The even Clifford structure of the fourth Severi variety

Abstract

The Hermitian symmetric space $M=\mathrm{EIII}$ 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 $\phi :{\mathrm{Cl}}^0(E)\to \mathrm{End}(TM)$ mapping ${\Lambda }^{2}E$ into skew-symmetric endomorphisms, and the existence of a metric connection on $E$ compatible with $\phi$. We give an explicit description of such a vector bundle $E$ as a sub-bundle of $\mathrm{End}(TM)$. From this we construct a canonical differential 8-form on $\mathrm{EIII}$, associated with its holonomy $\mathrm{Spin}(10)\cdot \mathrm{U}(1)\subset \mathrm{U}(16)$, that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at $\mathrm{EIII}$ as the smooth projective variety $V_{(4)}\subset \mathbb{C}{P}^{26}$ known as the fourth Severi variety.