1
$\begingroup$

Let $k$ be a perfect field of characteristic $p>0$ and consider an ordinary liftable Calabi-Yau threefold $X_{0}/k$. By this I mean that $H^{i}(X_{0},B_{X_{0}/k}^{j})=0$ for all $i\geq 0$ and $j\geq 1$, where $B_{X_{0}/k}^{j}:=\text{im}(d:\Omega_{X_{0}/k}^{j-1}\rightarrow\Omega_{X_{0}/k}^{j})$, and $\mathfrak{X}/W(k)$ is a lifting of $X_{0}$ to characteristic zero.

Now, let $X/R$ be a proper smooth lifting of $X_{0}$ over an artinian local algebra $R$ and let $A$ be an artinian local $R$-algebra with residue field $k$. I would like to show that \begin{equation*} H_{\text{fl}}^{2}(X\otimes_{R}A,\mu_{p^{\infty}})=0 \end{equation*} I have two questions...

a) Is this true?

b) If so, how do I show it?

Remarks:-

1) This is true if instead $X_{0}$ was a K3 surface and I was looking at $H_{\text{fl}}^{1}(X\otimes_{R}A,\mu_{p^{\infty}})$.

2) Under our hypotheses on $X_{0}$, $Br(X_{0})$ has no $p$-torsion, so I can show that \begin{equation*} H_{\text{fl}}^{2}(X\otimes_{R}A,\mu_{p^{r}})\cong\frac{Pic(X\otimes_{R}A)}{Pic(X\otimes_{R}A)^{p^{r}}} \end{equation*} but I don't know how to attack the right hand side.

Thank you very much for your help, and please don't hesitate to point out mistakes or ask for clarifications.

Improve this question
$\endgroup$
3
  • $\begingroup$ This seems false, because the Picard group might be $\mathbb Z$, in which case your formula shows that the cohomology group is $\mathbb Q_p/\mathbb Z_p$. $\endgroup$
    – Will Sawin
    Commented Jul 28, 2015 at 4:10
  • $\begingroup$ @WillSawin Yes, I had come to the same conclusion but I was rather hoping that somebody here would tell me that the hypotheses excludes this sort of thing. For example, in the K3 surface case I mentioned, this all works out because $k$ is perfect. Unfortunately I tend to agree with you that this seems false, which is rather a nuisance for the construction I had in mind... $\endgroup$
    – Oli Gregory
    Commented Jul 28, 2015 at 7:53
  • $\begingroup$ Thinking about it, I guess that a generic quintic threefold satisfies the hypotheses and provides a counter-example. Thanks to Daniel Litt for pointing this out; I definitely should've thought through that case. $\endgroup$
    – Oli Gregory
    Commented Sep 1, 2015 at 11:08

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .