ON HYPER KÄHLER MANIFOLDS ASSOCIATED TO LAGRANGEAN KÄHLER SUBMANIFOLDS OF T^{∗}ℂ^{n}

VICENTE CORTÉS

Abstract. For any Lagrangean Kähler submanifold M ⊂ T^{∗}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).