Vector bundles of degree zero over an elliptic curve.
To every vector bundle \(V\) of finite rank over a complete, connected, reduced scheme \(X\) over a perfect field \(k\) there exist principal \(G\)-bundles (for some group scheme \(G\)) over which the pullback of \(V\) is trivial. Let \(G_V\) denote such a group scheme of smallest dimension. On the other hand, using the Grothendieck ring \(K(X)\) of isomorphism classes of vector bundles over \(X\), to every \(V\) one can associate a \(\mathbb Q\)-algebra \(R(V) \subset K(X) \otimes \mathbb Q\). \textit{M. V. Nori} [Compos. Math. 33, 29--41 (1976; Zbl 0337.14016)] showed that for any finite vector bundle \(V\) one has \(\dim R(V) = \dim G_V = 0\). The author proves that for every vector bundle \(V\) of degree \(0\) over an elliptic curve \(X\) again \[ \dim R(V) = \dim G_V \] holds. An example shows that these dimensions may differ for curves \(X\) of genus~\(2\).
scientific article; zbMATH DE number 1809873
scientific article