This post is about a computation every algebraic geometry student should do, but that none of my courses covered. Let be a smooth, projective curve over . Then is a one dimensional -vector space. If you’ve read Hartshorne III.7 carefully, you’ll remember that there is a canonical isomorphism: . Explicitly, let and be two points of ; consider the open cover of and let be a holomorphic -form on . Let be the cocycle . Then sends the cocycle to the residue, at , of . (A good question which I might ask on a qual one day: Why is it OK that this is asymmetric in and ?)
On the other hand, suppose that . Then is isomorphic to . An element of is a -closed form, modulo -exact -forms. But, because only has two real dimensions, this simplifies: Every form is a -form and every -form is closed because there are no -forms. So an element of is a -form modulo -exact -forms. It turns out that this is the same as a -form modulo -exact two forms. In other words, is the same as the deRham cohomology group that we learn about in differential geometry. And we know a canonical map : Take the integral!
The point of this post is to compute the relation between and . I invite you to try it yourself, then meet me on the other side to see if we got the same answer.
UPDATE: I claimed earlier that this was easy to show for curves. As Akhil Mathew points out, it seems to only be easy to show that there is a well defined surjection . Since I only need the map to exist, I’ll leave it at that for now.
In order to do this computation, we need to remember exactly how the isomorphism is built. From now on, all my sheaf cohomology computations will be with respect to the analytic topology. (Also known as “the picture I am drawing when I draw a Riemmann surface”.) By GAGA, for coherent sheaves on , sheaf cohomology gives the same answers with respect to the analytic and Zariski topologies.
Let be the sheaf of -forms. So a section of locally looks like where is a smooth, complex valued function, while a section of locally looks like where is a complex-analytic function. See my earlier post for a review of these notions.
We have a short exact sequence
Locally, the first map sends a -form of the form , with analytic, to itself, considered as an example of a -form of the form , with smooth. The second map sends a -form of the form , with smooth, to . Exactness on the right is by a variant of the Poincare lemma.
Now, is a module over the sheaf of functions, and that sheaf has partitions of unity. A standard argument then shows that vanishes for . See Voisin, Hodge Theory and Algebraic Geometry I Proposition 4.36, for the details.
So, we have a long exact sequence of groups:
Our task is to take the class of in and lift to a volume form in , then compare to .
We first refine the open cover . Let be a small open discs centred at and let be a smaller closed disc, contained in and containing . Choose and similarly, with and disjoint. Let . Our new open cover will be . We want to pull the Cech cocycle back to a cocycle on the new cover.
A technical note: To pull back , we need to decide how the new cover is a refinement of the old one. In other words, for each open set in the new cover, we need to choose a specific set in the old cover which contains it. Our choice is that is regarded as a subset of .
With the above choice, the pullback of to the new cover is on and on . Call this cocycle .
We now look for a preimage of in . Remembering how the boundary map in Cech cohomology is defined, we first find sections of on , and whose difference on the overlaps is . We will take on , on and on . Here must be a smooth form, whose restriction to is . A standard trick with bump functions shows that such a smooth extension exists. (Exercise!)
Let be the form which is on and on . We check that this is well defined: On , we have so which is because is holomorphic. By the definition of the boundary map in Cech cohomology, is a preimage of . Our job is to compute .
At this point, the sheaf cohomology is done, and we switch to complex analysis. Everything is supported on , so we can work locally. The question is this: Let , in the complex plane, be a point, a closed disc, and an open disc. Let be a holomorphic -form on . Let be a smooth -form, which agreees with on . What is ?
Here is a nonrigourous way I like to think about this process. Let be the disc of radius and let . Suppose that, instead of requiring that and coincided on an annulus, we only had to make them match up on the boundary of . Then we could use . In general, turning into is like noticing that equals on the unit circle, but smeared out over an annulus.
The integral is crying out for an application of Stokes’ theorem. Since is a -form, is . So, after shrinking a little to make sure extends to the boundary, we have
But the last is simply the . So The computation is done.