The even Clifford structure of the fourth Severi variety

TL;DR

This paper explores the even Clifford structure of the Hermitian symmetric space EIII, explicitly describing associated vector bundles, constructing a canonical 8-form, and relating it to the fourth Severi variety's geometric and cohomological properties.

Contribution

It provides an explicit description of the vector bundle E on EIII, constructs a canonical 8-form linked to its holonomy, and connects this form to the Schubert cycle structure of the fourth Severi variety.

Findings

Explicit vector bundle E as a sub-bundle of End(TM)

Construction of a canonical differential 8-form on EIII

Relation of the 8-form to the cohomology and Schubert cycles of the 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 $\varphi:\mathrm{Cl}^0(E) \rightarrow \mathrm{End}(TM)$ mapping $\Lambda^2 E$ into skew-symmetric endomorphisms, and the existence of a metric connection on $E$ compatible with $\varphi$. 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.