Abstract:
Hyperkaehler (also called irreducible holomorphic symplectic) manifolds form an important class of manifolds with trivial canonical bundle. One fundamental aspect of their structure theory is the question whether a given hyperkaehler manifold admits a Lagrangian fibration. I will report on a joint project with Daniel Greb and Christian Lehn investigating the following question of Beauville: if a hyperkaehler manifold contains a complex torus $T$ as a Lagrangian submanifold, does it admit a (meromorphic) Lagrangian fibration with fibre $T$? I will describe a complete positive answer to Beauville's Question for non-algebraic hyperkaehler manifolds, and give explicit necessary and sufficient conditions for a positive solution in the general case using the deformation theory of the pair $(X,T)$.
