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.

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Volumes of p-adic Schemes

I came across this idea a long time ago, but I needed the result that uses it in its proof again, so I was curious about figuring out what in the world is going on. It turns out that you can make “{p}-adic measures” to integrate against on algebraic varieties. This is a pretty cool idea that I never would have guessed possible. I mean, maybe complex varieties or something, but over {p}-adic fields?

Let’s start with a pretty standard setup in {p}-adic geometry. Let {K/\mathbb{Q}_p} be a finite extension and {R} the ring of integers of {K}. Let {\mathbb{F}_q=R_K/\mathfrak{m}} be the residue field. If this scares you, then just take {K=\mathbb{Q}_p} and {R=\mathbb{Z}_p}.

Now let {X\rightarrow Spec(R)} be a smooth scheme of relative dimension {n}. The picture to have in mind here is some smooth {n}-dimensional variety over a finite field {X_0} as the closed fiber and a smooth characteristic {0} version of this variety, {X_\eta}, as the generic fiber. This scheme is just interpolating between the two.

Now suppose we have an {n}-form {\omega\in H^0(X, \Omega_{X/R}^n)}. We want to say what it means to integrate against this form. Let {|\cdot |_p} be the normalized {p}-adic valuation on {K}. We want to consider the {p}-adic topology on the set of {R}-valued points {X(R)}. This can be a little weird if you haven’t done it before. It is a totally disconnected, compact space.

The idea for the definition is the exact naive way of converting the definition from a manifold to this setting. Consider some point {s\in X(R)}. Locally in the {p}-adic topology we can find a “disk” containing {s}. This means there is some open {U} about {s} together with a {p}-adic analytic isomorphism {U\rightarrow V\subset R^n} to some open.

In the usual way, we now have a choice of local coordinates {x=(x_i)}. This means we can write {\omega|_U=fdx_1\wedge\cdots \wedge dx_n} where {f} is a {p}-adic analytic on {V}. Now we just define

\displaystyle \int_U \omega= \int_V |f(x)|_p dx_1 \cdots dx_n.

Now maybe it looks like we’ve converted this to another weird {p}-adic integration problem that we don’t know how to do, but we the right hand side makes sense because {R^n} is a compact topological group so we integrate with respect to the normalized Haar measure. Now we’re done, because modulo standard arguments that everything patches together we can define {\int_X \omega} in terms of these local patches (the reason for being able to patch without bump functions will be clear in a moment, but roughly on overlaps the form will differ by a unit with valuation {1}).

This allows us to define a “volume form” for smooth {p}-adic schemes. We will call an {n}-form a volume form if it is nowhere vanishing (i.e. it trivializes {\Omega^n}). You might be scared that the volume you get by integrating isn’t well-defined. After all, on a real manifold you can just scale a non-vanishing {n}-form to get another one, but the integral will be scaled by that constant.

We’re in luck here, because if {\omega} and {\omega'} are both volume forms, then there is some non-vanishing function such that {\omega=f\omega'}. Since {f} is never {0}, it is invertible, and hence is a unit. This means {|f(x)|_p=1}, so since we can only get other volume forms by scaling by a function with {p}-adic valuation {1} everywhere the volume is a well-defined notion under this definition! (A priori, there could be a bunch of “different” forms, though).

It turns out to actually be a really useful notion as well. If we want to compute the volume of {X/R}, then there is a natural way to do it with our set-up. Consider the reduction mod {\mathfrak{m}} map {\phi: X(R)\rightarrow X(\mathbb{F}_q)}. The fiber over any point is a {p}-adic open set, and they partition {X(R)} into a disjoint union of {|X(\mathbb{F}_q)|} mutually isomorphic sets (recall the reduction map is surjective here by the relevant variant on Hensel’s lemma). Fix one point {x_0\in X(\mathbb{F}_q)}, and define {U:=\phi^{-1}(x_0)}. Then by the above analysis we get

\displaystyle Vol(X)=\int_X \omega=|X(\mathbb{F}_q)|\int_{U}\omega

All we have to do is compute this integral over one open now. By our smoothness hypothesis, we can find a regular system of parameters {x_1, \ldots, x_n\in \mathcal{O}_{X, x_0}}. This is a legitimate choice of coordinates because they define a {p}-adic analytic isomorphism with {\mathfrak{m}^n\subset R^n}.

Now we use the same silly trick as before. Suppose {\omega=fdx_1\wedge \cdots \wedge dx_n}, then since {\omega} is a volume form, {f} can’t vanish and hence {|f(x)|_p=1} on {U}. Thus

\displaystyle \int_{U}\omega=\int_{\mathfrak{m}^n}dx_1\cdots dx_n=\frac{1}{q^n}

This tells us that no matter what {X/R} is, if there is a volume form (which often there isn’t), then the volume

\displaystyle Vol(X)=\frac{|X(\mathbb{F}_q)|}{q^n}

just suitably multiplies the number of {\mathbb{F}_q}-rational points there are by a factor dependent on the size of the residue field and the dimension of {X}. Next time we’ll talk about the one place I know of that this has been a really useful idea.

Riemann Hypothesis

Just because a good deal of my readers probably don’t look at any of the blogs on the blogroll under the category of Math, I will share some news that is spreading like wildfire. It seems to be the cool thing to do right now to point out that someone submitted a proof of the Riemann Hypothesis to the arxiv.

Hmm…some terms should be defined I guess. The arxiv is a preprint server for math and physics (and computer science, etc?). It allows you to “publish” your work before it gets picked up by a journal which could take years.

The Riemann Hypothesis is probably the most well-known problem in math. It has to do with the zeros of the Riemann zeta function. Blah, blah, blah…I know you don’t care, but!!! You get a million dollars if you solve it. (It is one of the seven Millenium Problems).

So the supposed solution has already had two people find holes. This type of thing happens all the time. The main reason for the blowing up of this is that the submitter is Xian-Jin Li, a professor at BYU. Usually the “solvers” are amateur mathematicians trying for a million dollars. A move like this is very embarrassing to an established research mathematician.