Let’s construct the example today as a quick post. First, we’ll need a theorem that will be used to show that HdR doesn’t degenerate. Let be a smooth variety over an algebraically closed field of positive characteristic with the property that as a group scheme. Then HdR does not degenerate for . We’ll use a lot of things we haven’t talked about or proved, but the purpose of this post is to give the example and a flavor of the why it is true. Later we’ll come back and look at the parts that go into it more carefully.
Here is how this is proved. It is well-known that for HdR to degenerate it must be the case that all global -forms are closed. So we assume this is the case otherwise we are done. Now Oda has a theorem that says when we are in this case where is the Dieudonne module. By assumption that we get that the first de Rham Betti number .
Since is isomorphic to the tangent space to the Picard scheme we get that , but also gives a global -form on the Picard scheme, so . Thus we have a contradiction if HdR degenerates and hence this does not happen.
This gives our example if we can come up with something that satisfies those properties. Let be the extension of of ramification index . Let and be a finite flat group scheme over such that . It is a theorem of Tate and Oort that if there are elements and in such that , then such a exists.
Now it is a theorem of Raynaud that there is a projective space over with a linear action of which contains a relative complete intersection surface which is stabilized by and such that acts freely on and is smooth over . It follows that . Thus for any characteristic we have an example of a smooth variety that lifts to characteristic over a very small ramified extension of but the HdR spectral sequence does not degenerate! We’ll try to unpack this better next time.