On the quaternionic manifolds whose twistor spaces are Fano manifolds

Abstract

Let $M$ be a quaternionic manifold, $\dim M=4k$, whose twistor space is a Fano manifold. We prove the following:

(a) $M$ admits a reduction to ${\rm Sp}(1)\times{\rm GL}(k,\mathbb{H})$ if and only if $M=\mathbb{H} P^k$,

(b) either $b_2(M)=0$ or $M={\rm Gr}_2(k+2,\mathbb{C})$.

This generalizes results of S. Salamon and C. R. LeBrun, respectively, who obtained the same conclusions under the assumption that $M$ is a complete quaternionic-Kähler manifold with positive scalar curvature.