Tag: Shizhang Li

p-adic Kahler manifolds

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.

scratch

(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?