Homogeneous ACM bundles on Grassmannians of exceptional types (Q6564641)

The paper under review presents a contribution to the study of vector bundles over projective varieties, particularly on Grassmannians of exceptional types.\N\NThe authors' motivation is to classify all irreducible homogeneous ACM bundles over generalized Grassmannians (rational homogeneous varieties with Picard group \(\mathbb{Z}\)), which is a generalization of previous work on usual Grassmannians [\textit{L. Costa} and \textit{R. M. Miró-Roig}, Adv. Math. 289, 95--113 (2016; Zbl 1421.14006)] and isotropic Grassmannians [\textit{R. Du} et al., Forum Math. 35, No. 3, 763--782 (2023; Zbl 1524.14093)]. This is an interesting problem in algebraic geometry as ACM bundles have various applications and connections to other areas of mathematics.\N\NThe main result of the paper, which is based on the Borel-Bott-Weil theorem, provides a necessary and sufficient condition for an irreducible homogeneous vector bundle to be an ACM bundle in terms of its associated datum. This theorem can be used to characterize ACM bundles and has interesting consequences, such as Corollary 1.2, which states that only finitely many irreducible homogeneous ACM bundles up to tensoring a line bundle exist on a given generalized Grassmannian.\N\NThe paper also presents a detailed analysis of the concrete form of the associated datum on generalized Grassmannians of different types, especially on Grassmannians of exceptional types. This analysis is carried out through explicit calculations of positive roots and the use of the Killing form, which leads to concrete examples and classifications of ACM bundles in these cases.\N\NFurthermore, the authors determine the representation type of some Grassmannians of exceptional types. By constructing families of indecomposable ACM bundles, they show that certain Grassmannians are of wild representation type.