An Application of p-adic Volume to Minimal Models

Today I’ll sketch a proof of Ito that birational smooth minimal models have all of their Hodge numbers exactly the same. It uses the {p}-adic integration from last time plus one piece of heavy machinery.

First, the piece of heavy machinery: If {X, Y} are finite type schemes over the ring of integers {\mathcal{O}_K} of a number field whose generic fibers are smooth and proper, then if {|X(\mathcal{O}_K/\mathfrak{p})|=|Y(\mathcal{O}_K/\mathfrak{p})|} for all but finitely many prime ideals, {\mathfrak{p}}, then the generic fibers {X_\eta} and {Y_\eta} have the same Hodge numbers.

If you’ve seen these types of hypotheses before, then there’s an obvious set of theorems that will probably be used to prove this (Chebotarev + Hodge-Tate decomposition + Weil conjectures). Let’s first restrict our attention to a single prime. Since we will be able to throw out bad primes, suppose we have {X, Y} smooth, proper varieties over {\mathbb{F}_q} of characteristic {p}.

Proposition: If {|X(\mathbb{F}_{q^r})|=|Y(\mathbb{F}_{q^r})|} for all {r}, then {X} and {Y} have the same {\ell}-adic Betti numbers.

This is a basic exercise in using the Weil conjectures. First, {X} and {Y} clearly have the same Zeta functions, because the Zeta function is defined entirely by the number of points over {\mathbb{F}_{q^r}}. But the Zeta function decomposes

\displaystyle Z(X,t)=\frac{P_1(t)\cdots P_{2n-1}(t)}{P_0(t)\cdots P_{2n}(t)}

where {P_i} is the characteristic polynomial of Frobenius acting on {H^i(X_{\overline{\mathbb{F}_q}}, \mathbb{Q}_\ell)}. The Weil conjectures tell us we can recover the {P_i(t)} if we know the Zeta function. But now

\displaystyle \dim H^i(X_{\overline{\mathbb{F}_q}}, \mathbb{Q}_\ell)=\deg P_i(t)=H^i(Y_{\overline{\mathbb{F}_q}}, \mathbb{Q}_\ell)

and hence the Betti numbers are the same. Now let’s go back and notice the magic of {\ell}-adic cohomology. If {X} and {Y} are as before over the ring of integers of a number field. Our assumption about the number of points over finite fields being the same for all but finitely many primes implies that we can pick a prime of good reduction and get that the {\ell}-adic Betti numbers of the reductions are the same {b_i(X_p)=b_i(Y_p)}.

One of the main purposes of {\ell}-adic cohomology is that it is “topological.” By smooth, proper base change we get that the {\ell}-adic Betti numbers of the geometric generic fibers are the same

\displaystyle b_i(X_{\overline{\eta}})=b_i(X_p)=b_i(Y_p)=b_i(Y_{\overline{\eta}}).

By the standard characteristic {0} comparison theorem we then get that the singular cohomology is the same when base changing to {\mathbb{C}}, i.e.

\displaystyle \dim H^i(X_\eta\otimes \mathbb{C}, \mathbb{Q})=\dim H^i(Y_\eta \otimes \mathbb{C}, \mathbb{Q}).

Now we use the Chebotarev density theorem. The Galois representations on each cohomology have the same traces of Frobenius for all but finitely many primes by assumption and hence the semisimplifications of these Galois representations are the same everywhere! Lastly, these Galois representations are coming from smooth, proper varieties and hence the representations are Hodge-Tate. You can now read the Hodge numbers off of the Hodge-Tate decomposition of the semisimplification and hence the two generic fibers have the same Hodge numbers.

Alright, in some sense that was the “uninteresting” part, because it just uses a bunch of machines and is a known fact (there’s also a lot of stuff to fill in to the above sketch to finish the argument). Here’s the application of {p}-adic integration.

Suppose {X} and {Y} are smooth birational minimal models over {\mathbb{C}} (for simplicity we’ll assume they are Calabi-Yau, Ito shows how to get around not necessarily having a non-vanishing top form). I’ll just sketch this part as well, since there are some subtleties with making sure you don’t mess up too much in the process. We can “spread out” our varieties to get our setup in the beginning. Namely, there are proper models over some {\mathcal{O}_K} (of course they aren’t smooth anymore), where the base change of the generic fibers are isomorphic to our original varieties.

By standard birational geometry arguments, there is some big open locus (the complement has codimension greater than {2}) where these are isomorphic and this descends to our model as well. Now we are almost there. We have an etale isomorphism {U\rightarrow V} over all but finitely many primes. If we choose nowhere vanishing top forms on the models, then the restrictions to the fibers are {p}-adic volume forms.

But our standard trick works again here. The isomorphism {U\rightarrow V} pulls back the volume form on {Y} to a volume form on {X} over all but finitely primes and hence they differ by a function which has {p}-adic valuation {1} everywhere. Thus the two models have the same volume over all but finitely many primes, and as was pointed out last time the two must have the same number of {\mathbb{F}_{q^r}}-valued points over these primes since we can read this off from knowing the volume.

The machinery says that we can now conclude the two smooth birational minimal models have the same Hodge numbers. I thought that was a pretty cool and unexpected application of this idea of {p}-adic volume. It is the only one I know of. I’d be interested if anyone knows of any other.

3 thoughts on “An Application of p-adic Volume to Minimal Models

  1. Regarding your “piece of heavy machinery” from the beginning: why is just knowing that \left| X(\mathcal{O}_K/\mathfrak{p}) \right| = \left| Y(\mathcal{O}_K/\mathfrak{p}) \right| enough? In your proof sketch you are using that \left| X(\mathbb{F}_{q^r}) \right| = \left| Y(\mathbb{F}_{q^r}) \right| for all r, not just r=1, which doesn’t seem to follow so easily… Or do you perhaps want to require \left| X(\mathcal{O}_L/\mathfrak{q}) \right| = \left| Y(\mathcal{O}_L/\mathfrak{q}) \right| for “almost all” L \supset K and \mathfrak{q} \in Spec(\mathcal{O}_L)?

  2. Yes. You are quite right that the hypothesis that |X(\mathbb{F}_{q^r})|=|Y(\mathbb{F}_{q^r})| is much stronger than the claimed hypotheses. The outline of the proof is basically the same, but you just have to be more careful because you can’t a priori argue that the dimensions of those cohomology groups are the same.

    It turns out not to matter because Chebotarev density doesn’t require the Galois representations to have the same dimension. You merely get that they have the same semi-simplifications, and then machinery from p-adic Hodge theory gives you enough to get to the end. As you point out, this result should still apply to varieties of the same dimension in characteristic 0 that have the same number of points over \mathbb{F}_q, but different Zeta functions.

    The full proof is given in section 4.4 of Ito’s paper: . It is pretty interesting and not very long at all.

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