Timeline for Flag variety type Beilinson resolution
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 12 at 18:14 | comment | added | Sasha | It is conjectured that an exceptional collection exists for any flag variety, see the introduction to [Kuznetsov, Alexander; Polishchuk, Alexander. Exceptional collections on isotropic Grassmannians. J. Eur. Math. Soc. (JEMS) 18 (2016), no. 3, 507--574] for a survey. However, for your purposes a numerically exceptional collection should be enough, and this, I think, can be obtained with a torus localization technique. | |
| Mar 12 at 13:07 | comment | added | fool rabbit | @Sasha This seems not enough for me. In fact, I want to do same explicit calculation (e.g. convolution) on the K group of some variety relate to flag variety. Use some technique, it is enough for me to know the class (sum of line bundle over B×B) of direct image sheaf of line bundle (on \mathcal{B} ). Then I can do my calculation. So I need a explicit formulas of such classes, which parameterized by integral weight. So I wonder if a Beilinson type resolution exist for any line bundle over B, then I can determine the class what I need in the K^0(B×B). So I want to a "canonical resolution" | |
| Mar 12 at 12:33 | comment | added | Sasha | Beilinson (and Kapranov) construct resolutions by means of exceptional collections; these resolutions are very special. On the other hand, for any coherent sheaf on a smooth projective variety one can construct a locally free resolution just by iteratively covering the sheaf by a sufficiently large multiple of a sufficiently negative line bundle. Is the latter enough for you, or do you need the former? | |
| Mar 12 at 11:04 | comment | added | fool rabbit | @Sasha, Thanks for your suggestion ! I have seen Kapranov's paper just now. In his paper, he considered the special case of flag variety, i.e. take G=GL(V)(or SL(V)). But I hope a resolution for arbitrary simple type of algebraic group G, maybe his result can be extended to the another type. Thanks for your suggestion again! | |
| Mar 12 at 10:57 | history | edited | fool rabbit | CC BY-SA 4.0 |
added 1 character in body
|
| Mar 12 at 10:07 | comment | added | Sasha | Have you seen the following paper? Kapranov, M. M. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), no. 3, 479--508. | |
| Mar 12 at 9:48 | history | asked | fool rabbit | CC BY-SA 4.0 |