On hyper Kähler manifolds associated to Lagrangian Kähler submanifolds of

For any Lagrangian Kähler submanifold , there exists a canonical hyper Kähler metric on . A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, an interesting class of pseudo hyper Kähler manifolds of complex signature is constructed. For any manifold in this class a group of automorphisms with a codimension one orbit on is specified. Finally, it is shown that the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper Kähler metric of complex signature .