On the classification of positive quaternionic Kähler manifolds with b 4 = 1

Let M be a positive quaternionic Kähler manifold of dimension 4m. We already showed that if the symmetry rank is greater than or equal to [m2]+2 and the fourth Betti number b 4 is equal to one, then M is isometric to .P m. The goal of this paper is to report that we can improve the lower bound of the symmetry rank by one for higher even-dimensional positive quaternionic Kähler manifolds. Namely, it is shown in this paper that if the symmetry rank of M with b 4(M) = 1 is greater than or equal to m2+1 for m . 10, then M is isometric to .P m. One of the main strategies of this paper is to apply a more delicate argument of Frankel type to positive quaternionic Kähler manifolds with certain symmetry rank.