The even Clifford structure of the fourth Severi variety

• Main Authors: Parton Maurizio, Piccinni Paolo
• Format: Article
• Language: English
• Published: De Gruyter 2015-08-01
• Series: Complex Manifolds
• Subjects: Clifford structure; exceptional symmetric space; octonions; canonical differential form; Primary 53C26; 53C27; 53C38
• Online Access: http://www.degruyter.com/view/j/coma.2015.2.issue-1/coma-2015-0008/coma-2015-0008.xml?format=INT

TheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl0(E) → End(TM) mapping Ʌ2E into skew-symmetric endomorphisms, and the existence of a metric connection on E compatible with φ. We give an explicit description of such a vector bundle E as a sub-bundle of End(TM). From this we construct a canonical differential 8-form on EIII, associated with its holonomy Spin(10) · U(1) ⊂ U(16), that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at EIII as the smooth projective variety V(4) ⊂ CP26 known as the fourth Severi variety.