eigen-bundles of a trivial vector bundle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:20:42Z http://mathoverflow.net/feeds/question/100338 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100338/eigen-bundles-of-a-trivial-vector-bundle eigen-bundles of a trivial vector bundle IMeasy 2012-06-22T10:08:23Z 2012-06-22T10:49:23Z <p>Suppose I have a trivial vector bundle $V\cong \mathcal{O}_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on the vector bundle $V$ that gives a decomposition of $V$ into eigenbundles, indexed by the characters of $G$. Are the eigen-bundles trivial as well? Why?</p> http://mathoverflow.net/questions/100338/eigen-bundles-of-a-trivial-vector-bundle/100341#100341 Answer by a-fortiori for eigen-bundles of a trivial vector bundle a-fortiori 2012-06-22T10:18:34Z 2012-06-22T10:49:23Z <p>No. Let $C$ be an open affine part of an elliptic curve over the complex numbers and $L$ a non-trivial line bundle on $C$. Now, $L\oplus L^{-1}$ is trivial, so let $G=\mathbf Z/2\mathbf Z$ operate by $1$ on $L$ and by $-1$ on $L^{-1}$.</p> <p>For complete varieties (for simplicity, say having a rational point), all endomorphisms of a free bundle are given by constant matrices, so all direct summands are free again.</p>