Timeline for What are the point-like objects in $D^b(X)$ when $X$ is an abelian variety?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 2, 2016 at 14:55 | comment | added | Yosemite Sam | @ZhaotingWei Cor 3.15 of Huybrechts tells you that any E in D(C) splits as a direct sum of Ei[i] with Ei a sheaf. If E is pointlike and has two summands Ei[i], Ej[j] then the identity of both would contribute to Hom(P,P) which would have to have dimension at least two. i.e. P must be a shifted sheaf. And now t3suji's and Sasha's comments make it clear it's a nightmare in general. Maybe an alternative question is what are the pointlike objects when the (anti)canonical is almost ample, like big&nef. | |
| Nov 2, 2016 at 14:16 | comment | added | t3suji | @YosemiteSam By the way, even when X is an elliptic curve, P does not need to be a line bundle --- any indecomposable vector bundle would do. | |
| Nov 2, 2016 at 6:57 | comment | added | Sasha | Any autoequivalence of $D(X)$ takes a point-like object to a point-like object. And in case of abelian variety there are many autoequivalences, some of them take line bundles to complexes with several cohomology sheaves. | |
| Nov 2, 2016 at 3:34 | comment | added | Zhaoting Wei | @YosemiteSam I'm still not sure why $P$ must be a sheaf up to shift also it seems so. Could you give more reason? | |
| Nov 2, 2016 at 2:34 | comment | added | Yosemite Sam | do you know what happens for X an elliptic curve? I guess P has to be a sheaf, by homological dimension of D(X), but it's not clear to me why it must be a line bundle. | |
| Nov 2, 2016 at 2:31 | history | edited | Zhaoting Wei | CC BY-SA 3.0 |
edited title
|
| Nov 2, 2016 at 2:12 | history | asked | Zhaoting Wei | CC BY-SA 3.0 |