Homogeneous bundles over abelian varieties (Q2907430)

Let \(A\) be an abelian variety over an algebraically closed field \(k\); and let \(E\) be a principal \(G\)-bundle over \(A\), where \(G\) is an algebraic group. \(E\) is called \textit{homogeneous} if it is isomorphic to its pull-back under any translation. In this paper, the author proves a result about the characterization and structure of homogeneous principal \(G\)-bundles over \(A\). According to this, a principal \(G\)-bundle \(\pi:X \to A\) is homogeneous if and only if there exists a commutative subgroup \(H \subset G\) and an extension \(0 \to H \to \mathcal{G} \to A \to 0\) such that \(\pi\) can be obtained as the extension of the \(H\)-bundle \(\mathcal{G} \to A\) to \(G\). This \(H\) can be chosen to be smallest, up to conjugation. Another equivalent condition for \(\pi:X \to A\) to be homogeneous is for the associated bundle \(X/R_u(G) \to X/G_{aff}\) to be homogeneous; where \(R_u(G)\) is the unipotent radical of \(G\) and \(G_{aff}\) is the largest connected affine subgroup of \(G\).NEWLINENEWLINEThese results generalize former results of [\textit{M. Miyanishi}, in: Number Theory, algebr. Geom., commut. Algebra, in Honor of Yasno Akizuki, 71--93 (1973; Zbl 0268.14017)] and [\textit{S. Mukai}, J. Math. Kyoto Univ. 18, 239--272 (1978; Zbl 0417.14029)] on homogeneous vector bundles over an abelian variety. In characteristic 0, an additional characterization is that \(\pi\) is homogeneous if and only if it admits a connection, which can be taken to be integrable.