Codimension one foliations on homogeneous varieties

TL;DR

This paper investigates codimension one foliations on rational homogeneous spaces, showing that low-degree foliations on various Grassmannians and special varieties are restrictions of ambient projective space foliations, with potential for broader applicability.

Contribution

It demonstrates that certain low-degree codimension one foliations on homogeneous varieties are restrictions of foliations from ambient projective spaces, using equivariant techniques.

Findings

Foliations on Grassmannians of lines are restrictions from projective space.

Identifies foliations on spinor and Lagrangian Grassmannians as restrictions.

Provides evidence for extending these results beyond studied cases.

Abstract

The aim of this paper is to study codimension one foliations on rational homogeneous spaces, with a focus on the moduli space of foliations of low degree on Grassmannians and cominuscule spaces. Using equivariant techniques, we show that codimension one degree zero foliations on (ordinary, orthogonal, symplectic) Grassmannians of lines, some spinor varieties, some Lagrangian Grassmannians, the Cayley plane (an $E_6$-variety) and the Freudenthal variety (an $E_7$-variety) are identified with restrictions of foliations on the ambient projective space. We also provide some evidence that such results can be extended beyond these cases.