Minimal-Degree Foliations on Cominuscule Grassmannians

Abstract

Abstract Let X be a cominuscule Grassmannian (equivalently, an irreducible Hermitian symmetric space), and let p be an integer. We compute the minimum l ( p ) such that $$H^0(\Omega _X^p(l(p))) \ne 0.$$ H 0 ( Ω X p ( l ( p ) ) ) ≠ 0 . This allows us to conclude that any codimension-one foliation of degree zero on a cominuscule Grassmannian is a pencil of hyperplanes, improving a result of the first and third authors with D. Faenzi. We also deduce the structure of codimension-one foliations of degree one. Finally, we provide families of examples of high codimensional foliations of minimal degree on classical Grassmannians, Lagrangian Grassmannians, Spinor varieties, and the Cayley plane.