# When does Hodge-de Rham Degenerate?

I wrote up half of the follow up to the last post almost immediately, but then got really stuck. I wasn’t sure how to avoid the technical details and still have it be a worthwhile post. I’ve tried twice to do it, and now I’ve decided it just isn’t worth going down that road right now. Instead I’ve thought of something else to do which I find quite amazing and shocking. It also fits in really well with the past series of posts. In this post I’ll overview the example and why it is shocking. Next time I’ll give the construction. Then we’ll end with a closer examination of what is going on and why it shouldn’t be shocking that it exists.

Let’s recall some facts and theorems about the Hodge-de Rham spectral sequence. We’ve thought quite a bit about this over the past six months or so. Let ${X}$ be a smooth projective variety over a field of characteristic ${0}$. Deligne and Illusie came up with a pretty crazy way to prove that HdR degenerates at ${E_1}$. First off, degeneration is somewhat rare in positive characteristic, but they decided to reduce mod ${p}$ and prove that if ${Y}$ is a scheme over a perfect field of characteristic bigger than ${\dim Y}$ and ${Y}$ admits a lift to ${W_2(k)}$, then HdR SS degenerates at ${E_1}$. Since this always happens for varieties that came from characteristic ${0}$, it is true for ${Y}$. To finish it off we can extrapolate back.

The moral of this story is that we have degeneration in characteristic ${0}$, and we have degeneration in positive characteristic if we have even just a single step in trying to lift to characteristic ${0}$. Note that there are varieties that lift to ${W_2(k)}$ but do not lift to ${W_3(k)}$, and there are varieties that admit a formal lift by being able to lift each stage from ${W_n(k)}$ to ${W_{n+1}(k)}$ all the way up and then don’t algebraize to get an actual lift to characteristic ${0}$. Both of these types of varieties have HdR degenerate at ${E_1}$! They don’t have to lift to get the degeneration. In some sense they merely need to exhibit some sort of “evidence” that a lift is possible.

Lastly, we have a theorem that relates degeneration of HdR of the generic and closed fibers in a flat family over a DVR. It says that if ${\frak{X}\rightarrow S=\mathrm{Spec}(R)}$ is a flat family over a DVR and ${R/m\simeq k}$ is of characteristic ${p}$ and ${\mathrm{Frac}(R)=K}$ is of characteristic ${0}$, then if ${\dim H^q(X_0, \Omega^p)=\dim H^q(X_\eta, \Omega^p)}$ then HdR degenerates at ${E_1}$ for ${X_0}$. In other words, if we have a lift of ${X_0}$ to characteristic ${0}$ (over ANY ring, not necessarily ${W(k)}$), and the Hodge numbers match the lifted Hodge numbers this guarantees degeneration. This argument is quite simple and just involves counting dimensions and upper-semicontinuity, so we actually didn’t need characteristic ${0}$. It can be generalized to say if HdR degenerates on the generic fiber (of whatever characteristic) then it degenerates on the special fiber.

Let’s just recap here. HdR degenerates for characteristic ${0}$ things. Things that merely exhibit evidence of a lift to characteristic ${0}$ by lifting to ${W_2(k)}$ have HdR degenerate (this includes things that provably can’t be lifted all the way to characteristic ${0}$). We can relate degeneration of HdR on the generic fiber to degeneration on the special fiber of a flat family. One might make the guess that if you have an actual honest lift to characteristic ${0}$ (rather than just “evidence”) that HdR must degenerate. You may even think you have a proof by just saying lift it ${\frak{X}\rightarrow S}$ and now since the generic fiber degenerates the special fiber must also.

You would be wrong. William Lang came up with an example of a smooth variety (over any characteristic that you want) that lifts to characteristic ${0}$ but has non-degenerate HdR spectral sequence. This should be sort of shocking in light of the above theorems and “proof” outline. I’m going to partially give away the punchline now. There are two things going on here. First, we needed to be able to say that the Hodge numbers matched up after doing the lifting to say that degeneration of one implied degeneration of the other. This example will not have that property. The other thing going on is that it seems that degeneration of HdR somehow really has something to do with liftability to ${W_2(k)}$ and is not something that is merely about characteristic ${0}$. This example will be a lift over a ramified (of degree ${2}$, so it is as close to ${W(k)}$ as possible) extension of ${W(k)}$.