On the quaternionic manifolds whose twistor spaces are Fano manifolds

TL;DR

This paper characterizes quaternionic manifolds with Fano twistor spaces, showing they are either quaternionic projective spaces or certain Grassmannians, extending previous results under broader conditions.

Contribution

It proves new classification results for quaternionic manifolds with Fano twistor spaces, generalizing earlier work to non-Kähler cases.

Findings

M admits a reduction to Sp(1) x GL(k,H) iff M=HP^k

Either b2(M)=0 or M=Gr_2(k+2,C)

Extends classification to non-Kähler quaternionic 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 $Sp(1) \times GL(k,H)$ if and only if $M=HP^k$, (b) either $b_2(M)=0$ or $M=Gr_2(k+2,C)$. 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-Kaehler manifold with positive scalar curvature.