On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

Abstract

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type $A_n, \tilde{A_n}$ or $\tilde{D_n}$ . Moreover, we show all types of the above singular fibres actually occur.