Timeline for Why study Higher Sheaf Cohomology?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 26, 2016 at 1:24 | comment | added | Steven Gubkin | An example computation. Let $C$ be the unit circle, and $D$ the unit disc. $\int_C \frac{1}{z} dz = \int_C \bar{z} dz = \int_D d\bar{z} \wedge d z = \int_D (dx-i dy) \wedge (dx+idy) = \int_D 2i dx \wedge dy = 2\pi i$, where I have used that $\frac{1}{z} = \bar{z}$ on the unit circle in the first equality, and Stoke's theorem in the second equality. | |
| Oct 26, 2016 at 1:22 | comment | added | Steven Gubkin | You could say that you are really just integrating things like $(x_1^2+iy_3) dx_1 \wedge dy_2$ (complex functions of the real coordinate functions times wedges of differentials of the coordinate functions). These are sort of the most general sort of things you could integrate. The $dz$ and $d\bar{z}$ are just a more convenient basis for lots of computations. | |
| Oct 26, 2016 at 1:08 | comment | added | finnlim | Also, I'm a little ashamed to ask this, but I never really understood the point of considering conjugate-differential $d\bar z$ (I know that $\bar\partial f=0\iff $f is holomorphic, but still don't really see what's the point of $d\bar z$). Could you tell me more about what makes their integration meaningful? | |
| Oct 26, 2016 at 0:57 | comment | added | finnlim | Indeed I was interested in sheaves like this (holomorphic forms, etc.) but I'd say that generally I am not interested in structure for the sake of structure. I mean, self-intertwining outgrowth of abstraction is always an exciting side of mathematics, but I'd still say "that doesn't have much of a point" if they didn't point at something outside the formalism. I found Will Sawin's answer very helpful because it was pointed out that manifold classification problem gets a huge help from cohomology, which is a problem that can be posed independently of any cohomological concerns. | |
| Oct 25, 2016 at 23:18 | comment | added | Steven Gubkin | @BenMcKay I agree that this does not motivate sheaf cohomology for many different sheaves, but it seems like these were the primary sheaves OP was interested in. | |
| Oct 25, 2016 at 23:17 | history | edited | Steven Gubkin | CC BY-SA 3.0 |
deleted 63 characters in body
|
| Oct 25, 2016 at 23:16 | comment | added | Steven Gubkin | @BenMcKay Whoops, cannot believe I made that mistake! Editing. | |
| Oct 25, 2016 at 15:47 | comment | added | Ben McKay | This doesn't motivate the sheaf cohomology of very many different sheaves. | |
| Oct 25, 2016 at 15:46 | comment | added | Ben McKay | Closed forms are interesting because their integrals are not always 0. | |
| Oct 25, 2016 at 14:57 | history | answered | Steven Gubkin | CC BY-SA 3.0 |