# Linear system of curves on K3 surfaces and Matsushita's conjecture

# Linear system of curves on K3 surfaces and Matsushita's conjecture

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**Zoom link:**** https://princeton.zoom.us/j/91248028438**

The geometry of hyperkaehler manifolds is known to be quite restrictive. For instance any fibration from a hyperkaehler to a lower dimensional variety is known to be Lagrangian with abelian varieties as general fibre. In this light, Matsushita conjectured that a general fibre of a Lagrangian fibration of a hyperkaehler manifold is either isomorphic to all other nearby fibres or there are at most finitely many nearby fibres that are isomorphic to it. In this talk, I will present some evidence toward this conjecture for certain Lagrangian fibrations induced by linear systems of curves on K3 surfaces. The results were obtained in recent joint work with D. Huybrechts.