Rigidity of projective symmetric manifolds of Picard number 1 associated to composition algebras

TL;DR

This paper proves the rigidity of certain projective symmetric manifolds associated with complex composition algebras, showing they are uniquely determined within smooth families.

Contribution

It establishes the rigidity of manifolds linked to composition algebras, extending understanding of their deformation properties.

Findings

Varieties are rigid under smooth deformations.

Rigidity applies to manifolds associated with specific composition algebras.

All fibers in a family are isomorphic if one is.

Abstract

To each complex composition algebra $\mathbb{A}$, there associates a projective symmetric manifold $X(\mathbb{A})$ of Picard number one, which is just a smooth hyperplane section of the following varieties ${\rm Lag}(3,6), {\rm Gr}(3,6), \mathbb{S}_6, E_7/P_7.$ In this paper, it is proven that these varieties are rigid, namely for any smooth family of projective manifolds over a connected base, if one fiber is isomorphic to $X(\mathbb{A})$, then every fiber is isomorphic to $X(\mathbb{A})$.