Timeline for When can we cancel vector bundles from tensor products?
Current License: CC BY-SA 2.5
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| Apr 15, 2010 at 11:29 | comment | added | Angelo | To handle the case that $E$ and $F$ are direct sums of line bundles, it is easier to restrict to a line; sums of line bundles are determined by their restrictions to a line. Also, the case that $E$ is a direct sum of copies of the same line bundle follows by the same trick from Horrock's theorem that a vector bundle with splitting type $(d, \dots, d)$ on all lines splits. | |
| Apr 14, 2010 at 18:19 | comment | added | damiano | You might be able to use some kind of Harder-Narasimhan filtration argument to show that if one of the two bundles splits, then the other one must split as well. I do not know how to do this, it just seems a natural generalization of the above argument. | |
| Apr 14, 2010 at 17:58 | comment | added | Hailong Dao | A quick comment: to show that F splits, one can restrict to some hyperplane and reduce to the situation $n=2$. | |
| Apr 14, 2010 at 17:48 | comment | added | Hailong Dao | @damiano: That's right! I think the tricky part is: given E splits, showing F has to split too. I think one can do it, but it will not be very easy. | |
| Apr 14, 2010 at 17:05 | history | answered | damiano | CC BY-SA 2.5 |