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Projective planes, Severi varieties and spheres. (English) Zbl 1057.53040
Yau, S.-T. (ed.), Surveys in differential geometry. Lectures on geometry and topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, Cambridge, MA, USA, May 3–5, 2002. Somerville, MA: International Press (ISBN 1-57146-114-0/hbk). Surv. Differ. Geom. 8, 1-27 (2003).
The article is devoted to the geometry over graded algebras such as quaternion and octonion. Projective algebraic varieties such as projective planes are studied. Since the algebra of octonions is not associative, due to non-Desarguesian property it is not possible to construct projective plane. These projective spaces are embedded into the space of Hermitian $$3\times 3$$ matrices over quaternions or octonions respectively. The action of finite dimensional Lie groups on the varieties is considered. Some related fibre structures are outlined.
For the entire collection see [Zbl 1034.53003].

##### MSC:
 53C30 Differential geometry of homogeneous manifolds 53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 11R52 Quaternion and other division algebras: arithmetic, zeta functions 53C27 Spin and Spin$${}^c$$ geometry
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