Nondegenerate Hodge de Rham

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 ${Z}$ be a smooth variety over an algebraically closed field of positive characteristic with the property that ${\mathrm{Pic}^\tau(Z)\simeq \alpha_p}$ as a group scheme. Then HdR does not degenerate for ${Z}$. 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 ${1}$-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 ${H^1_{dR}(Z/k)\simeq D(pPic^\tau(Z))}$ where ${D(-)}$ is the Dieudonne module. By assumption that ${\mathrm{Pic}^\tau(Z)\simeq \alpha_p}$ we get that the first de Rham Betti number ${h^1_{dR}=1}$.

Since ${H^1(Z, \mathcal{O}_Z)}$ is isomorphic to the tangent space to the Picard scheme we get that ${1=\dim H^1(Z, \mathcal{O}_Z)}$, but also ${\alpha_p}$ gives a global ${1}$-form on the Picard scheme, so ${\dim H^0(Z, \Omega^1)\geq 1}$. 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 ${R}$ be the extension of ${W(k)}$ of ramification index ${2}$. Let ${S=\textrm{Spec}(R)}$ and ${G}$ be a finite flat group scheme over ${S}$ such that ${G_0\simeq \alpha_p}$. It is a theorem of Tate and Oort that if there are elements ${a}$ and ${c}$ in ${\frak{m}_R}$ such that ${ac=p}$, then such a ${G}$ exists.

Now it is a theorem of Raynaud that there is a projective space ${P}$ over ${S}$ with a linear action of ${G}$ which contains a relative complete intersection surface ${Y}$ which is stabilized by ${G}$ and such that ${G}$ acts freely on ${Y}$ and ${Y=X/G}$ is smooth over ${S}$. It follows that ${\textrm{Pic}^\tau(X_0)\simeq G_0^D\simeq \alpha_p}$. Thus for any characteristic we have an example of a smooth variety that lifts to characteristic ${0}$ over a very small ramified extension of ${W(k)}$ but the HdR spectral sequence does not degenerate! We’ll try to unpack this better next time.