Minimal-Degree Foliations on Cominuscule Grassmannians

TL;DR

This paper determines the minimal degrees of certain differential forms on cominuscule Grassmannians, classifies specific foliations, and constructs examples of high codimensional foliations with minimal degree.

Contribution

It computes minimal degrees for non-zero cohomology of differential forms, classifies degree-zero and degree-one codimension-one foliations, and provides new examples on classical and exceptional varieties.

Findings

Any degree-zero codimension-one foliation is a pencil of hyperplanes.

Classifies codimension-one foliations of degree one.

Constructs families of high codimensional minimal degree foliations.

Abstract

Given $X$ a cominuscule Grassmannian (or irreducible Hermitian symmetric space) and an integer $p,$ we compute the minimum $l(p)$ such that $H^0 (\Omega^p_X (l(p)))$ is not 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.