Geometric structures modeled on smooth projective horospherical varieties of Picard number one

TL;DR

This paper investigates geometric structures on horospherical varieties, showing that such structures on certain Fano manifolds are locally equivalent to standard models, thus extending rigidity results beyond homogeneous cases.

Contribution

It generalizes deformation rigidity results from rational homogeneous manifolds to quasihomogeneous horospherical varieties using Cartan geometry.

Findings

Geometric structures on smooth projective horospherical varieties of Picard number one are locally equivalent to standard structures.

The results extend rigidity properties to a broader class of quasihomogeneous varieties.

The study employs Cartan geometry to establish local equivalence on Fano manifolds.

Abstract

Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric structures modeled on horospherical varieties. Using Cartan geometry, we prove that a geometric structure modeled on a smooth projective horospherical variety of Picard number one is locally equivalent to the standard geometric structure when the geometric structure is defined on a Fano manifold of Picard number one.