On Hyper-Kähler Manifolds of Type A ∞ and D ∞

We shall use an infinite dimensional hyper-K?hler quotient method to obtain hyper-K?hler 4 manifolds of type A _∞ and D _∞. Hyper-K?hler manifolds of type A _∞ and D _∞ are constructed in terms of Dynkin diagrams of type A _∞ and D _∞ respectively. A hyper-K?hler manifold of type D _∞ is the minimal resolution of the quotient space of a hyper-K?hler manifold of type A _∞ by an involution. Finally we shall show that a hyper-K?hler manifold of type A _∞ can be considered as the universal cover of elliptic fibre space of type I _b . Received: 18 July 1997 / Accepted: 14 April 1998