Hyperplane section \({\mathbb {O}\mathbb {P}}^2_0\) of the complex Cayley plane as the homogeneous space \({\text F}_4/{\text P}_4\).

zbMATH Open1249.32019arXiv1006.3407MaRDI QIDQ2897373

Vít Tuček, Karel Pazourek, Peter Franek

Publication date: 10 July 2012

Published in: Commentationes Mathematicae Universitatis Carolinae (Search for Journal in Brave)

Abstract: We prove that the exceptional complex Lie group $F_4$ has a transitive action on the hyperplane section of the complex Cayley plane $mathbb{OP}^2$. Our proof is direct and constructive. We use an explicit realization of the vector and spin actions of $Spin(9,C)leq{F}_4$. Moreover, we identify the stabilizer of the $F_4$-action as a parabolic subgroup $P_4$ (with Levi factor $B_3{T}_1$) of the complex Lie group $F_4$. In the real case we obtain an analogous realization of $F_4^{(-20)}/P_4$.

Full work available at URL: https://arxiv.org/abs/1006.3407

zbMATH Keywords

hyperplane sectionCayley planeparabolic geometrytwistor fibrationSeveri varietyexceptional geometryoctonionic contact structure

Mathematics Subject Classification ID

Homogeneous spaces and generalizations (14M17) Almost homogeneous manifolds and spaces (32M12)

This page was built for publication: Hyperplane section \({\mathbb {O}\mathbb {P}}^2_0\) of the complex Cayley plane as the homogeneous space \({\text F}_4/{\text P}_4\).

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q2897373)