Positive quaternion Kähler manifolds with fourth Betti number equal to one (Q617728)

The main result of the paper is the following: Let \((M, g)\) be a \(4n\)-dimensional quaternion-Kähler manifold with positive scalar curvature, with \(b_1(M) = 1\) and \(2 \neq n \leq 6\). Then \((M, g)\) is homothetic to \(\mathbb H P^n\). This extends a result by Galicki and Salamon, which was stated for \(2 \neq n \leq 4\). By various results on quaternionic-Kähler manifolds and the Atiyah-Singer Index theorem [see \textit{S. M. Salamon}, Surv. Differ. Geom., Suppl. J. Differ. Geom. 6, 83--121 (1999; Zbl 1003.53039)], it is known that \(d = \dim \text{Iso}(M,g)\) is related with the even Betti numbers \(b_{2j}(M)\) of \(M\) by a system of polynomial equations with integer coefficients. The main argument essentially consists in proving that, when \(n = 5\) or \(6\), the integer solutions of such system can be realized only when \(d = \dim \text{Iso}(\mathbb H P^{n})\).