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On the complex Cayley Grassmannian. (English) Zbl 1471.14105

Let \(\mathbb{O}\) denote the complex octonion algebra, and let \(\Xi : \mathbb{O}\times \mathbb{O}\times \mathbb{O}\times \mathbb{O}\to \mathbb{R}\) denote the function that assigns the imaginary part to the cross product \(x\times u \times v \times w \in \mathbb{O}\times \mathbb{O}\times \mathbb{O}\times \mathbb{O}\). The Cayley Grassmannian, denoted \(X\), is the variety parametrizing four dimensional subspaces spanned by the vectors \(x,u,v,w\) such that \(\Xi(x,u,v,w)=0\). It is a 12-dimensional, singular, \(\text{Spin}(7,\mathbb{C})\)-stable subvariety of the Grassmann variety, \(Gr(4,\mathbb{O})\). In this article, the author analyzes the geometry of the natural \(\text{Spin}(7,\mathbb{C})\)-action on \(\mathbb{O}\) by restricting it to various subgroups. Briefly stated, the author obtains the following results.
1.
An explicit determination of the \(T\)-fixed points, where \(T\) is the maximal torus of \(\text{Spin}(7,\mathbb{C})\).
2.
An explicit characterization of the singular locus of \(X\); it turns out that \(\text{Sing}(X)\) is isomorphic to the partial flag variety corresponding to the long root of the exceptional simple algebraic group, \(\text{G}_2 (\mathbb{C})\).
These results can be seen as natural extensions of the results of the articles by S. Akbulut and M. B. Can [J. Gökova Geom. Topol. GGT 11, 56–79 (2017; Zbl 1390.22011)] and L. Manivel [J. Algebra 503, 277–298 (2018; Zbl 1423.14293)].

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
53C29 Issues of holonomy in differential geometry
53C38 Calibrations and calibrated geometries
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References:

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