In complex geometry, the most interesting class of complex manifolds is probably the Kahler class. In the non-archimedean world, say over a fixed p-adic base field $K$, the analogue of a compact complex manifold is a smooth proper rigid analytic space. In some ways, these are already surprisingly “close” to being Kahler – in particular, the Hodge-de Rham spectral sequence of such a space always degenerates at $E_1$. However, Hodge symmetry can definitely fail. A standard example is the non-archimedean Hopf surface $X = \mathbf{A}^2_{K} \smallsetminus \{ (0,0) \} / p^{\mathbf{Z}}$ where $p^n$ acts through diagonal multiplication. By a fun direct calculation, one checks that $H^0(X,\Omega^1_X)=0$ and $H^1(X,\mathcal{O}_X) = K$, so Hodge symmetry fails in degree one.

We now see a natural question: is there is some non-archimedean analogue of the Kahler condition which restores Hodge symmetry? Two years ago, Shizhang Li hit upon the following candiate condition:

A smooth proper rigid space $X$ satisfies (*) if it admits a formal model $\mathfrak{X}$ over $\mathcal{O}_K$ whose special fiber is projective (as opposed to merely proper).

Using fantastic ideas due to Shizhang, we managed to prove the following suggestive result.

Theorem. Let $X$ be a smooth proper rigid space satisfying (*). Then $h^{1,0}(X) = h^{0,1}(X)$.

Of course, one can then guess that (*) implies Hodge symmetry in all degrees. This speculation seems to have caught the imagination of others in the field, but until recently I personally regarded it as not much more than wishful thinking. However, my perspective completely changed a month ago, when I learned from Shizhang that, according to Robert Friedman, the archimedean analogue of “(*) implies Hodge symmetry” is a theorem! More precisely, we have the following result:

Theorem. Let $D$ be the complex disk, with $D^\times =D \smallsetminus \{0 \}$ the punctured disk. Let $f:Y \to D$ be a proper map of complex analytic spaces. Suppose that $f^{-1}(D^\times) \to D^\times$ is a submersion, and that the central fiber $Y_0=f^{-1}(0)$ is the analytification of a projective (and not necessarily smooth) algebraic variety. Then for all $t \in D^\times$ with $|t| \ll 1$, the fiber $Y_t$ satisfies Hodge symmetry and Hodge-de Rham degeneration.

Of course, the analogy is that $\mathfrak{X} \to \mathrm{Spf} \mathcal{O}_K$ is analogous to $Y \to D$, and $X$ is analogous to the “nearby” fibers $Y_t$ with $0<|t| \ll 1$.

The proof of this theorem uses the full power of mixed Hodge theory. In fact the claim about Hodge-de Rham degeneration is exactly Corollary 11.24 in the book of Peters-Steenbrink. Hodge symmetry is even more subtle, and the argument for this doesn’t seem to be written down anywhere; Friedman explained it to Shizhang, who explained it to me, but the details entailed such a horrible explosion of gradings, filtrations, and multi-indices that I can’t hope to reproduce it here.

Anyway, I’m now completely convinced that Shizhang’s condition (*) implies Hodge symmetry in all degrees, and that this is truly the “right” p-adic analogue of the Kahler condition.

## A counterexample

Let $C/\mathbf{Q}_p$ be a complete algebraically closed nonarchimedean field extension, and let $X$ be any proper rigid space over $C$. Let $\mathbf{L}$ be any $\mathbf{Z}_p$-local system on $X_{\mathrm{proet}}$. By the main results in Scholze’s p-adic Hodge theory paper, the pro-etale cohomology groups $H^i_{\mathrm{proet}}(X,\mathbf{L})$ are always finitely generated $\mathbf{Z}_p$-modules. It also seems likely that Poincare duality holds in this setting (and maybe someone has proved this?).

Suppose instead that we’re given a $\mathbf{Q}_p$-local system $\mathbf{V}$. By analogy, one might guess that the cohomology groups $H^i_{\mathrm{proet}}(X,\mathbf{V})$ are always finitely generated $\mathbf{Q}_p$-vector spaces. Indeed, this (and more) was claimed as a theorem by Kedlaya-Liu in a 2016 preprint. However, it is false. The goal of this post is to work out an explicit counterexample.

So, consider $X=\mathbf{P}^1$ as a rigid space over $C$. This is the target of the Gross-Hopkins period map $\pi_{\mathrm{GM}}: \mathcal{M} \to X$, where $\mathcal{M}$ is (the rigid generic fiber of the base change to $\mathcal{O}_C$ of) the Lubin-Tate deformation space of some fixed connected p-divisible group $G_0/\overline{\mathbf{F}_p}$ of dimension 1 and height 2. The rational p-adic Tate module of the universal p-divisible group $G/\mathcal{M}$ descends along $\pi_{\mathrm{GM}}$ to a rank two $\mathbf{Q}_p$ local system $\mathbf{V}_{LT}$ on $X$.

Theorem. Maintain the above setup. Then
i. For any $i \neq 1,2$ the group $H^i_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ is zero.

ii. The group $H^1_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ is a Banach-Colmez space over $C$ of Dimension $(1,-2)$.
iii. The group $H^2_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ is a Banach-Colmez space over $C$ of Dimension $(1,2)$.

Recall that a Banach-Colmez space is a special kind of topological $\mathbf{Q}_p$-vector space (really it’s a functor valued in such things, but I’ll be a little sloppy about this point). Morally, it’s something like a finite-dimensional $C$-vector space defined up to a finite-dimensional $\mathbf{Q}_p$-vector space. In particular, any such space has a well-defined Dimension, which is a pair in $\mathbf{Z}_{\geq 0} \times \mathbf{Z}$ whose entries record the $C$-dimension and the $\mathbf{Q}_p$-dimension of the space, respectively. So for example the space $C^2$ has Dimension $(2,0)$, and the space $C/\mathbf{Q}_p$ has Dimension $(1,-1)$. Unsurprisingly, any Banach-Colmez space whose $C$-dimension is positive will be disgustingly infinitely generated as a $\mathbf{Q}_p$-vector space, so the Theorem really does give us an example of the desired type. Note also that Poincare duality fails in this example.

Proof. Let $\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p}$ be the evident sheaf on $X_{\mathrm{proet}}$, where e.g. $\mathbf{B}_{\mathrm{crys}}^{+}$ is the crystalline period sheaf defined in Tan-Tong’s paper on crystalline comparison. The key observation is that there is a short exact sequence

$(\ast)\;\;\; 0 \to \mathbf{V}_{LT} \to \mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p} \to \mathrm{Lie}(G)[\tfrac{1}{p}] \otimes_{\mathcal{O}_X} \widehat{\mathcal{O}}_X \to 0$

of sheaves on $X_{\mathrm{proet}}$. This is a sheaf-theoretic version of the sequence (1.0.1) in Scholze-Weinstein, and it can be constructed using the methods in their paper. One can also construct it directly using the modern fancy-pants interpretation of $X$ in this setup as the period domain parametrizing admissible length one modifications of the bundle $\mathcal{O}(1/2)$ on the Fargues-Fontaine curve. (Nb. The mysterious middle term in the sequence is really the sheaf of $\varphi$-equivariant maps from $H^1_{\mathrm{crys}}(G_0/W(\overline{\mathbf{F}_p}))[\tfrac{1}{p}]$ to $\mathbf{B}_{\mathrm{crys}}^{+}$.)

Anyway, this reduces us to computing the groups $H^i_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p})$ and $H^i_{\mathrm{proet}}(X,\mathrm{Lie}(G)[\tfrac{1}{p}] \otimes_{\mathcal{O}_X} \widehat{\mathcal{O}}_X)$. This might look a bit terrifying, but it’s not so bad. In fact, the first of these can be handled by a general lemma.

Lemma. Let $\mathbf{M}$ be any Banach-Colmez space over $C$. For any proper rigid space $X/C$, we may regard $\mathbf{M}$ as a (pre)sheaf on $X_{\mathrm{proet}}$, so in particular we can talk about the pro-etale cohomology groups $H^i_{\mathrm{proet}}(X,\mathbf{M})$. In this notation, the natural map $H^i_{\mathrm{proet}}(X,\mathbf{Q}_p) \otimes_{\mathbf{Q}_p} \mathbf{M}(C) \to H^i_{\mathrm{proet}}(X,\mathbf{M})$ is an isomorphism.

(Proof sketch: Use the 5 lemma to reduce to the case of effective Banach-Colmez spaces, and then to the cases of the space $\mathbf{Q}_p$, where it’s a tautology, and the space Colmez notates $\mathbb{V}^1$, where it follows from the primitive comparison theorem, cf. Theorem 3.17 in Scholze’s CDM survey.)

Applied to our problem, this immediately gives that $H^i_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p}) \cong H^i_{\mathrm{proet}}(X,\mathbf{Q}_p) \otimes_{\mathbf{Q}_p} B_{\mathrm{crys}}^{+,\varphi^2 = p}$. By the standard easy computation of $H^i_{\mathrm{proet}}(X,\mathbf{Q}_p)$, we get that $H^i_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p})$ is a copy of $B_{\mathrm{crys}}^{+,\varphi^2=p}$ for either of $i \in \{ 0,2 \}$, and it vanishes otherwise. In particular, in degrees 0 and 2, it has Dimension $(1,2)$ by some of the calculations in Colmez’s original paper.

Next, we need to compute the pro-etale cohomology of $\mathcal{E} = \mathrm{Lie}(G)[\tfrac{1}{p}] \otimes_{\mathcal{O}_X} \widehat{\mathcal{O}}_X$. For this, we use the fact (already in Gross and Hopkins’s original article) that $\mathrm{Lie}(G)[\tfrac{1}{p}] \simeq \mathcal{O}_X(1)$. Let $\lambda : X_{\mathrm{proet}} \to X_{\mathrm{an}}$ be the evident projection of sites. Combining the description of $\mathrm{Lie}(G)[\tfrac{1}{p}]$ with an easy projection formula gives an isomorphism $E\overset{def}{=}R\lambda_{\ast} \mathcal{E} \cong \mathcal{O}_X (1) \otimes_{\mathcal{O}_X} R\lambda_{\ast} \widehat{\mathcal{O}}_X$. Moreover, $R^i \lambda_{\ast} \widehat{\mathcal{O}}_X \simeq \Omega_{X}^i$ identifies with $\mathcal{O}_X$ in degree zero and $\mathcal{O}_X(-2)$ in degree 1, and vanishes otherwise. In particular, the only nonvanishing cohomology sheaves of $E$ are $\mathcal{O}_X(1)$ in degree 0 and $\mathcal{O}_X(-1)$ in degree 1; moreover, the latter has no global cohomology in any degree. Feeding this information into the Leray spectral sequence for $\lambda$, we get that $H^i_{\mathrm{proet}}(X,\mathcal{E}) \simeq H^i(X,E) \simeq H^i(X, \tau^{\leq 0} E) \simeq H^i(X,\mathcal{O}_X(1))$, so this is $C^2$ for $i=0$ and zero otherwise.

Finally, we can put everything together and take the long exact sequence in pro-etale cohomology associated with $(\ast)$. It’s easy to check that $\mathbf{V}_{LT}$ doesn’t have any global sections, and the middle term of $(\ast)$ has no cohomology in degree one, so we get a short exact sequence $0 \to H^0_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p}) \to C^2 \to H^1_{\mathrm{proet}}(X,\mathbf{V}_{LT}) \to 0$. We’ve already identified the $H^0$ here as something of Dimension $(1,2)$, so by the additivity of Dimensions in short exact sequences, we deduce that $H^1_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ has Dimension $(1,-2)$, as desired. By a similar argument, we get an isomorphism $H^2_{\mathrm{proet}}(X,\mathbf{V}_{LT}) \simeq B_{\mathrm{crys}}^{+,\varphi^2=p}$, which we already observed has Dimension $(1,2)$. The vanishing of all the other cohomologies of $\mathbf{V}_{LT}$ also follows easily. $\square$

BTW, there is nothing special about height $2$ in this story; I just stuck with it for convenience. For any heght $h \geq 2$, there is an analogous rank $h$ $\mathbf{Q}_p$-local system $V_{LT,h}$ on $\mathbf{P}^{h-1}$, and one can check that e.g. $H^1_{\mathrm{proet}}(\mathbf{P}^{h-1},\mathbf{V}_{LT,h})$ has Dimension $(h-1,-h)$.

Let me briefly explain the genesis of these calculations. Several months ago Shizhang Li pointed out to me that the primitive comparison theorem doesn’t obviously generalize to $\mathbf{Q}_p$-local systems without globally defined lattices, and he also suggested that the cohomologies of $\mathbf{V}_{LT}$ might be strange. I promptly forgot about this conversation until Monday, when Shizhang reported that Ruochuan Liu had told him in Oberwolfach several weeks ago that the Kedlaya-Liu “theorem” was false. It was then natural to double down on $\mathbf{V}_{LT}$ as a possible counterexample. Immediately after I gave a lecture on the Tan-Tong paper the next day, the sequence $(\ast)$ entered my head randomly. Then everything just came out. Also, luckily, we were at a restaurant with paper tablecloths.

(I’m not sure whether Ruochuan also had this particular counterexample in mind.)

There are lots of interesting questions here, I think. Are there other natural examples of this type? Are the pro-etale cohomology groups of $\mathbf{Q}_p$-local systems on proper rigid spaces always Banach-Colmez spaces?

## What does an inadmissible locus look like?

Let $H/ \overline{\mathbf{F}_p}$ be some p-divisible group of dimension d and height h, and let $\mathcal{M}$ be the rigid generic fiber (over $\mathrm{Spa}\,\breve{\mathbf{Q}}_p$) of the associated Rapoport-Zink space. This comes with its Grothendieck-Messing period map $\pi: \mathcal{M} \to \mathrm{Gr}(d,h)$, where $\mathrm{Gr}(d,h)$ is the rigid analytic Grassmannian paramatrizing rank d quotients of the (covariant) rational Dieudonne module $M(H) /\breve{\mathbf{Q}}_p$. Note that $\mathrm{Gr}(d,h)$ is a very nice space: it’s a smooth connected homogeneous rigid analytic variety, of dimension d(h-d).

The morphism $\pi$ is etale and partially proper (i.e. without boundary in Berkovich’s sense), and so the image of $\pi$ is an open and partially proper subspace* of the Grassmannian, which is usually known as the admissible locus. Let’s denote this locus by $\mathrm{Gr}(d,h)^a$. The structure of the admissible locus is understood in very few cases, and getting a handle on it more generally is a famous and difficult problem first raised by Grothendieck (cf. the Remarques on p. 435 of his 1970 ICM article). About all we know so far is the following:

• When d=1 (so $\mathrm{Gr}(d,h) = \mathbf{P}^{h-1}$) and $H$ is connected, we’re in the much-studied Lubin-Tate situation. Here, Gross and Hopkins famously proved that $\pi$ is surjective, not just on classical rigid points but on all adic points, so $\mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h)$ is the whole space. This case (along with the “dual” case where h>2,d=h-1) turns out to be the only case where $\mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h)$, cf. Rapoport’s appendix to Scholze’s paper on the Lubin-Tate tower.
• When $H \simeq \mathbf{G}_m^{d} \oplus (\mathbf{Q}_p/\mathbf{Z}_p)^{h-d}$, i.e. when $H$ has no bi-infinitesimal component, it turns out that $\mathrm{Gr}(d,h)^a = \mathbf{A}^{d(h-d)}$ is isomorphic to rigid analytic affine space of the appropriate dimension, and can be identified with the open Bruhat cell inside $\mathrm{Gr}(d,h)$. This goes back to Dwork, who proved it when d=1,h=2. (I don’t know a citation for the general result, but presumably for arbitrary d,h this is morally due to Serre-Tate/Katz?)
• In general there’s also the so-called weakly admissible locus $\mathrm{Gr}(d,h)^{wa} \subset \mathrm{Gr}(d,h)$, which contains the admissible locus and is defined in some fairly explicit way. It’s also characterized as the maximal admissible open subset of the Grassmannian with the same classical points as the admissible locus. In the classical rigid language, the map $\mathrm{Gr}(d,h)^a \to \mathrm{Gr}(d,h)^{wa}$ is etale and bijective; this is the terminology used e.g. in Rapoport-Zink’s book.
• In general, the admissible and weakly admissible loci are very different.  For example, when $H$ is isoclinic and (d,h)=1 (i.e. when $M(H)$ is irreducible as a $\varphi$-module), $\mathrm{Gr}(d,h)^a$ contains every classical point, and $\mathrm{Gr}(d,h)^{wa} = \mathrm{Gr}(d,h)$, so the weakly admissible locus tells you zilch about the admissible locus in this situation (and they really are different for any $1 < d < h-1$).

That’s about it for general results.

To go further, let’s switch our perspective a little. Since $\mathrm{Gr}(d,h)^a$ is an open and partially proper subspace of $\mathrm{Gr}(d,h)$, the subset $|\mathrm{Gr}(d,h)^a| \subseteq |\mathrm{Gr}(d,h)|$ is open and specializing, so its complement is closed and generalizing.  Now, according to a very general theorem of Scholze, namely Theorem 2.42 here (for future readers, in case the numbering there changes: it’s the main theorem in the section entitled “The miracle theorems”), if $\mathcal{D}$ is any diamond and $E \subset |\mathcal{D}|$ is any locally closed generalizing subset, there is a functorially associated subdiamond $\mathcal{E} \subset \mathcal{D}$ with $|\mathcal{E}| = E$ inside $|\mathcal{D}|$. More colloquially, one can “diamondize” any locally closed generalizing subset of $|\mathcal{D}|$, just as any locally closed subspace of $|X|$ for a scheme $X$ comes from a unique (reduced) subscheme of $X$.

Definition. The inadmissible/nonadmissible locus $\mathrm{Gr}(d,h)^{na}$ is the subdiamond of $\mathrm{Gr}(d,h)^{\lozenge}$ obtained by diamondizing the topological complement of the admissible locus, i.e. by diamondizing the closed generalizing subset $|\mathrm{Gr}(d,h)^a|^c \subset |\mathrm{Gr}(d,h)| \cong |\mathrm{Gr}(d,h)^{\lozenge}|$.

It turns out that one can actually get a handle on $\mathrm{Gr}(d,h)^{na}$ in a bunch of cases!  This grew out of some conversations with Jared Weinstein – back in April, Jared raised the question of understanding the inadmissible locus in a certain particular period domain for $\mathrm{GL}_2$ with non-minuscule Hodge numbers, and we managed to describe it completely in that case (see link below). Last night, though, I realized we hadn’t worked out any interesting examples in the minuscule (i.e. p-divisible group) setting! Here I want to record two such examples, hot off my blackboard, one simple and one delightfully bizarre.

Example 1. Take h=4, d=2 and $H$ isoclinic. Then $|\mathrm{Gr}(d,h)^a|^c$ is a single classical point, corresponding to the unique filtration on $M(H)$ with Hodge numbers $0,0,1,1$ which is not weakly admissible. So $\mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h)^{wa}$ in this case.

Example 2. Take h=5, d=2 and $H$ isoclinic\$.  Now things are much stranger.  Are you ready?
Theorem. In this case, the locus $\mathrm{Gr}^{na}$ is naturally isomorphic to the diamond $(X \smallsetminus 0)^{\lozenge} / \underline{D^\times}$, where $X$ is an open perfectoid unit disk in one variable over $\breve{\mathbf{Q}}_p$ and $D=D_{1/3}$ is the division algebra over $\mathbf{Q}_p$ with invariant 1/3, acting freely on $X \smallsetminus 0$ in a certain natural way. Precisely, the disk $X$ arises as the universal cover of the connected p-divisible group of dimension 1 and height 15, and its natural $D$-action comes from the natural $D_{1/15}$-action on $X$ via the map $D_{1/3} \to D_{1/3} \otimes D_{-2/5} \simeq D_{-1/15} \simeq D_{1/15}^{op}$.

This explicit description is actually equivariant for the $D_{2/5}$-actions on $X$ and $Gr$. As far as diamonds go, $(X \smallsetminus 0)^{\lozenge}/\underline{D^{\times}}$ is pretty high-carat: it’s spatial (roughly, its qcqs with lots of qcqs open subdiamonds), and its structure morphism to $\mathrm{Spd}\,\breve{\mathbf{Q}}_p$ is separated, smooth, quasicompact, and partially proper in the appropriate senses. Smoothness, in particular, is meant in the sense of Definition 6.1 here (cf. also the discussion in Section 4.3 here). So even though this beast doesn’t have any points over any finite extension of $\breve{\mathbf{Q}}_p$, it’s still morally a diamondly version of a smooth projective curve!

The example Jared and I had originally worked out is recorded in section 5.5 here. The reader may wish to try adapting our argument from that situation to the cases mentioned above – this is a great exercise in actually using the classification of vector bundles on the Fargues-Fontaine curve in a hands-on calculation.

Anyway, here’s a picture of $(X \smallsetminus 0)^{\lozenge} / \underline{D^{\times}}$, with some other inadmissible loci in the background:

*All rigid spaces here and throughout the post are viewed as adic spaces: in the classical language, $\mathrm{Gr}(d,h)^a$ does not generally correspond to an admissible open subset of $\mathrm{Gr}(d,h)$, so one would be forced to say that there exists a rigid space $\mathrm{Gr}(d,h)^a$ together with an etale monomorphism $\mathrm{Gr}(d,h)^a \to \mathrm{Gr}(d,h)$. But in the adic world it really is a subspace.