Timeline for line bundles on smooth affine variety
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 14, 2010 at 5:31 | comment | added | Torsten Ekedahl | Yes, a line bundle on a compact Kähler variety has a connection iff it has a flat connection. This follows from the Hodge decomposition of de Rham cohomology, the obstruction for finding a flat connection lies in $H^1(X,\Omega^{\ge1})$ and the obstruction for finding a connection is its image in $H^1(X,\Omega^1)$. By the Hodge decomposition (and the fact that the obstruction is of type $(1,1)$), if the latter vanishes so does the former. | |
| Apr 13, 2010 at 22:52 | comment | added | Ben Wieland | Is this motivated the Atiyah class, which lives in coherent cohomology and thus vanishes for affines? The vanishing of this class implies the existence an algebraic connection. Moreover, one can compute the Chern classes in Dolbeault cohomology from the Atiyah class. If the variety is complete, so that Dolbeault cohomology is pretty close to de Rham cohomology, does the vanishing of the Atiyah class imply the vanishing of the rational Chern classes? or is something lost in the extension data? | |
| Apr 13, 2010 at 21:39 | answer | added | Torsten Ekedahl | timeline score: 14 | |
| Apr 13, 2010 at 21:27 | answer | added | Bugs Bunny | timeline score: 1 | |
| Apr 13, 2010 at 21:05 | history | asked | Vladimir | CC BY-SA 2.5 |