Abstract : When a complex surface $X$ admits a nowhere vanishing holomorphic 2-form, it determines a (holomorphic) symplectic structure on $X$. We consider the case when $X$ is an elliptic surface and study how the symplectic geometry is related to the underlying complex geometry of the elliptic fibration. This is based on a joint work with Guolei Zhong.

Keywords : elliptic fibration; K3 surface; holomorphic lagrangian fibration