Arithmetically Cohen--Macaulay bundles on homogeneous varieties of Picard rank one

TL;DR

This paper characterizes arithmetically Cohen-Macaulay bundles on homogeneous varieties of Picard rank one, extending previous results to exceptional types and showing finiteness of such bundles up to twisting.

Contribution

It generalizes the classification of ACM bundles to all homogeneous varieties of Picard rank one, including exceptional types, and provides explicit lists of highest weights.

Findings

Finitely many irreducible homogeneous ACM bundles exist up to twisting.

Complete classification of ACM bundles on classical types.

Explicit highest weight lists for exceptional types like Cayley Plane.

Abstract

In this paper, we study arithmetically Cohen--Macaulay (ACM) bundles on homogeneous varieties $G/P$. Indeed we characterize the homogeneous ACM bundles on $G/P$ of Picard rank one in terms of highest weights. This is a generalization of the result on $G/P$ of classical types presented by Costa and Mir\'{o}-Roig for type $A$, and Du, Fang, and Ren for types $B,C$ and $D$. As a consequence we prove that only finitely many irreducible homogeneous ACM bundles, up to twisting line bundles, exist over all such $G/P$. Moreover, we derive the list of the highest weights of the irreducible homogeneous ACM bundles on particular homogeneous varieties of exceptional types such as the Cayley Plane and the Freudenthal variety.