The even Clifford structure of the fourth Severi variety

The Hermitian symmetric space M = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ mapping E-valued 2-forms 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) ⊂ CP^26 known as the fourth Severi variety.