Timeline for Splitting of vector bundles on a complex torus
Current License: CC BY-SA 3.0
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| Nov 16, 2011 at 13:39 | comment | added | Jason Starr | No, that's not true. Let $X$ be a $2$-dimensional complex torus. Use Serre's construction to produce a rank $2$ vector bundle $E$ whose first Chern class equals $0$ and whose second Chern class equals $1$ times the Poincare dual of the class of a point. If $E$ were a direct sum of two line bundles, then by using the Whitney sum formula we would have the second Chern class is $-c^2$, where $c$ is the first Chern class of either line bundle. But it is easy to write out a general element $c$ in $H^2(X,\mathbb{Z})$ and see that its square is always divisible by $2$. | |
| Nov 16, 2011 at 12:51 | history | asked | Xandi Tuni | CC BY-SA 3.0 |