Tohoku Mathematical Journal

On the quaternionic manifolds whose twistor spaces are Fano manifolds

Radu Pantilie

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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.

Article information

Source
Tohoku Math. J. (2), Volume 67, Number 4 (2015), 507-511.

Dates
First available in Project Euclid: 22 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1450798069

Digital Object Identifier
doi:10.2748/tmj/1450798069

Mathematical Reviews number (MathSciNet)
MR3436538

Zentralblatt MATH identifier
1344.53039

Subjects
Primary: 53C28: Twistor methods [See also 32L25]
Secondary: 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry

Keywords
Quaternionic manifolds

Citation

Pantilie, Radu. On the quaternionic manifolds whose twistor spaces are Fano manifolds. Tohoku Math. J. (2) 67 (2015), no. 4, 507--511. doi:10.2748/tmj/1450798069. https://projecteuclid.org/euclid.tmj/1450798069


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