H.-Kaletha-Weinstein study guide and FAQ

Tasho Kaletha, Jared Weinstein and I have just posted a joint paper, On the Kottwitz conjecture for local shtuka spaces. This is a heavily revised and rewritten version of a preprint by Tasho and Jared which has been available since 2017 (henceforth “KW”).

Is the main result here the same as in KW?
More or less, yes. The main theorem confirms the Kottwitz conjecture for the cohomology of moduli spaces of local shtukas, ignoring the Weil group action and allowing for an “error term” consisting of a non-elliptic virtual representation.

Is the rough idea of the proof still the same as in KW?
Yes: the idea is still to compute the cohomology by applying a suitable Lefschetz-Verdier trace formula, and explicitly computing the “local terms” at all of the elliptic fixed points.

OK, what’s new, then?
Well, basically all of the details are different. First and foremost, the discussion of the relevant Lefschetz-Verdier trace formula for v-stacks has been completely rewritten, using Lu-Zheng’s magical formalism of symmetric monoidal 2-categories of cohomological correspondences. This leads to huge conceptual simplifications, both in the statements and the proofs. Actually, we go beyond the formalism of Lu-Zheng, introducing certain 2-categories of “based cohomological correspondences”. This formalism play a crucial role in the proof of Theorem 4.5.3, which is a key ingredient in the proof of the main theorem. It’s also hard to imagine proving Theorem 4.6.1 without this formalism.

Secondly, a key calculation of local terms on the B_{\mathrm{dR}}-affine Grassmannian, Theorem 5.1.3, now has a completely different and correct (!) proof, via a degeneration to a similar calculation on the Witt vector affine Grassmannian. This degeneration argument relies on the observation that the local terms appearing in the Lefschetz-Verdier trace formula are compatible with any base change, Proposition 5.3.1; this seems to be a new observation even for schemes. (My first proof of Proposition 5.3.1 involved checking the commutativity of 500 diagrams, but eventually I realized that it follows immediately from the Lu-Zheng formalism!) Once we’ve degenerated to the Witt vector affine Grassmannian, we make an intriguing global-to-local argument using the trace formula and known properties of the weight functors in geometric Satake, and relying crucially on a recent result of Varshavsky.

There are some other notable improvements:

  • All assumptions in KW of the form “assume that some representation admits an invariant \overline{\mathbf{Z}_{\ell}}-lattice” have now been eliminated. This is far from formal, since we definitely don’t have a 6-functor formalism with \overline{\mathbf{Q}_{\ell}}-coefficients at hand. Instead, the idea is to reduce “by a continuity argument” to the case where things do admit lattices. The key ingredient here is Theorem 6.5.4, formerly a conjecture of Taylor, which should have lots of other applications.
  • The final part of the main theorem, giving practical criteria for the error term to vanish, is new.
  • Theorem 1.0.3 is new.

On a practical note, the Fargues-Scholze paper is now finished and available, so we can properly build on the powerful machinery and results in that paper.

Finally, the margins are no longer gigantic.

Any advice on how to read the paper?
Sure. The main Theorem 1.0.2/6.5.1 in the paper is essentially a combination of two separate results, Theorem 6.5.2 and Theorem 3.2.9. Theorem 3.2.9 is conditional on the refined local Langlands correspondence, and the material needed to formulate and prove this result is developed in Chapter 3 and Appendix A. The bulk of the paper (sections 4-6 and Appendices B-C) is devoted to building up enough material to prove Theorem 6.5.2.

Each of sections 3, 4, and 5 is fairly self-contained, although section 5 depends slightly on section 4. Section 6 builds heavily on sections 4 and 5, and also on [FS21]. Section 5 is actually devoted to proving a single result, Theorem 5.1.3 (and its reinterpretation Theorem 5.1.4), and can be treated as a black box. 

One thing you could do is read the introduction, then jump to Proposition 6.4.7 and work your way backwards as needed until all the notation makes sense. This proposition is the technical heart of the paper. The key inputs into the proof of Proposition 6.4.7 are Theorem 5.1.4, Theorem 4.5.3, and Corollary 4.3.8. The use of “the trace formula” in this whole argument is actually somewhat hidden: it is entirely encoded in the equality sign labelled “Cor. 4.3.8” at the bottom of p. 66.

From Proposition 6.4.7, Theorem 6.4.9 (a weak form of Theorem 6.5.2) follows quickly, and then Theorem 6.5.2 follows from Theorem 6.4.9 by a continuity argument which relies crucially on Theorem 6.5.4.

This sounds cool, but you didn’t punt any key ingredients into a separate and currently nonexistent paper, did you?
I’m glad you asked. In working out the trace formula formalism in enough generality to give a conceptual proof of the main theorem, it became clear at some point that we needed the existence and expected properties of the functors Rf_! and Rf^! in etale cohomology for “smooth-locally nice” maps f between Artin v-stacks. For maps which are actually “nice”, and in particular non-stacky, these functors were constructed by Scholze, but extending them to the setting of stacky maps is not formal. Already in the analogous setting of schemes versus Artin stacks, these functors for Artin stacks were only constructed recently by Liu-Zheng, making heavy use of \infty-categories. In a forthcoming paper, Jared and I together with my postdoc Dan Gulotta will explain how to construct Rf_! and Rf^! for smooth-locally nice maps of Artin v-stacks. The arguments will very closely follow the arguments of Liu-Zheng, with the exception of one key non-formal piece of input (which took me three years of intermittent work).

Why do you only prove an explicit formula for the virtual character of \mathrm{Mant}_{b,\mu}(\rho) restricted to elliptic elements of G(F)? What’s so special about elliptic elements in this context?
The fixed points of elliptic elements lie in the “admissible locus” inside the relevant closed Schubert cell. This is discussed in detail in the introduction.

Fine, but maybe there’s still an obvious explicit formula for the virtual character of \mathrm{Mant}_{b,\mu}(\rho) at any strongly regular semisimple element.
Here’s an example to illustrate why I think this is a hard problem. I’ll be slightly heuristic about the details; if you want more precision, please leave a comment.

Let’s take G= \mathrm{GL}_2 and \mu=(1,0), so we’re in the Lubin-Tate/Drinfeld setting with G_b(F)=D^\times the units in the quaternion algebra over F. Let \rho be the trivial representation of D^\times. Then \mathrm{Mant}_{b,\mu}(\rho) = \mathrm{St}-\mathbf{1} as a virtual representation of G(F), by an old calculation of Schneider-Stuhler. Note that \mathrm{St}+\mathbf{1} is a principal series representation, hence non-elliptic, so the virtual character of \mathrm{St}-\mathbf{1} on elliptic elements of G(F) is the constant function -2. This matches perfectly with the fact that any elliptic g\in G(F) has two fixed points in \mathbf{P}^1, both contained in \Omega^{1}, and the “naive” local terms of the relevant sheaf j_!\mathbf{Z}_{\ell}[1] at both these points are -1. Here j:\Omega^1 \to \mathbf{P}^1 is the evident open immersion.

On the other hand, if g \in G(F) is regular semisimple and nonelliptic, then it’s conjugate to some t=\mathrm{diag}(t_1,t_2) with t_1 \neq t_2 \in F^\times. In this case there are still two fixed points, but they both lie in the “boundary” \mathbf{P}^1-\Omega^{1}. Since j_!\mathbf{Z}_{\ell}[1] restricted to the boundary is identically zero, its naive local terms at both fixed points are zero. Nevertheless, the trace formula still works, so the contribution of the true local terms at these two points must add up to the virtual character of \mathrm{St}-\mathbf{1} evaluated at t. This character value can be computed explicitly by van Dijk’s formula, and turns out to be -2+\tfrac{|t_1| + |t_2|}{|t_1-t_2|}. So this slightly strange expression needs to emerge from the sum of these two local terms. 

Putting these observations together, we see that in this example, the true local terms do not equal the naive local terms at the non-elliptic fixed points. Morally, the analysis in Section 3 of the paper can and should be interpreted as an unravelling of naive local terms (at elliptic fixed points). Hopefully this makes the difficulty clear.

Can you give an example where the error term in Theorem 1.0.2 is nonzero?
Sure, look at the two-dimensional Lubin-Tate local Shimura datum and take \rho trivial again, so \rho \in \Pi_{\phi}(G_b) with \phi the Steinberg parameter. Then \mathrm{Mant}_{b,\mu}(\rho) = \mathrm{St}-\mathbf{1} as discussed in the previous answer, while the sum on the right-hand side in Theorem 1.0.2 is just 2 \mathrm{St}. So in this case \mathrm{err}= -(\mathrm{St}+\mathbf{1}) is, up to sign, a reducible principal series representation.

It would be very interesting to formulate an extension of the Kottwitz conjecture which covered all discrete L-parameters. Since the Kottwitz conjecture is best regarded as a reflection of the Hecke eigensheaf property in Fargues’s conjecture, this is closely related to understanding Fargues’s conjecture for general discrete parameters.

It seems like the Kottwitz conjecture should also make sense for local shtukas over Laurent series fields. Why doesn’t the paper cover this case too? 
Good question! The short answer is that all of the p-adic geometry should work out uniformly over Laurent series fields and over finite extensions of \mathbf{Q}_p, and indeed Fargues-Scholze handles these two situations uniformly. However, the literature on local Langlands and harmonic analysis for reductive groups over positive characteristic local fields case is pretty thin. This is the main reason we stuck to the mixed characteristic case. For more on the state of the literature, take a look at section 1.6 in Kal16a.

Do the methods of this paper give any information about \mathrm{Mant}_{b,\mu} for non-basic b?
Yes: If b is non-basic, or b is basic and \rho is parabolically induced, our methods can be applied to prove that \mathrm{Mant}_{b,\mu}(\rho) is always a virtual combination of representations induced from proper parabolic subgroups of G. This is a soft form of the Harris-Viehmann conjecture. The details of this argument will be given elsewhere (in the sense of Deligne-Mumford).

Any surprising subtleties?
Lemma 6.5.5 was a bracing experience. My first “proof” was absolute nonsense, due to some serious misconceptions I had about harmonic analysis on p-adic groups – big thanks to JT for setting me straight. My second proof didn’t work because I cited a result in a famous article (>100 mathscinet citations) whose proof turns out to be fallacious – big thanks to JFD for setting me straight. Tasho and I devised the current proof, which in hindsight is actually a very natural argument. (I was hoping, apparently unreasonably, for a shortcut.) 

Closing thoughts:
-At least for me, the proof of Theorem 6.5.4 is a very striking illustration of the raw Wagnerian power of the machinery developed in [FS21]. This result was totally out of reach a few years ago, but with the machinery of [FS21] in hand, the proof is less than a page! Note that this argument crucially uses the D_{\mathrm{lis}} and Hecke operator formalism of [FS21] with general coefficient rings.
-Let me again emphasize how heavily the paper depends on the recent preprints [LZ20], [Var20], and (of course) [FS21].
-The paper depends, more significantly than most, on a choice of isomorphism \mathbf{C} \simeq \overline{\mathbf{Q}_{\ell}}. It’s naturally of interest to wonder how everything depends on this choice. Imai has formulated a conjectural answer to this question. Of course, “independence of \ell” results in the etale cohomology of diamonds are probably very hard.


Several more things

Next year
I’m very pleased that several outstanding young p-adic geometers are coming to Bonn soon: Haoyang Guo, Emanuel Reinecke, and Bogdan Zavyalov will all begin postdoc positions at the MPIM in September, and Ian Gleason will also begin a postdoc at the university in the fall. The group of p-adic people here was already pretty strong, but now it’ll just be off the charts.  And as a bonus, JB will be here starting in July!

Next week
Starting next week, I’m giving three lectures in the Arithmetic Monday seminar on local Shimura varieties and their cohomology. The goal here is to give a gentle introduction to local Langlands and local Jacquet-Langlands correspondences, and the idea that they should be realized in the cohomology of local Shimura varieties (whatever those are). I’ll try to keep everything down to earth and example-based, and I won’t really prove anything. Nevertheless, I hope these lectures might be useful to people starting out in this area.

Several things

A rant about valuations
If you look at any detailed introductory text on adic spaces (e.g. the notes of Conrad, Morel, Wedhorn, etc.) you’ll find lots and lots and lots of preliminary material on valuation theory. On the one hand, this isn’t so crazy, since adic spaces are built from valuation spectra, and you have to eat your spinach before you get to have cake. On the other hand, I think this is pretty unfortunate, since valuation theory is incredibly boring and dry, and (more importantly) when you actually work with analytic adic spaces in real life, you never need to worry about most of this material. How often have I worried about horizontal specializations versus vertical specializations, etc.? Essentially never.

Open problems
Suppose you want to study representation theory of p-adic reductive groups with coefficients in some Noetherian ring R with p \in R^\times. You might be surprised to learn that the following basic results are all unknown in general (as Jean-Francois Dat kindly pointed out to me), even when R is a DVR:
-parabolic induction preserves finitely generated objects,
-Jacquet modules preserve admissible objects,
-products of cuspidal objects are cuspidal,
-the category is Noetherian,
-second adjointness holds.
I believe the state of the art here is a paper of Dat from 2009, which explains the interrelations between these problems and solves them for many groups. Does anyone have in mind a strategy to solve these problems completely? I would love to know.

The dangers of naming things after people

I spent part of last weekend reading Alice Silverberg’s blog, which is simultaneously depressing and hilarious. Everyone should read it, but you probably won’t enjoy reading it unless you enjoy Coen brothers movies or the short stories of Kafka. Anyway, the following thoughts have been in my head for a few months, but I decided to record them here after reading this.

The cohomology of non-basic local Shimura varieties is described by the “Harris-Viehmann conjecture”, which is formally stated as Conjecture 8.4 in Rapoport-Viehmann’s paper. This story started with a daring and beautiful conjecture of Harris (conjecture 5.2 here), whose formulation however turned out to be slightly incorrect in general, cf. Example 8.3 in RV. The conjecture was then modified by Viehmann, and Rapoport named this modified conjecture the Harris-Viehmann conjecture (footnote 5 in RV).

Unfortunately, Conjecture 8.4 in RV is still not correct as stated: the Weil group action on the summands appearing on the right-hand side needs to be modified by certain half-integral Tate twists. As far as I know, Alexander Bertoloni-Meli is the only person who has publicly pointed out the need for this modification, and Conjecture 3.2.1 in his very cool paper is the only correct formulation of the Harris-Viehmann conjecture in print. 

Since the need for these Tate twists was overlooked by a lot of very smart people, it only seems fair to me that Alexander should get credit for his contribution here. The obvious way to do this would be to refer to the Harris–Viehmann–Bertoloni-Meli conjecture, or the Bertoloni-Meli–Harris–Viehmann conjecture. You could pick the second option if you’re a stickler for alphabetical name orders in mathematics, or the first option if you feel (as I do) that Harris’s contribution here deserves priority.

But it gets even worse, because Harris also formulated another conjecture along similar lines (conjecture 5.4 in his article linked above), which has gotten somewhat less attention but which is nevertheless extremely interesting.* It turns out that one can formulate a unified conjecture which encompasses both Harris’s conjecture 5.4 and the Harris–Viehmann–Bertoloni-Meli conjecture. What should it be called? The Harris–Viehmann–Bertoloni-Meli—Harris conjecture? I guess not.

*Here’s a comment from MH: “I was (and am) much more attached to this conjecture than to the one that is called the Harris-Viehmann conjecture, because it required some work to find the right formalism (the parabolics that transfer between inner forms), whereas the other conjecture (independently of the incorrect formulation in my paper) was just the obvious extension of Boyer’s result.”

Remarks on Fargues-Scholze, part 2

Today I want to talk about section VII.3 in the manuscript. Here they define and study a functor f_{\natural} on solid sheaves which is left adjoint to the usual pullback functor. But why is this called relative homology?

To explain this name, you have to remember that in the usual formalism of etale cohomology, homology is realized as the compactly supported cohomology of the dualizing complex. Symbolically, if X is a variety with structure map f:X \to \mathrm{Spec} k, then the homology of X is given by Rf_! Rf^! \Lambda. Now, you might ask whether this works in families: if f:X \to Y is some map of varieties, maybe I can find some complex on Y whose stalk at y realizes the homology of X_y? If the constant sheaf is f-ULA, then formation of Rf^! \Lambda commutes with any base change, and exactly the same formula works, but in general there is no naive sheaf with this property.

The punchline now is that f_{\natural} \Lambda does have this property: when the constant sheaf is f-ULA (e.g. if Y is a point) it agrees with Rf_! Rf^! \Lambda by Proposition VII.5.2, and its formation commutes with arbitrary base change, so it really does give a complex on Y whose stalks realize the homology of the fibers of f. The only twist is that f_{\natural}\Lambda is a solid sheaf in general, not a classical etale sheaf.

Remarks on Fargues-Scholze

The Fargues-Scholze geometrization paper is available! In this post, and probably some future posts also, I’ll make some random comments on this paper. These won’t mean anything unless you’ve read (at the very least) the beautifully written introduction to the paper. To be clear, I have nothing of substance to say about the “big picture” – these will be purely technical remarks.

First of all, at the bottom of p. 324, one finds the slightly cryptic claim that although there are no general i_! functors in the D_{lis} setting (for i a locally closed immersion), one can define functors i^{b}_! in the D_{lis} setting, where i^b: \mathrm{Bun}_{G}^b \to \mathrm{Bun}_G is the inclusion of any Harder-Narasimhan stratum into \mathrm{Bun}_G. This is stated without proof. However, if you read carefully, you will notice that these lisse i^{b}_! functors are actually used in the paper, in the proof of Proposition VII.7.6. So maybe it’s worth saying something about how to construct them.

To build i^{b}_! in the D_{lis} setting, factor i^b as the composition \mathrm{Bun}_{G}^b \overset{i}{\to}  \mathrm{Bun}_{G}^{ \leq b} \overset{j}{\to} \mathrm{Bun}_G. Here \mathrm{Bun}_{G}^{ \leq b} is the open substack of bundles which are “more semistable” than \mathcal{E}_b. It will also be convenient to write \mathrm{Bun}_{G}^{ < b} = \mathrm{Bun}_{G}^{ \leq b} - \mathrm{Bun}_{G}^{ b}. Note that i is a closed immersion, and j is an open immersion, so j_! = j_{\natural} clearly preserves D_{lis}. The subtlety is in making sense of i_!, since then we can write i_{!}^{b} = j_! i_! as usual.

For i_!, we need the local chart \pi_{b}: \mathcal{M}_b \to \mathrm{Bun}_{G}^{\leq b} and its punctured version \pi_{b}^\circ : \mathcal{M}_{b}^\circ = \mathcal{M}_{b} \times_{\mathrm{Bun}_{G}^{\leq b}} \mathrm{Bun}_{G}^{< b} \to \mathrm{Bun}_{G}^{\leq b}. Recall that these charts also come with compatible maps q_b: \mathcal{M}_b \to [\ast / G_b(E)] and q_b^{\circ}: \mathcal{M}_{b}^{\circ} \to [\ast / G_b(E)]. Then for any A \in D_{lis}(\mathrm{Bun}_{G}^{b},\Lambda) \cong D_{lis}([\ast / G_b(E)],\Lambda), the correct definition turns out to be

i_! A = \mathrm{Cone}(\pi_{b \natural}^{\circ} q_{b}^{\circ \ast}A \to \pi_{b \natural} q_{b}^{\ast}A)\,\,\,\,(1).

The point here is that in the lisse world, the only pushforward functors which come for free are the functors f_{\natural} for cohomologically smooth maps f. Since \pi_{b} and \pi_{b}^{\circ} are cohomologically smooth – one of the hardest theorems in the paper! – the above construction preserves D_{lis}. Moreover, it’s easy to check that the formula above has the right properties. Indeed, the *-restriction of the RHS of (1) to \mathrm{Bun}_{G}^{b} is just A, by Proposition VII.7.2, while its complementary restriction to \mathrm{Bun}_{G}^{ < b} clearly vanishes.

Families of perverse sheaves

In this post I want to talk about some ongoing joint work with Peter Scholze. Since this came up in Scholze’s geometrization lectures, I thought it would be fun to go into a little more detail here. All inaccuracies below are entirely due to me, and the standard caveats about blog-level rigor apply.

The goal, broadly speaking, is to define a relative notion of perversity in etale cohomology, with respect to any finite type morphism f:X \to S of schemes. In order to not make slightly false statements, I will take my coefficient ring to be \mathbf{F}_\ell for some prime \ell invertible on S. Everything below also works with more general torsion coefficients killed by an integer invertible on S, but then one has to be mindful of the difference between D^{b}_{c} and D^{b}_{ctf}. With mild assumptions on S, everything below also works with \mathbf{Q}_\ell-coefficients.

When S=\mathrm{Spec}k is a point, X is just a finite type k-scheme, and we have the familiar perverse t-structure ( \phantom{}^p D^{\leq 0}(X), \phantom{}^p D^{\geq 0}(X)) on D(X)=D(X,\mathbf{F}_\ell), with all its wonderful properties as usual. The key new definition is the following.

Definition. Given a finite type map of schemes f:X \to S, let \phantom{}^{p/S}D^{\leq 0}(X) \subset D(X) be the full subcategory of objects A such that A|X_{\overline{s}} \in \phantom{}^p D^{\leq 0}(X_{\overline{s}}) for all geometric points \overline{s} \to S.

It is easy to see that \phantom{}^{p/S}D^{\leq 0}(X) is stable under extensions and (after upgrading to derived \infty-categories) under filtered colimits, and is set-theoretically reasonable, so it defines the left half of a t-structure on D(X) by Proposition in Lurie’s Higher Algebra. We denote the right half of this t-structure, unsurprisingly, by \phantom{}^{p/S}D^{\geq 0}(X), and call it the relative perverse t-structure (relative to X\to S, of course). We write \phantom{}^{p/S}\tau^{\leq n} and \phantom{}^{p/S}\tau^{\geq n} for the associated truncation functors.

This t-structure satisfies a number of good and fairly obvious formal properties which I won’t get into here (it can be glued from any open-closed decomposition of X, various operations are obviously left- or right- t-exact, etc.). Less formally, if S is a finite-dimensional excellent Noetherian scheme, then the relative perverse truncation functors preserve D^{b}_{c}(X) \subset D(X), so we get an induced relative perverse t-structure on D^{b}_{c}(X). This follows from some results of Gabber: roughly, one can check that the relative perverse t-structure is the t-structure associated with the weak perversity function p(x)=-\mathrm{tr.deg}k(x)/k(f(x)), and that the conditions in Theorem 8.2 are satisfied for excellent S. (Nb. Gabber’s methods also reprove the existence of the relative perverse t-structure for any Noetherian S, without appealing to \infty-categories.)

However, the right half \phantom{}^{p/S}D^{\geq 0}(X) is defined in a very inexplicit way, and it isn’t clear how to get your hands on this at all. The really shocking theorem, then, is the following result.

Key Theorem. An object A \in D(X) lies in \phantom{}^{p/S}D^{\geq 0}(X) if and only if A|X_{\overline{s}} \in \phantom{}^p D^{\geq 0}(X_{\overline{s}}) for all geometric points \overline{s} \to S.

Note that I really am taking *-restrictions to geometric fibers here, just as in the definition of \phantom{}^{p/S}D^{\leq 0}(X). One might naively guess that !-restrictions should be appearing instead, but no!

This theorem has a number of corollaries.

Corollary 1. The heart \mathrm{Perv}(X/S) of the relative perverse t-structure consists of objects A \in D(X) which are perverse after restriction to any geometric fiber of f. In particular, the objects with this property naturally have the structure of an abelian category.

This fully justifies the choice of name for this t-structure, and shows that the heart of the relative perverse t-structure gives a completely reasonable notion of a “family of perverse sheaves parameterized by S”.

Corollary 2. For any map T\to S, the pullback functor D(X) \to D(X_T) is t-exact for the relative perverse t-structures (relative to S and T, respectively). In particular, relative perverse truncations commute with any base change on S, and pullback induces an exact functor \mathrm{Perv}(X/S) \to \mathrm{Perv}(X_T / T).

Corollary 3. If X\to S is any finitely presented morphism of qcqs schemes, then the relative perverse truncation functors on D(X) preserve D^{b}_{c}(X).

Corollaries 1 and 2 are immediate consequences of the Key Theorem. Corollary 3 then follows from the case where S is Noetherian excellent finite-dimensional by Noetherian approximation arguments, using Corollary 2 crucially.

To prove the key theorem, we make some formal reductions to the situation where S is excellent Noetherian finite-dimensional and A \in D^{b}_{c}(X). In this situation, we argue by induction on \dim S, with the base case \dim S=0 being obvious. In general, this induction is somewhat subtle, and involves playing off the relative perverse t-structure on D(X) against the perverse t-structures on D(X_{\overline{s}}) and the (absolute) perverse t-structure on D(X) (which exists once you pick a dimension function on S).

However, when S is the spectrum of an excellent DVR, one can give a direct proof of the key theorem, and this is what I want to do in the rest of this post. Let i: s \to S and j: \eta \to S be the inclusions of the special and generic points, with obvious base changes \tilde{i}:X_s \to X and \tilde{j}: X_\eta \to X. By definition, A \in D(X) lies in \phantom{}^{p/S}D^{\leq 0}(X) iff \tilde{j}^{\ast}A \in \phantom{}^{p}D^{\leq 0}(X_\eta) and \tilde{i}^{\ast}A \in \phantom{}^{p}D^{\leq 0}(X_s). By standard results on gluing t-structures (see chapter 1 in BBDG), this implies that A lies in \phantom{}^{p/S}D^{\geq 0}(X) iff \tilde{j}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_\eta) and R\tilde{i}^{!}A \in \phantom{}^{p}D^{\geq 0}(X_s). Thus, to prove the key theorem in this case, we need to show that for any A \in D(X) with \tilde{j}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_\eta), the conditions \tilde{i}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_s) and R\tilde{i}^{!}A \in \phantom{}^{p}D^{\geq 0}(X_s) are equivalent.

To show this, consider the triangle R\tilde{i}^{!}A \to \tilde{i}^{\ast}A \to \tilde{i}^{\ast}R\tilde{j}_{\ast} \tilde{j}^{\ast} A \to . The crucial observation is that \tilde{j}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_\eta) by assumption, and that \tilde{i}^{\ast}R\tilde{j}_{\ast} carries \phantom{}^{p}D^{\geq 0}(X_\eta) into \phantom{}^{p}D^{\geq 0}(X_s). The italicized result follows from some theorems of Gabber generalizing the classical Artin-Grothendieck vanishing theorem for affine varieties, and is closely related to the well-known fact that nearby cycles are perverse t-exact. This immediately gives what we want: we now know that \tilde{i}^{\ast}R\tilde{j}_{\ast} \tilde{j}^{\ast} A only can only have nonzero perverse cohomologies in degrees \geq 0, so R\tilde{i}^{!}A and \tilde{i}^{\ast}A have the same perverse cohomologies in degrees <0.

The six functors for Zariski-constructible sheaves in rigid geometry

In this post I want to talk about my recent paper with Bhargav Bhatt, which you can find here. This paper was a lot of fun to write, and I hope the toolkit we built will be useful for other researchers in this area. In this post I want to make some random remarks on this paper, which probably won’t mean anything if you don’t go read the real introduction to the paper first.

One funny point is that the proof of Theorem 1.6 leans on Theorem 1.7 fairly heavily, but in fact you can prove Theorem 1.6 without appealing to Theorem 1.7, at the price of much more intricate arguments. This was actually the state of the manuscript until mid-December, when we finally figured out how to prove Theorem 1.7.

Another funny point is that the discussion of the “standard” / “constructible” t-structure on D^{(b)}_{zc}(X,\mathbf{Z}_{\ell}) turned out to be surprisingly subtle, cf. Theorem 3.39. Note that D^{(b)}_{zc}(X,\mathbf{Z}_{\ell}) is by definition a full subcategory of D(X_v,\mathbf{Z}_{\ell}), and the latter carries an obvious t-structure. Nevertheless, we weren’t able to settle the question of whether these t-structures are compatible:

Question. Do the cohomological functors ^c \mathcal{H}^n(-) on D^{(b)}_{zc}(X,\mathbf{Z}_{\ell}) produced by Theorem 3.39 agree with the usual cohomology sheaves on D(X_v,\mathbf{Z}_{\ell})?

I would be extremely interested to know the answer to this.

One thing missing from the paper is any discussion of ULA sheaves. (See Fargues-Scholze for the foundations of ULA sheaves in p-adic geometry. In what follows I take \ell \neq p, but the case \ell = p should actually also be OK.) The first basic point to make is that for any rigid space X/K, any object A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell}) is ULA for the structure map X \to \mathrm{Spa}K. Sketch: The claim is local on X, so we can assume X is quasicompact. By Proposition 3.6 and stability of ULA sheaves under proper pushforward, we reduce to the special case where A = \mathbf{F}_{\ell} is constant. By an argument with resolution of singularities, we now reduce further to the case where A is constant and X is smooth, which is handled in Fargues-Scholze. Identical remarks apply with \mathbf{Z}_{\ell}-coefficients, or with general \mathbf{Z}/n coefficients (but then only for objects of “finite tor-dimension”).

This is already enough to show that the Lefschetz trace formula works as expected for proper rigid spaces in characteristic zero. More precisely, suppose given a correspondence c=(c_1,c_2): C \to X \times X of proper rigid spaces over an algebraically closed field, and a cohomological correspondence u: c_1^{\ast}A \to Rc_2^{!}A on some A \in D^b_{zc}(X,\mathbf{Z}_{\ell}). Then the usual recipe to define local terms applies, and the expected equality \mathrm{tr}(u|R\Gamma(X,A)) = \sum_{\beta \in \pi_0 \mathrm{Fix}(c)} \mathrm{loc}_{\beta}(u,A) holds true. (Note that R\Gamma(X,A) is a perfect \mathbf{Z}_{\ell}-complex by Theorem 3.35.(3).)  This can be proved by imitating the unpleasant arguments with giant diagrams in SGA5, or by the amazing categorical magic of Lu-Zheng. Of course, the local terms \mathrm{loc}_{\beta}(u,A) are just as mysterious as in the case of schemes.

It’s also natural to guess that an analogue of Deligne’s generic ULA theorem holds in this setting. Let me formally state this as a conjecture.

Conjecture. Let f:X \to Y be a proper map of characteristic zero rigid spaces, and let A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell}) be any given object. Then there is a dense Zariski-open subset of Y over which A is f-ULA.

This should be within reach, but I didn’t think about it very much.

Finally, I want to highlight the open problems mentioned in Remark 4.10, Remark 4.14 and Section 4.5. Conjecture 4.16 is probably (for whatever reason) my favorite open problem right now. Actually, I don’t even know how to prove that IH^{\ast}(X_C,\mathbf{Q}_p) is Hodge-Tate, or the even weaker statement that it has integral Hodge-Tate-Sen weights.

What does a general proper rigid space look like?

As the title says. Consider proper rigid spaces X over some nonarchimedean field K. The “standard” examples of such things which don’t come from algebraic geometry are i) the Hopf surface (\mathbf{A}^2 - 0)/p^\mathbf{Z}, ii) non-algebraizable deformations of K3 surfaces over the residue field of K, and iii) generic abeloid varieties (which are analogous to generic compact complex tori).  But there must be gazillions of other examples, right? A “random” proper rigid space is hard to write down, sort of by definition. But there are certainly some natural questions one can ask:

-For every n \geq 2, does there exist a proper n-dimensional rigid space with no non-constant meromorphic functions, and admitting a formal model whose special fiber has components of general type? Can we find examples of such spaces with arbitrarily large dimension dimension which don’t come from lower-dimensional examples by simple operations (products, quotients by finite groups, etc.)? Same question but with “no non-constant meromorphic functions” replaced by the weaker requirement that \mathrm{tr.deg}K(X)/K is small compared to \dim X.

-Do there exist non-algebraizable proper rigid spaces with “arbitrarily bad” singularities?

-Do there exist rigid analytic analogues of Kodaira’s class VII0 surfaces?

Euler characteristics and perverse sheaves

Let X be a variety over a separably closed field k, and let A be some object in D^b_c(X,\mathbf{Q}_{\ell}). Laumon proved the beautiful result that the usual and compactly supported Euler characteristics \chi(X,A) and \chi_c(X,A) are always equal. Recently while trying to do something else, I found a quick proof of Laumon’s result, as well as a relative version, and I want to sketch this here.

Pick an open immersion into a compactification j:X \to X'; after a blowup, we can assume that Z=X' - X is an effective Cartier divisor. Write i:Z \to X' for the inclusion of the boundary. By the usual triangle j_!A \to Rj_*A \to i_*i^* Rj_*A \to , we reduce to showing that \chi(X',i_*i^* Rj_*A)=0. Filtering A by its perverse cohomology sheaves, we reduce further to the case where also A is perverse. Cover X' by open affines X_n' such that Z_n= Z \cap X_n' is the divisor of a function f_n. By an easy Mayer-Vietoras argument, it’s now enough to show that for every open U contained in some X_n', \chi(U,(i_* i^{\ast}Rj_{\ast}A)|U) = 0.

But now we win: for any choice of such U \subset X_n', there is an exact triangle R\psi_{f_n}(A|U \cap X) \to R\psi_{f_n}(A|U \cap X) \to (i_* i^{\ast}Rj_{\ast}A)|U \to in D^b_c(U,\mathbf{Q}_{\ell}) where R\psi_{f_n}:\mathrm{Perv}(U \cap X) \to \mathrm{Perv}( U \cap Z) is the unipotent nearby cycles functor associated with f_n, and the first arrow is the logarithm of the unipotent part of the monodromy. Since \chi(U, -) is additive in exact triangles and the first two terms agree, we’re done.

A closer reading of this argument shows that you actually get the following stronger statement: for any A, the class [i_*i^* Rj_*A] \in K_0\mathrm{Perv}(X') is identically zero. From here it’s easy to get a relative version of Laumon’s result.

Theorem. Let f:X \to Y be any map of k-varieties. Then for any A\in D^b_c(X,\mathbf{Q}_\ell), there is an equality [Rf_! A]=[Rf_\ast A] in K_0\mathrm{Perv}(Y).