\phi \in O(H^2(X, \mathbb Z), q_X)\) which is an isomorphism of Hodge structures and preserves a Kähler class is induced by an automorphism of \(X\). For higher-dimensional symplectic manifolds we are far from having such a complete picture, even in the special case of the Hilbert scheme \(S^{[n]}\) of \(n\) points on a \(K3\) surface \(S\). The main result of this paper is a characterisation of those automorphisms of \(S^{[n]}\) which are induced by an automorphism of \(S\). Such an automorphism preserves the cohomology class \([E] \in H^2(S^{[n]}, \mathbb Z)\) of the exceptional divisor \(E\) of the birational map \(S^{[n]} \rightarrow S^{(n)}\). The authors show that this property is already sufficient: any automorphism of \(S^{[n]}\) preserving the exceptional divisor \(E\) is induced by an automorphism of \(S\). The proof is based on the Torelli theorem for \(K3\) surfaces and a relation between the Kähler cones of \(S\) and \(S^{[n]}\). Using recent work by \textit{E. Markman} and \textit{M. Verbitsky} the authors also show that the group of birational automorphisms of any projective holomorphic symplectic manifold is finitely generated.