On Hyper Kähler manifolds associated to Lagrangean Kähler submanifolds of $T^*{\Bbb C}^n$

Abstract

For any Lagrangean Kähler submanifold $M \subset T^*{\Bbb C}^n$, there exists a canonical hyper Kähler metric on $T^*M$. 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, a class of pseudo hyper Kähler manifolds of complex signature $(2,2n)$ is constructed. For any hyper Kähler manifold $N$ in this class a group of automorphisms with a codimension one orbit on $N$ 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 $(2,2n)$.