Splitting of vector bundles on a complex torus - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:22:45Z http://mathoverflow.net/feeds/question/81063 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81063/splitting-of-vector-bundles-on-a-complex-torus Splitting of vector bundles on a complex torus Xandi Tuni 2011-11-16T12:51:46Z 2011-11-16T12:51:46Z <p><em>Let $X$ be a complex torus (a finite dimensional complex vector space modulo a lattice) and let $E$ be a smooth (not necessarily holomorphic) complex vector bundle over $X$. Is it true $E$ is isomorphic to a sum of smooth line bundles? If not, what would be a minimal counterexample?</em></p> <p>A theorem of Atyiah and Hirzebruch tells me that the Chern-character is an isomorphism of $\mathbb Q$-algebras $K(X)\otimes\mathbb Q \to H^{2\ast}(X,\mathbb Q)$. Since $X$ is a torus, its even cohomology ring is generated by Chern-classes of complex line bundles, hence so is $K(X)\otimes\mathbb Q$. It follows that some power of $E$ is stably isomorphic to a sum of line bundles. If one can get rid of the denominators, it would follow that $E$ itself is stably isomorphic to a sum of line bundles, but that's probably as far as one gets with $K$-theory. </p>