# Rigid-analytic varieties with projective reduction violating Hodge symmetry

Alexander Petrov  Harvard University, USA<EMAIL_ADDRESS>
###### Abstract.

We construct examples of smooth proper rigid-analytic varieties admitting
formal model with projective special fiber and violating Hodge symmetry for
cohomology in degrees $\geq 3$. This answers negatively the question raised by
Hansen and Li in [HL17].

##  1\. Introduction

A powerful tool in algebraic geometry over the field of complex numbers is
viewing smooth projective varieties as complex manifolds and henceforth using
complex analysis to study them. An important role for that technique is played
by Kähler manifolds: this is a subclass of complex manifolds large enough to
contain all smooth projective algebraic varieties and small enough to be
amenable to Hodge theory. In particular, for any compact Kähler manifold $X$
the Hodge-to-de Rham spectral sequence
$E_{1}^{i,j}=H^{j}(X,\Omega^{i}_{X})\Rightarrow
H^{i+j}_{\mathrm{dR}}(X/{\mathbb{C}})$ degenerates at the first page and the
Hodge numbers satisfy Hodge symmetry:
$\dim_{{\mathbb{C}}}H^{j}(X,\Omega^{i}_{X})=\dim_{{\mathbb{C}}}H^{i}(X,\Omega^{j}_{X})$.

The situation in $p$-adic geometry is somewhat different: $p$-adic Hodge
theory has been developed for all smooth proper rigid-analytic varieties,
without any “Kähler” assumption. It was proven by Scholze [Sch13] that for any
such variety the Hodge-to-de Rham spectral sequence degenerates. There are,
however, examples of smooth proper rigid-analytic varieties that fail Hodge
symmetry. This might suggest that there should be a natural narrower class of
rigid-analytic varieties for which Hodge symmetry does hold.

Let $K$ be a discretely valued $p$-adic field with ring of integers
${\mathcal{O}}_{K}$ and perfect residue field $k$. In [HL17] David Hansen and
Shizhang Li raised the following question (see also Conjecture 2.4 in [Sch18])
suggesting a candidate for that narrower class:

######  Question.

Let $X$ be a smooth proper rigid-analytic variety over $K$ admitting a formal
model over ${\mathcal{O}}_{K}$ with projective special fiber. Is it true that
$\dim_{K}H^{i}(X,\Omega^{j}_{X/K})=\dim_{K}H^{j}(X,\Omega^{i}_{X/K})$ for all
$i,j$?

Hansen and Li answered this question positively for $i+j=1$ in Theorem 1.2 of
[HL17]. The main result of this text is that the answer is in general negative
for $i+j\geq 3$:

######  Theorem (Corollary 4.4).

For every pair of positive integers $i\neq j$ with $i+j\geq 3$ there exists a
smooth proper rigid-analytic variety $X$ over ${\mathbb{Q}}_{p}$ admitting a
smooth formal model ${\mathfrak{X}}$ over
$\operatorname{{Spf}}{\mathbb{Z}}_{p}$ with projective special fiber
${\mathfrak{X}}_{{\mathbb{F}}_{p}}$ such that

$\dim_{{\mathbb{Q}}_{p}}H^{i}(X,\Omega^{j}_{X/{\mathbb{Q}}_{p}})\neq\dim_{{\mathbb{Q}}_{p}}H^{j}(X,\Omega^{i}_{X/{\mathbb{Q}}_{p}}).$

The idea behind the examples comes from the following difference between
complex Hodge theory and $p$-adic Hodge theory. For any compact Kähler
manifold $X$ not only there is an equality of numbers $h^{i,j}(X)=h^{j,i}(X)$
but there is a canonical isomorphism of ${\mathbb{C}}$-vector spaces:

$H^{j}(X,\Omega^{i}_{X})\simeq\overline{H^{i}(X,\Omega^{j}_{X})}$

In particular, for a finite group $G$ acting on $X$ the $G$-representations
$H^{j}(X,\Omega^{i}_{X})$ and $H^{i}(X,\Omega^{j}_{X})$ are dual to each
other.

We show that the analogous statement in $p$-adic geometry fails already for
abeloid varieties. We construct a formal abelian scheme ${\mathfrak{A}}$
(necessarily a non-algebraizable one) with an action of a finite group $G$ of
order prime to $p$ such that the $G$-representations $H^{0}(A,\Omega^{1}_{A})$
and $H^{1}(A,{\mathcal{O}})$ on the cohomology of the abeloid generic fiber
$A$ of ${\mathfrak{A}}$ are not dual to each other.

Moreover, for $i+j=3$ we can arrange (Proposition 3.1) that the dimensions of
$G$-invariants on the spaces $H^{i}(A,\Omega_{A}^{j})$ and
$H^{j}(A,\Omega_{A}^{i})$ are different (note that these $G$-representations
are obtained from $H^{0}(A,\Omega^{1}_{A})$ and $H^{1}(A,{\mathcal{O}})$ by
taking exterior powers and tensor products). Taking the direct product with an
auxiliary smooth projective formal scheme to make the $G$-action free and
taking the quotient by $G$ gives the desired example. The examples for $i+j>3$
are obtained simply by taking the direct product with an appropriate
projective scheme to move the asymmetry to higher cohomology via the Künneth
formula.

This strategy does not work for $i+j=2$ for a good reason: any smooth proper
rigid-analytic variety that admits a smooth (not necessarily with a projective
special fiber) formal model satisfies Hodge symmetry on first and second
cohomology:

######  Proposition 1.1 (Corollary 2.2).

If a smooth proper rigid-analytic variety $X$ over $K$ admits a smooth formal
model over ${\mathcal{O}}_{K}$ then $\dim
H^{0}(X,\Omega^{i}_{X/K})=\dim_{K}H^{i}(X,{\mathcal{O}})$ for $i=1,2$.

Proposition 1.1 is a consequence of a certain self-duality of Frobenius action
on the crystalline cohomology of the special fiber and weak admissibility (in
the sense of filtered $\varphi$-modules) of de Rham cohomology of $X$ coming
from the crystalline comparison theorem. As mentioned above, Hansen and Li
proved this statement by a different method for $i=1$ in the case of
(arbitrarily singular) projective reduction. Piotr Achinger has independently
obtained Proposition 1.1 in the case of smooth projective reduction by the
same method, see [Ach20] for a treatment of a number of situations where Hodge
symmetry does follow from the existence of a model with projective special
fiber, including some cases when reduction is singular.

Notation. Let $p$ be a prime number fixed throughout the text. If $X$ is an
object of any of the three types $\\{$smooth proper rigid-analytic variety
over $K$, smooth proper formal scheme over
$\operatorname{{Spf}}{\mathcal{O}}_{K}$, smooth proper algebraic variety over
an arbitrary field $F\\}$, denote by $h^{i,j}(X)$ the number
$\dim_{K}H^{j}(X,\Omega^{i}_{X/K}),\operatorname{{rk}}_{{\mathcal{O}}_{K}}H^{j}(X,\Omega^{i}_{X/{\mathcal{O}}_{K}})$
or $\dim_{F}H^{j}(X,\Omega^{i}_{X/F})$ respectively. Denote also by
$\delta^{i,j}(X)$ the number $h^{i,j}(X)-h^{j,i}(X)$ in any of the three
situations. For the convenience of notation, we declare $h^{i,j}(X)=0$ if $i$
or $j$ is negative. Note that $h^{i,j}(X)=h^{i,j}({\mathfrak{X}})$ if $X$ is
the generic fiber of a smooth proper formal scheme ${\mathfrak{X}}$ by Lemma
4.1. We say that an object $X$ “satisfies Hodge symmetry in degree $d$” if
$h^{i,j}(X)=h^{j,i}(X)$ for all $i,j$ with $i+j=d$ and $X$ “satisfies Hodge
symmetry” if it does so for all $d$.

Acknowledgments. I am grateful to Piotr Achinger, Shizhang Li, Vadim
Vologodsky, Zijian Yao and Bogdan Zavyalov for their interest and several
useful discussions and to Piotr Achinger and Borys Kadets for comments on the
drafts of this text. I am indebted to Vasily Rogov whose talk on the work
[Rog19] motivated the examples presented here.

##  2\. Hodge symmetry in degree $2$

For the duration of this section let $K$ be a discretely valued field of
characteristic zero with perfect residue field $k$ of characteristic $p$.
Denote by $K_{0}$ the subfield $W(k)[\frac{1}{p}]$, and let $\sigma$ be its
automorphism induced by Frobenius on $k$.

The goal of this section is to observe that results of $p$-adic Hodge theory
impose the following relation between Hodge numbers of a rigid-analytic
variety with good reduction. A similar idea has been used in [FM87] I.4.4 to
prove Hodge symmetry for smooth proper algebraic varieties over $K$.

######  Proposition 2.1.

If ${\mathfrak{X}}$ is a smooth proper formal scheme over ${\mathcal{O}}_{K}$
with projective special fiber, then for its generic fiber $X$ and every $n$
the following relation among $h^{i,j}:=h^{i,j}(X)$ holds:

(2.1) $h^{1,n-1}+2h^{2,n-2}+\dots+n\cdot h^{n,0}=n\cdot h^{0,n}+(n-1)\cdot
h^{1,n-1}+\dots+h^{n-1,1}.$

######  Corollary 2.2.

For a smooth proper formal scheme ${\mathfrak{X}}$ over
$\operatorname{{Spf}}{\mathcal{O}}_{K}$ we have the following for the generic
fiber $X={\mathfrak{X}}_{K}$

(i) $h^{1,0}(X)=h^{0,1}(X)$ and $h^{2,0}(X)=h^{0,2}(X)$

(ii) For odd $n$ the Betti numbers
$\dim_{{\mathbb{Q}}_{p}}H^{n}_{\acute{e}t}(X_{\overline{K}},{\mathbb{Q}}_{p})=\dim_{K}H^{n}_{\mathrm{dR}}(X/K)$
are even.

###### Proof.

(i) The equality 2.1 takes forms $h^{1,0}=h^{0,1}$ and
$h^{1,1}+2h^{2,0}=2h^{0,2}+h^{1,1}$ for $n=1,2$ respectively.

(ii) The sum of the left-hand side and the right-hand sied of 2.1 is, of
course, even and is also equal to
$n\cdot(h^{0,n}+h^{1,n-1}+\dots+h^{n,0})=n\cdot\dim_{K}H^{n}_{\mathrm{dR}}(X/K)$
so $\dim_{K}H^{n}_{\mathrm{dR}}(X/K)$ is even for odd $n$. ∎

We first recall the setup of rational $p$-adic Hodge theory. Let
$\mathrm{MF}^{\varphi}_{K}$ be the category of filtered $\varphi$-modules over
$K$ [Fon79]. Its objects are finite-dimensional vector spaces $D$ over $K_{0}$
equipped with a semi-linear automorphism $\varphi_{D}:D\to D$ and a deceasing
filtration $F^{i}D_{K}$ on the $K$-vector space $D_{K}:=D\otimes_{K_{0}}K$. To
every such object we can attach two integers

(2.2)
$t_{N}(D)=\sum\limits_{\alpha\in{\mathbb{Q}}}\alpha\cdot\dim_{K_{0}}D_{\alpha}\text{
and
}t_{H}(D)=\sum\limits_{i\in{\mathbb{Z}}}i\cdot\dim_{K}\operatorname{{gr}}^{i}D_{K}$

Here $D_{\alpha}$ is the slope $\alpha$ direct summand of the $\varphi$-module
$D$ under the Dieudonne-Manin decomposition and
$\operatorname{{gr}}^{i}D_{K}:=F^{i}D_{K}/F^{i+1}D_{K}$ are the graded pieces
of the filtration. Note that $t_{N}$ depends only on the semi-linear
endomorphism and we will also use the notation $t_{N}(D)$ for a
$\varphi$-module $D$ not equipped with a filtration. A filtered
$\varphi$-module $D\in\mathrm{MF}^{\varphi}_{K}$ is called weakly admissible
if $t_{N}(D)=t_{H}(D)$ and for every subobject $D^{\prime}\subset D$ we have
$t_{N}(D^{\prime})\geq t_{H}(D^{\prime})$. For an integer $i$ denote by
$K_{0}(i)$ the filtered $\varphi$-module given by a one-dimensional vector
space $D=K_{0}$ with $\varphi_{d}=p^{-i}\sigma$ and the filtration defined by
$F^{-i}D_{K}=D_{K},F^{-i+1}=0$. For a filtered $\varphi$-module $D$ define its
$i$-fold twist $D(i)$ as the tensor product $D\otimes_{K_{0}}K_{0}(i)$, we use
the same notation for twisting $\varphi$-modules not equipped with a
filtration.

We have the following symmetry of Frobenius slopes, due to [Suh12]:

######  Lemma 2.3 (Corollary 2.2.4 in [Suh12]).

Let $Y$ be a smooth proper variety over $k$. For the $\varphi$-module
$D=H^{n}_{\mathrm{cris}}(Y/W(k))[\frac{1}{p}]$ we have
$t_{N}(D)=n\cdot\dim_{K_{0}}D-t_{N}(D)$.

###### Proof.

For the convenience of the reader we recall the proof in the case when $Y$ is
projective over an arbitrary perfect field or arbitrary proper defined over a
finite field.

If $Y$ is projective, denote by $L\in H^{2}_{\mathrm{cris}}(Y/W(k))$ the
crystalline Chern class of an ample line bundle on $Y$. The hard Lefschetz
theorem for crystalline cohomology [KM74] shows that cup-product with
$L^{d-\min(n,2d-n)}$ induces an isomorphism
$H^{n}_{\mathrm{cris}}(Y/W(k))[\frac{1}{p}](n-d)\simeq
H^{2d-n}(Y/W(k))[\frac{1}{p}]$ of $\varphi$-modules over $K_{0}$. On the other
hand, Poincare duality provides an isomorphism
$H^{2d-n}_{\mathrm{cris}}(Y/W(k))[\frac{1}{p}]\simeq
H^{n}_{\mathrm{cris}}(Y/W(k))^{\vee}[\frac{1}{p}](-d)$. Combining these
isomorphisms we get $H^{n}_{cris}(Y/W(k))[\frac{1}{p}](n)\simeq
H^{n}_{\mathrm{cris}}(Y/W(k))^{\vee}$. Since
$t_{N}(D^{\vee}(-n))=-t_{N}(D(n))=-t_{N}(D)+n\cdot\dim_{K_{0}}D$ we get the
desired equality.

If $Y$ is not projective but is defined over a finite field
$k={\mathbb{F}}_{q}$ the result follows from Weil conjectures: the multiset of
$q$-power Frobenius eigenvalues consists of algebraic integers and is stable
under complex conjugation. Hence, it is stable under the operation
$\alpha\mapsto q^{n}\alpha^{-1}=\overline{\alpha}$ so the multiset of slopes
is stable under $\lambda\mapsto n-\lambda$, as desired. ∎

###### Proof of Proposition 2.1.

Consider the filtered $\varphi$-module given by
$D=H^{n}_{\mathrm{cris}}({\mathfrak{X}}_{k}/W(k))[\frac{1}{p}]$ with the
Frobenius induced by
$\operatorname{Fr}:{\mathfrak{X}}_{k}\times_{k,\sigma}k\to{\mathfrak{X}}_{k}$
and the filtration on $D_{K}$ is given by the Hodge filtration on the de Rham
cohomology of the generic fiber $H^{n}_{\mathrm{dR}}(X/K)\simeq
H^{n}_{\mathrm{cris}}({\mathfrak{X}}/W(k))\otimes_{W(k)}K=D_{K}$.

By the crystalline comparison theorem proved in [BMS18] Theorem 1.1(i) for
smooth proper formal schemes, the filtered $\varphi$-module $D$ is obtained by
applying the functor $D_{\mathrm{cris}}$ to the $p$-adic Galois representation
$H^{n}_{\text{et}}(X_{\overline{K}},{\mathbb{Q}}_{p})$. In particular, $D$ is
weakly admissible by Proposition 4.4.5 [Fon79].

Hence, $t_{H}(D)=t_{N}(D)$ so by Lemma 2.3 we get
$t_{H}(D)=n\cdot\dim_{K}H^{n}_{\mathrm{dR}}(X/K)-t_{H}(D)$. As
$t_{H}(D)=h^{1,{n-1}}+2h^{2,n-2}+\dots+n\cdot h^{n,0}$ and
$\dim_{K}H^{n}_{\mathrm{dR}}(X/K)=h^{0,n}+h^{1,n-1}+\dots+h^{n,0}$ we get the
desired equality. ∎

######  Remark 2.4.

The above proof used only that the endpoints of the Newton and Hodge polygons
of $D$ coincide. The full strength of weak admissibility, however, does not
put any further restrictions on the Hodge numbers. For any tuple of non-
negative integers $h^{0,n},h^{1,n-1},\dots,h^{n,0}$ satisfying (2.1) equip the
vector space $D:={\mathbb{Q}}_{p}^{b}$ of dimension
$b=h^{0,n}+h^{1,n-1}+\dots+h^{n,0}$ with an endomorphism $\varphi$ that has
irreducible characteristic polynomial with all roots having valutation $a/b$
where $a=h^{1,n-1}+2h^{2,n-2}\dots+n\cdot h^{n,0}$. Endowing $D$ with an
arbitrary filtration $F^{i}D$ such that
$\dim_{{\mathbb{Q}}_{p}}F^{i}D=h^{i,n-i}+h^{i+1,n-i-1}+\dots+h^{n,0}$ turns
$D$ into an object of $\mathrm{MF}_{{\mathbb{Q}}_{p}}^{\varphi}$ with Hodge
numbers $h^{i,n-i}$ which is weakly admissible just because there are no
proper non-zero submodules stable under $\varphi$. Alternatively, we can apply
Theorem 1 of [FR05] that shows the existence of a weakly admissible module
over $W(\overline{k})[\frac{1}{p}]$ with any given Newton polygon and any
Hodge polygon having the same endpoints and lying below it.

##  3\. Cyclic group acting on a formal abelian scheme

Fix once and for all a prime number $l$ satisfying the following condition:

(3.1) $l\neq p\textit{ and the order of }p\textit{ in the multiplicative group
}({\mathbb{Z}}/l)^{\times}\textit{ is divisible by }4$

This condition is equivalent to $l$ being a prime divisor of a number of the
form $p^{2r}+1$. Denote by $k$ the finite field ${\mathbb{F}}_{p}(\mu_{l})$
obtained by adjoining to ${\mathbb{F}}_{p}$ all $l$-th roots of unity and by
$\kappa\subset k$ the finite field $\kappa={\mathbb{F}}_{p^{2}}$. Let $G$
denote the cyclic group ${\mathbb{Z}}/l$ with a chosen generator $\sigma\in
G$. We will denote by ${\mathbb{Z}}[\mu_{l}]$ the ring of integers of the
cyclotomic extension ${\mathbb{Q}}(\mu_{l})$ inside a chosen algebraic closure
$\overline{\mathbb{Q}}$.

The goal of this section is to exploit the existence of abelian varieties over
a finite field with multiplication by ${\mathbb{Z}}[\mu_{l}]$ such that the
resulting CM type is different from those appearing over characteristic zero
fields. This allows us to construct a formal abelian scheme with an action of
$G$ having asymmetric dimensions of invariant spaces on Hodge cohomology
groups in degree $3$:

######  Proposition 3.1.

There exists a formal abelian scheme ${\mathfrak{Z}}$ with an action of $G$
over $\operatorname{{Spf}}W(k^{\prime})$ for some finite field $k^{\prime}$
such that

(3.2)
$\operatorname{{rk}}H^{0}({\mathfrak{Z}},\Omega^{3}_{{\mathfrak{Z}}/W(k^{\prime})})^{G}<\operatorname{{rk}}H^{3}({\mathfrak{Z}},{\mathcal{O}})^{G}.$

######  Remark 3.2.

The proof also produces examples with inequalities going in the other
direction, we record here this particular version for the convenience of
applying it to Lemma 5.1.

We first collect the necessary facts about abelian varieties in Lemmas 3.3-3.8
and then use them together with the combinatorial Lemma 3.10 to prove the
Proposition 3.1 at the end of this section. All the facts about abelian
varieties used in this section are contained in [CCO14] (especially, cf.
Example 4.1.2 in loc. cit). We include here the proofs in an attempt to make
the exposition relatively self-contained.

######  Lemma 3.3.

There exists an abelian variety $A$ over $\kappa$ of dimension $\frac{l-1}{2}$
equipped with an action of $G$ such that the eigenvalues of $\sigma\in G$ on
$H^{1}_{\mathrm{cris}}(A/W(\kappa))\otimes_{W(\kappa)}W(k)[\frac{1}{p}]$ are
the $l-1$ pairwise different nontrivial roots of unity of order $l$ and the
$p^{2}$-Frobenius endomorphism of $A$ is given by $\sigma\circ[p]_{A}$.

###### Proof.

Let $\zeta_{l}$ be a primitive $l$-th root of unity in
$\overline{\mathbb{Q}}$. By Honda–Tate theory, there exists an abelian variety
$A^{\prime}$ (unique up to isogeny) over $\kappa$ with eigenvalues of the
$p^{2}$-Frobenius on first etale cohomology given by the conjugates of the
Weil number $p\cdot\zeta_{l}$. Since
$[{\mathbb{Q}}_{p}(\mu_{l}):{\mathbb{Q}}_{p}]=\operatorname{{ord}}_{({\mathbb{Z}}/l)^{\times}}p$
is even we have $2\dim A^{\prime}=[{\mathbb{Q}}(\mu_{l}):{\mathbb{Q}}]=l-1$ by
Theoreme 1(ii) of [Tat71]. If $\varphi_{A^{\prime}}$ denotes the
$p^{2}$-Frobenius endomorphism of $A^{\prime}$ then $\varphi_{A^{\prime}}^{l}$
acts by $p^{l}$ on
$H^{1}_{\text{et}}(A^{\prime}_{\overline{{\mathbb{F}}}_{p}},{\mathbb{Z}}_{t})$
(where $t$ is any prime different from $p$) so $\varphi_{A^{\prime}}^{l}$ is
equal to the multiplication-by-$p^{l}$ endomorphism of $A^{\prime}$. Hence,
$\varphi_{A^{\prime}}$ induces the action of the subalgebra
${\mathbb{Z}}[p\zeta_{l}]\subset{\mathbb{Z}}[\mu_{l}]$ on the abelian variety
$A^{\prime}$ with $p\zeta_{l}$ acting by $\varphi_{A^{\prime}}$.

Proposition 1.7.4.4 of [CCO14] shows that the Serre’s tensor product
$A:={\mathbb{Z}}[\zeta_{l}]\otimes_{{\mathbb{Z}}[p\zeta_{l}]}A^{\prime}$ is an
abelian variety isogenous to $A^{\prime}$ that has an action of $G$ with
$\sigma$ acting as $\zeta_{l}$. The eigenvalues of $\sigma$ on
$H^{1}_{\mathrm{cris}}(A/W(\kappa))$ are equal to the eigenvalues of
$\varphi_{A}$ divided by $p$ and, since characteristic polynomials of
$\varphi_{A}$ on crystalline and etale cohomology are equal, the eiegnvalues
of $\sigma$ are the $l-1$ Galois conjugates of $\zeta_{l}$, as desired. ∎

######  Remark 3.4.

This particular construction does not require the full power of Honda–Tate
theory: we can take $A^{\prime}$ to be the Weil restriction of scalars
$\operatorname{{Res}}^{{\mathbb{F}}_{p^{2l}}}_{{\mathbb{F}}_{p^{2}}}E$ of a
supersingular elliptic curve $E$ defined over ${\mathbb{F}}_{p}$ with
vanishing trace of $p$-Frobenius, cf. Lemme 1 in §3 of [Tat71].

The map $g\mapsto g^{p}$ is an automorphism of the group $G$. For a
representation $\rho:G\to\operatorname{{Aut}}(V)$ on a module $V$ we denote by
$V^{\tau}$ the representation on the same module in which $g\in G$ acts by
$\rho(g^{p})$.

######  Lemma 3.5.

For any $A$ as in Lemma 3.3 we have

(i) The representations $H^{1}(A,{\mathcal{O}})$ and
$H^{0}(A,\Omega^{1}_{A/\kappa})^{\tau}$ are isomorphic.

(ii) $A$ lifts to a formal abelian scheme ${\mathfrak{A}}$ over
$\operatorname{{Spf}}W(\kappa)$ together with an action of $G$ such that the
representations $H^{1}({\mathfrak{A}},{\mathcal{O}})$ and
$H^{0}({\mathfrak{A}},\Omega^{1}_{{\mathfrak{A}}/W(\kappa)})^{\tau}$ are
isomorphic.

###### Proof.

(i) Denote by $A^{(1)}=A\times_{\kappa,\operatorname{Fr}}\kappa$ the
Frobenius-twist of $A$. For a vector space $W$ over $\kappa$ denote as well by
$W^{(1)}:=W\otimes_{\kappa,\operatorname{Fr}}\kappa$ its twist. Note that the
double twist $A^{(2)}:=(A^{(1)})^{(1)}$ is canonically identified with $A$.

The de Rham cohomology $H^{1}_{\mathrm{dR}}(A/\kappa)$ carries the Hodge
filtration $H^{1}_{\mathrm{dR}}(A/\kappa)=F^{0}\supset F^{1}\supset F^{2}=0$
and the conjugate filtration $H^{1}_{\mathrm{dR}}(A/\kappa)=G_{2}\supset
G_{1}\supset G_{0}=0$ satisfying
$F^{0}/F^{1}=H^{0}(A,\Omega^{1}_{A/\kappa}),F^{1}/F^{2}=H^{1}(A,{\mathcal{O}}),G_{2}/G_{1}=H^{0}(A^{(1)},\Omega^{1}_{A^{(1)}/\kappa}),G_{1}/G_{0}=H^{1}(A^{(1)},{\mathcal{O}})$,
see §7 of [Kat70]. The compositions

$H^{1}(A^{(1)},{\mathcal{O}})=G_{1}\to H^{1}_{\mathrm{dR}}(A/\kappa)\to
F^{0}/F^{1}=H^{1}(A,{\mathcal{O}})$

and

$H^{1}_{\mathrm{dR}}(A^{(1)}/\kappa)=H^{1}_{\mathrm{dR}}(A/\kappa)^{(1)}\to(F^{0}/F^{1})^{(1)}=H^{1}(A,{\mathcal{O}})^{(1)}=G_{1}/G_{0}\to
H^{1}_{\mathrm{dR}}(A/\kappa)$

are both induced by the relative Frobenius $\operatorname{Fr}_{A}:A\to
A^{(1)}$.

Since $\operatorname{Fr}_{A}^{2}$ is equal to the multiplication by $p$
endomorphism $[p]_{A}$ up to composing with an automorphism,
$\operatorname{Fr}_{A}^{2}$ induces the zero map on
$H^{1}_{\mathrm{dR}}(A/k)$. Equivalently,
$\ker(\operatorname{Fr}_{A}^{*}:H^{1}_{\mathrm{dR}}(A^{(1)}/\kappa)\to
H^{1}_{\mathrm{dR}}(A/\kappa))$ contains
$\operatorname{{Im}}(\operatorname{Fr}_{A}^{*}:H^{1}_{\mathrm{dR}}(A/\kappa)=H^{1}_{\mathrm{dR}}(A^{(2)}/\kappa)\to
H^{1}_{\mathrm{dR}}(A^{(1)}/\kappa))=G_{1}^{(1)}$. The sum of the dimensions
of these two vector space is $\dim_{\kappa}H^{1}_{\mathrm{dR}}(A/\kappa)=2g$
and dimension of the image is equal to $g$, so the containment is in fact
equality and we get $G_{1}=F^{1}$.

This gives a $G$-equivariant $\kappa$-linear isomorphism
$H^{1}(A^{(1)},{\mathcal{O}})\simeq H^{0}(A,\Omega^{1}_{A/\kappa})$. If $V$ is
a representation of $G$ on a vector space over a field of characteristic $p$
then the character of Frobenius-twisted representation $V^{(1)}$ is equal to
that of $V^{\tau}$ so the first assertion of the lemma follows.

(ii) By Theorem V.1.10 of [Mes72] the category of formal abelian schemes over
$W(\kappa)$ is equivalent to the category of pairs $(A_{0},\widetilde{F})$
where $A_{0}$ is an abelian variety over $\kappa$ and $\widetilde{F}$ is a
$W(\kappa)$-submodule of $H^{1}_{\mathrm{cris}}(A_{0}/W(\kappa))$ such that
the quotient $H^{1}_{\mathrm{cris}}(A_{0}/W(\kappa))/\widetilde{F}$ is
torsion-free and $\widetilde{F}/p$ is equal to the first step of Hodge
filtration $F^{1}\subset
H^{1}_{\mathrm{dR}}(A/\kappa)=H^{1}_{cris}(A/W(\kappa))/p$.

Since the eigenvalues of $\sigma$ on $H^{1}_{\mathrm{cris}}(A/W(\kappa))$ are
pair-wise different, the same is true for the action of $\sigma$ on
$H^{1}_{\mathrm{dR}}(A/\kappa)$ so there is a unique isomorphism
$H^{1}_{\mathrm{dR}}(A/\kappa)\simeq H^{0}(A,\Omega^{1}_{A/\kappa})\oplus
H^{1}(A,{\mathcal{O}})$ of $G$-representations splitting the Hodge filtration
on $H^{1}_{\mathrm{dR}}(A/\kappa)$. Since the isomorphism class of a
representation of $G$ on a finite free $W(\kappa)$-module is completely
determined by its reduction modulo $p$, there is a unique $G$-equivariant
decomposition
$H^{1}_{\mathrm{cris}}(A/W(\kappa))=\widetilde{F}\oplus\widetilde{F}^{\prime}$
lifting the decomposition of the de Rham cohomology. The formal lift
${\mathfrak{A}}$ of $A$ defined by $\widetilde{F}$ then comes equipped with a
lift of the action of $G$ because $\widetilde{F}\subset
H^{1}_{\mathrm{cris}}(A/W(\kappa))$ is stable under $G$. The assertion about
the action of $G$ on Hodge cohomology groups of ${\mathfrak{A}}$ is a formal
consequence of (i). ∎

The result of Lemma 3.5(i) is specific to positive characteristic: for an
abelian variety $B$ with an action of $G$ over a field $F$ of characteristic
zero the representations $H^{0}(B,\Omega^{1}_{B/F})$ and
$H^{1}(B,{\mathcal{O}})^{\vee}$ have to be isomorphic by the next lemma,
whereas for $A$ as above the representations $H^{0}(A,\Omega^{1}_{A/\kappa})$
and $H^{1}(A,{\mathcal{O}})^{\vee}\simeq
H^{0}(A,\Omega^{1}_{A/\kappa})^{\tau\vee}$ are self-dual and are not
isomorphic to each other (in fact, any representation of $G$ on a
${\mathbb{F}}_{p^{2}}$-vector space has to be self-dual because $-1=p^{2r}$ in
$({\mathbb{Z}}/l)^{\times}$ for some $r$ by the assumption on $l$). The same
is happening for the lift ${\mathfrak{A}}$, so, in particular, it has to be
non-algebraizable.

######  Lemma 3.6.

For an abelian variety $B$ over a field $F$ of characteristic zero equipped
with an action of a finite group $H$ the representations
$H^{0}(B,\Omega^{1}_{B/F})$ and $H^{1}(B,{\mathcal{O}})^{\vee}$ are
isomorphic.

###### Proof.

Choose an ample line bundle $L$ on $B$. Then $M:=\bigotimes\limits_{h\in
H}h^{*}(L)$ is an ample line bundle inducing a $H$-equivariant polarization
$\lambda_{M}:B\to B^{\vee}$ that is separable because $\mathrm{char}\,F=0$.
The induced map $\lambda_{M}^{*}:H^{0}(B^{\vee},\Omega^{1}_{B^{\vee}/F})\to
H^{0}(B,\Omega^{1}_{B/F})$ provides a $H$-equivariant isomophism between
$H^{0}(B,\Omega^{1}_{B/F})$ and $H^{1}(B,{\mathcal{O}})^{\vee}\simeq
H^{0}(B^{\vee},\Omega^{1}_{B^{\vee}/F})$ ∎

The abelian schemes constructed in Lemma 3.5 do not yet provide the desired
${\mathfrak{Z}}$ because for all $i,j$ the module
$H^{i}({\mathfrak{A}},\Omega^{j}_{{\mathfrak{A}}/W(\kappa)})$ is the
$\tau$-twist of $H^{j}({\mathfrak{A}},\Omega^{i}_{{\mathfrak{A}}/W(\kappa)})$
and, in particular, these modules have equal ranks of invariants. We will
break this symmetry by taking the direct product with an appropriate algebraic
abelian scheme with complex multiplication by ${\mathbb{Q}}(\mu_{l})$.

Recall that $k$ is an extension of ${\mathbb{F}}_{p}$ containing all $l$-th
roots of unity, so any representation of $G$ on a finite free module over
$W(k)$ is isomorphic to a direct sum of characters $\chi_{\zeta}$ given by
sending $\sigma$ to a degree $l$ root of unity $\zeta\in\mu_{l}\subset
W(k)^{\times}$.

######  Definition 3.7.

We call a representation $U$ of $G$ on a finite free $W(k)$-module typical if
it has the form $\bigoplus\limits_{i=1}^{d}\bigoplus\limits_{\zeta\in
S_{i}}\chi_{\zeta}$ where each $S_{i}$ is a subset of $\mu_{l}$ of cardinality
$\frac{l-1}{2}$ such that $S_{i}\cap S_{i}^{-1}=\varnothing$ and $d$ is some
number.

If ${\mathfrak{B}}$ is an algebraic abelian scheme over $W(k)$ of dimension
$\frac{l-1}{2}$ with a non-trivial action of $G$ then the representation
$H^{0}({\mathfrak{B}},\Omega^{1}_{{\mathfrak{B}}/W(k)})$ is typical: for the
base change of ${\mathfrak{B}}$ to ${\mathbb{C}}$ singular cohomology
$H^{1}({\mathfrak{B}}_{{\mathbb{C}}},{\mathbb{Z}})$ is a free abelian group of
rank $l-1$ with a non-trivial action of $G$. Hence, all non-trivial characters
appear in $H^{1}({\mathfrak{B}}_{{\mathbb{C}}},{\mathbb{Z}})$ exactly once and
by Lemma 3.6 these characters have to be distributed between
$H^{0}({\mathfrak{B}}_{{\mathbb{C}}},\Omega^{1}_{{\mathfrak{B}}_{{\mathbb{C}}}/{\mathbb{C}}})=H^{0}({\mathfrak{B}},\Omega^{1}_{{\mathfrak{B}}/W(k)})\otimes_{W(k)}{\mathbb{C}}$
and $H^{1}({\mathfrak{B}}_{{\mathbb{C}}},{\mathcal{O}})$ in such a way that
$\chi_{\zeta}$ and $\chi_{\zeta^{-1}}$ never occur in the same piece of the
Hodge decomposition. The next lemma deduces from the theory of complex
multiplication that every typical representation arises from an abelian scheme
over the ring of integers in an unramified extension of ${\mathbb{Q}}_{p}$:

######  Lemma 3.8.

For any typical representation of $G$ on a finite free $W(k)$-module $U$ there
exists an abelian scheme ${\mathfrak{B}}$ with an action of $G$ over the ring
of integers ${\mathcal{O}}_{L}=W(k^{\prime})$ of a finite unframified
extension $L=W(k^{\prime})[\frac{1}{p}]\supset W(k)[\frac{1}{p}]$ such that
$H^{0}({\mathfrak{B}},\Omega^{1}_{{\mathfrak{B}}/{\mathcal{O}}_{L}})\simeq
U\otimes_{W(k)}{{\mathcal{O}}_{L}}$ and
$H^{1}({\mathfrak{B}},{\mathcal{O}})\simeq
U^{\vee}\otimes_{W(k)}{\mathcal{O}}_{L}$ as $G$-representations.

###### Proof.

It is enough to prove the lemma for typical representations of rank
$\frac{l-1}{2}$ because we have
$H^{0}({\mathfrak{B}}_{1}\times_{{\mathcal{O}}_{L}}{\mathfrak{B}}_{2},\Omega^{1}_{{\mathfrak{B}}_{1}\times_{{\mathcal{O}}_{L}}{\mathfrak{B}}_{2}/{\mathcal{O}}_{L}})\simeq
H^{0}({\mathfrak{B}}_{1},\Omega^{1}_{{\mathfrak{B}}_{1}/{\mathcal{O}}_{L}})\oplus
H^{0}({\mathfrak{B}}_{2},\Omega^{1}_{{\mathfrak{B}}_{2}/{\mathcal{O}}_{L}})$
and likewise
$H^{1}({\mathfrak{B}}_{1}\times_{{\mathcal{O}}_{L}}{\mathfrak{B}}_{2},{\mathcal{O}})\simeq
H^{1}({\mathfrak{B}}_{1},{\mathcal{O}})\oplus
H^{1}({\mathfrak{B}}_{2},{\mathcal{O}})$ for all abelian schemes
${\mathfrak{B}}_{1},{\mathfrak{B}}_{2}$, so the general case can be obtained
by taking the products.

Fix a non-trivial root of unity $\zeta_{l}\in{\mathbb{Q}}(\mu_{l})$ and an
embedding ${\mathbb{Q}}(\mu_{l})\subset W(k)[\frac{1}{p}]$, giving a bijection
between roots of unity in ${\mathbb{Q}}(\mu_{l})$ and $W(k)$. Each non-trivial
root of unity $\zeta\in\mu_{l}(W(k))$ thus defines an embedding of the
cyclotomic field
$\varphi_{\zeta}:{\mathbb{Q}}(\mu_{l})\to\overline{{\mathbb{Q}}}$ into the
algebraic closure given by $\zeta_{l}\mapsto\zeta$.

Let $U$ be a typical representation of rank $\frac{l-1}{2}$ over $W(k)$. We
associate to $U$ the set $\Phi=\\{\varphi_{\zeta}|\chi_{\zeta^{-1}}\text{
appears in }U\\}$ of embeddings. By the definition of typical representation
$\Phi$ is a CM type of the field ${\mathbb{Q}}(\mu_{l})$. Since
${\mathbb{Q}}(\mu_{l})$ is Galois over ${\mathbb{Q}}$, the reflex field of
$\Phi$ is contained in ${\mathbb{Q}}(\mu_{l})$. By [CCO14] Corollary A.4.6.5
applied with $N=p$ (beware that the symbol ”$p$” bears its own meaning in loc.
cit.) there exists an abelian variety $B^{\prime}$ with complex multiplication
by ${\mathbb{Q}}(\mu_{l})$ of type $\Phi$ defined over an unramified extension
$F$ of the reflex field that has good reduction at all primes above $p$. Take
$L=F_{{\mathfrak{p}}}$ for some prime ${\mathfrak{p}}\subset{\mathcal{O}}_{F}$
above $p$ and consider the good model ${\mathfrak{B}}^{\prime}$ of
$B^{\prime}$ over ${\mathcal{O}}_{L}$. The isogeny action of
${\mathbb{Q}}(\mu_{l})$ then extends onto ${\mathfrak{B}}^{\prime}$. In
particular, embedding $G$ into ${\mathbb{Q}}(\mu_{l})$ via
$\sigma\mapsto\zeta_{l}$ gives an action of $G$ on ${\mathfrak{B}}^{\prime}$
in the isogeny category. By our choice of $\Phi$ the $G$-representation
$H^{0}({\mathfrak{B}}^{\prime},\Omega^{1}_{{\mathfrak{B}}^{\prime}/{\mathcal{O}}_{L}})\otimes_{{\mathcal{O}}_{L}}L$
is isomorphic to $U\otimes_{W(k)}L$.

Analogously to the proof of Lemma 3.3, Serre’s tensor product construction
gives an isogeny ${\mathfrak{B}}^{\prime}\to{\mathfrak{B}}$ to an abelian
scheme ${\mathfrak{B}}$ that carries a genuine action of
${\mathbb{Z}}[\mu_{l}]$. In particular, $G$ acts on ${\mathfrak{B}}$ such that
$H^{0}({\mathfrak{B}},\Omega^{1}_{{\mathfrak{B}}/{\mathcal{O}}_{L}})$ is
isomorphic to $U\otimes_{W(k)}{\mathcal{O}}_{L}$, as desired. To see that
$H^{1}({\mathfrak{B}},{\mathcal{O}})$ is isomorphic to
$U^{\vee}\otimes_{W(k)}{\mathcal{O}}_{L}$ it is enough to check the analogous
statement for the generic fiber ${\mathfrak{B}}_{L}$ and this is given by
Lemma 3.6. ∎

######  Remark 3.9.

We have appealed to stronger results of [CCO14] to make sure that there exists
${\mathfrak{B}}$ defined over the ring of integers in an unramified extension
of ${\mathbb{Q}}_{p}$. If we did not care about the field of definition of the
ultimate counterexamples to Hodge symmetry we could have applied a cruder form
of CM theory given e.g. by [Tat71] Lemme 4.

######  Lemma 3.10.

Let $V$ be a representation of $G$ on a finite free $W(k)$-module such that
$V$ is isomorphic to its dual $V^{\vee}$ but is not isomorphic to the twist
$V^{\tau}$. Then there exists a typical representation $U$ such that

(3.3) $\operatorname{{rk}}\Lambda^{3}(V\oplus
U)^{G}\neq\operatorname{{rk}}\Lambda^{3}(V^{\tau}\oplus U^{\vee})^{G}$

###### Proof.

Suppose on the contrary that these ranks of invariant subspaces are equal for
all typical $U$. Using that $\Lambda^{3}(V\oplus
U)\simeq\Lambda^{3}V\oplus\Lambda^{2}V\otimes U\oplus
V\otimes\Lambda^{2}U\oplus\Lambda^{3}U$ and the fact that twisting by an
automorphism of $G$ does not change the invariant submodule of a
representation we get

$\operatorname{{rk}}(\Lambda^{2}V\otimes U^{\oplus
r})^{G}+\operatorname{{rk}}(V\otimes\Lambda^{2}(U^{\oplus
r}))^{G}=\operatorname{{rk}}(\Lambda^{2}V^{\tau}\otimes U^{\vee\oplus
r})^{G}+\operatorname{{rk}}(V^{\tau}\otimes\Lambda^{2}(U^{\vee\oplus r}))^{G}$

for all typical $U$ and any multiplicity $r$ (by definition, the class of
typical representations is closed under direct sums). Since
$\Lambda^{2}(U^{\oplus r})\simeq(\Lambda^{2}U)^{\oplus r}\oplus(U^{\otimes
2})^{\oplus\binom{r}{2}}$ both sides of the equality are quadratic polynomials
in $r$ for a fixed $U$ and comparing the leading coefficients we get

$\operatorname{{rk}}(V\otimes U^{\otimes
2})^{G}=\operatorname{{rk}}(V^{\tau}\otimes U^{\vee\otimes 2})^{G}$

As $V$ is assumed to be self-dual this equality is equivalent to
$\operatorname{{rk}}(V\otimes U^{\otimes
2})^{G}=\operatorname{{rk}}(V^{\tau}\otimes U^{\otimes 2})^{G}$. In other
words, the difference of characters $\chi_{V}-\chi_{V^{\tau}}$ is orthogonal
to any character $\chi_{U^{\otimes 2}}$ for a typical $U$ with respect to the
pairing $\langle\chi_{1},\chi_{2}\rangle:=\sum\limits_{g\in
G}\chi_{1}(g)\chi_{2}(g)$ on the character ring $W(k)[G]$. Moreover, this
implies that $\chi_{V}-\chi_{V^{\tau}}$ is orthogonal to $\chi_{U_{1}\otimes
U_{2}}=\chi_{U_{1}}\cdot\chi_{U_{2}}$ for any two typical representations
$U_{1},U_{2}$ because $2\chi_{U_{1}\otimes U_{2}}=\chi_{(U_{1}\oplus
U_{2})^{\otimes 2}}-\chi_{U_{1}^{\otimes 2}}-\chi_{U_{2}^{\otimes 2}}$.

If $\zeta$ is any non-trivial root of unity, then for a set
$\zeta_{1},\dots,\zeta_{\frac{l-3}{2}}$ of pairwise different roots such that
$\zeta_{i}\neq\zeta_{j}^{-1}$ and $\zeta_{i}\neq\zeta^{\pm 1}$ for all $i,j$
both representations
$\chi_{\zeta}\oplus\chi_{\zeta_{1}}\oplus\dots\oplus\chi_{\zeta_{\frac{l-3}{2}}}$
and
$\chi_{\zeta^{-1}}\oplus\chi_{\zeta_{1}}\oplus\dots\oplus\chi_{\zeta_{\frac{l-3}{2}}}$
are typical. Hence the span of characters of typical representations in
$W(k)[G]$ contains all characters of the form $\chi_{\zeta}-\chi_{\zeta^{-1}}$
for a non-trivial root of unity $\zeta$. By linearity of the pairing,
$\chi_{V}-\chi_{V^{\tau}}$ is hence also orthogonal to
$\chi^{2}_{\zeta}+\chi^{2}_{\zeta^{-1}}=(\chi_{\zeta}-\chi_{\zeta^{-1}})^{2}+2\chi_{1}$.
However,
$\langle\chi_{V},\chi^{2}_{\zeta}+\chi^{2}_{\zeta^{-1}}\rangle=2\langle\chi_{V},\chi^{2}_{\zeta}\rangle$
by self-duality of $V$ and likewise for $V^{\tau}$ so we get a contradiction
since, by assumption, there exists $\zeta$ such that multiplicities of
$\chi_{\zeta^{-2}}$ in $V$ and $V^{\tau}$ are different. ∎

###### Proof of Proposition 3.1.

Let ${\mathfrak{A}}$ be a formal abelian scheme with an action of $G$ provided
by Lemma 3.5(ii). Consider the representation
$V=H^{0}({\mathfrak{A}},\Omega^{1}_{{\mathfrak{A}}/W(\kappa)})$. We have
$G$-equivariant isomorphisms $V\oplus V^{\tau}\simeq
H^{0}({\mathfrak{A}},\Omega^{1}_{{\mathfrak{A}}/W(\kappa)})\oplus
H^{1}({\mathfrak{A}},{\mathcal{O}})\simeq
H^{1}_{\mathrm{dR}}({\mathfrak{A}}/W(\kappa))\simeq
H^{1}_{\mathrm{cris}}(A/W(\kappa))$, so every non-trivial character
$\chi_{\zeta}$ appears in exactly one of $V$ and $V^{\tau}$ with multiplicity
one. It follows that $V\simeq(V^{\tau})^{\tau}$. Since $V^{\vee}$ is obtained
from $V$ by applying the twist $\tau$ an even number of times, the
representation $V\otimes_{W(\kappa)}W(k)$ satisfies the assumptions of Lemma
3.10 and there exists a typical representation $U$ such that non-equality
(3.3) holds. Let ${\mathfrak{B}}$ be an abelian scheme over
${\mathcal{O}}_{L}=W(k^{\prime})$ provided by Lemma 3.8 applied to $U$. The
product ${\mathfrak{Z}}^{\prime}={\mathcal{A}}\times_{W(k)}{\mathfrak{B}}$
equipped with the diagonal action then has
$H^{0}({\mathfrak{Z}}^{\prime},\Omega^{1}_{{\mathfrak{Z}}/{\mathcal{O}}_{L}})\simeq(V\oplus
U)\otimes_{W(k)}{{\mathcal{O}}_{L}}$ and
$H^{1}({\mathfrak{Z}}^{\prime},{\mathcal{O}})\simeq(V^{\tau}\oplus
U^{\vee})\otimes_{W(k)}{{\mathcal{O}}_{L}}$ as $G$-representations.

Since
$H^{0}({\mathfrak{Z}}^{\prime},\Omega^{3}_{{\mathfrak{Z}}/{\mathcal{O}}_{L}})\simeq\Lambda^{3}H^{0}({\mathfrak{Z}}^{\prime},\Omega^{1}_{{\mathfrak{Z}}^{\prime}/{\mathcal{O}}_{L}})$
and
$H^{3}({\mathfrak{Z}}^{\prime},{\mathcal{O}})\simeq\Lambda^{3}H^{1}({\mathfrak{Z}}^{\prime},{\mathcal{O}})$
we get
$\operatorname{{rk}}H^{0}({\mathfrak{Z}}^{\prime},\Omega^{3}_{{\mathfrak{Z}}/{\mathcal{O}}_{L}})^{G}\neq\operatorname{{rk}}H^{3}({\mathfrak{Z}}^{\prime},{\mathcal{O}})^{G}$.
If
$\operatorname{{rk}}H^{0}({\mathfrak{Z}}^{\prime},\Omega^{3}_{{\mathfrak{Z}}/{\mathcal{O}}_{L}})^{G}>\operatorname{{rk}}H^{3}({\mathfrak{Z}}^{\prime},{\mathcal{O}})^{G}$
then ${\mathfrak{Z}}={\mathfrak{Z}}^{\prime}$ gives the desired inequality
3.2. If the inequality goes the other way, take
${\mathfrak{Z}}={\mathfrak{Z}}^{\prime\vee}$ using that there are
$G$-equivariant isomorphisms
$H^{0}({\mathfrak{Z}}^{\prime\vee},\Omega^{1}_{{\mathcal{O}}_{L}})\simeq
H^{1}({\mathfrak{Z}}^{\prime},{\mathcal{O}})^{\vee}$ and
$H^{1}({\mathfrak{Z}}^{\prime\vee},{\mathcal{O}})\simeq
H^{0}({\mathfrak{Z}}^{\prime},\Omega^{1}_{{\mathfrak{Z}}^{\prime}/{\mathcal{O}}_{L}})^{\vee}$.
∎

######  Remark 3.11.

(i) The same argument works just as well to provide an abelian scheme with
different ranks of invariants
$\operatorname{{rk}}H^{i}({\mathfrak{Z}},\Omega^{j}_{{\mathfrak{Z}}/W(k)})^{G}\neq\operatorname{{rk}}H^{i}({\mathfrak{Z}},\Omega^{j}_{{\mathfrak{Z}}/W(k)})^{G}$
for any $i\neq j,i+j\geq 3$. We chose to treat here only the case
$(i,j)=(3,0)$ and obtain counterexamples to symmetry in higher degrees by
taking products with auxiliary varieties because that simplifies the argument
in the next section.

(ii) I am not aware of a simple way of choosing ${\mathfrak{Z}}$ that would
work uniformly for all $p$. However, for any fixed $l$ everything can be made
more explicit for primes $p$ for which $l$ satisfied condition (3.1). For
instance, for $p$ congruent to $2$ or $3$ modulo $5$ we can take $l=5$ so that
$\dim{\mathfrak{A}}=2$ and ${\mathfrak{B}}$ can also be chosen to be
$2$-dimensional. In this case (for any of the particular choices of
${\mathfrak{A}}$ and ${\mathfrak{B}}$ that give non-symmetric dimensions of
invariants) $H^{3}_{\mathrm{dR}}(A\times B/L)^{G}$ is a filtered
$\varphi$-module with Hodge numbers
$(h^{3,0},h^{2,1},h^{1,2},h^{0,3})=(0,5,2,1)$ or $(1,2,5,0)$ and single slope
$\frac{3}{2}$ (the slopes can be computed by the Shimura-Taniyama formula).
Note that such Hodge numbers satisfy the relation (2.1)

##  4\. Main example

######  Theorem 4.1.

For every pair of distinct positive integers $i\neq j$ with $i+j\geq 3$ there
exists a smooth proper formal scheme ${\mathfrak{X}}$ over ${\mathbb{Z}}_{p}$
with projective special fiber ${\mathfrak{X}}_{{\mathbb{F}}_{p}}$ such that
the Hodge cohomology groups
$H^{i}({\mathfrak{X}},\Omega^{j}_{{\mathfrak{X}}/{\mathbb{Z}}_{p}})$ and
$H^{j}({\mathfrak{X}},\Omega^{i}_{{\mathfrak{X}}/{\mathbb{Z}}_{p}})$ are free
${\mathbb{Z}}_{p}$-modules of different rank.

We first record two standard general facts from formal geometry. The main
reference on rigid-analytic generic fibers of formal schemes is [BL93]. Let
$K$ be a discretely valued field of characteristic zero with ring of integers
${\mathcal{O}}_{K}$, maximal ideal ${\mathfrak{m}}\subset{\mathcal{O}}_{K}$
and perfect residue field $k={\mathcal{O}}_{K}/{\mathfrak{m}}$ of
characteristic $p$.

######  Lemma 4.1.

If ${\mathfrak{X}}$ is a smooth proper formal scheme over
$\operatorname{{Spf}}{\mathcal{O}}_{K}$ with generic fiber $X$, then there is
a canonical isomophism $H^{i}(X,\Omega^{j}_{X/K})\simeq
H^{i}({\mathfrak{X}},\Omega^{j}_{{\mathfrak{X}}/{\mathcal{O}}_{K}})[\frac{1}{p}]$.
In particular, $h^{i,j}(X)=h^{i,j}({\mathfrak{X}})$.

###### Proof.

Let $({\mathfrak{U}}_{i})_{i\in I}$ be a finite cover of ${\mathfrak{X}}$ by
affine open formal subschemes. Their generic fibers $U_{i}$ induce a covering
of $X$ by affinoid subdomains. For any coherent sheaf ${\mathcal{F}}$ on
${\mathfrak{X}}$ its cohomology is computed by the Čech complex
$\bigoplus_{i\in I}{\mathcal{F}}({\mathfrak{U}}_{i})\to\bigoplus_{i,j\in
I}{\mathcal{F}}({\mathfrak{U}}_{i}\cap{\mathfrak{U}}_{j})\to\dots$. The
cohomology of the sheaf $F$ on $X$ associated to ${\mathcal{F}}$ is computed
by the Čech complex $\bigoplus_{i\in I}F(U_{i})\to\bigoplus_{i,j\in
I}F(U_{i}\cap U_{j})\to\dots$ since higher cohomology of coherent sheaves on
affinoids vanish. By definition,
$F(U_{i})={\mathcal{F}}({\mathfrak{U}}_{i})[\frac{1}{p}]$ so the Čech complex
on the generic fiber is obtained from that of the formal scheme by inverting
$p$ and we get the isomorphism $H^{i}(X,F)\simeq
H^{i}({\mathfrak{X}},{\mathcal{F}})$ which implies the statement by using
${\mathcal{F}}=\Omega^{j}_{{\mathfrak{X}}/{\mathcal{O}}_{K}}$. ∎

######  Lemma 4.2.

Let ${\mathfrak{Y}}$ be a smooth proper formal scheme over ${\mathcal{O}}_{K}$
with an action of a finite group $\Gamma$ of order prime to $p$. If the
special fiber ${\mathfrak{Y}}_{k}$ is projective over $k$ and the action of
$\Gamma$ on it is free there exists a quotient smooth proper formal scheme
${\mathfrak{X}}$ over ${\mathcal{O}}_{K}$ with projective special fiber such
that

$H^{i}({\mathfrak{X}},\Omega^{j}_{{\mathfrak{X}}/{\mathcal{O}}_{K}})\simeq
H^{i}({\mathfrak{Y}},\Omega^{j}_{{\mathfrak{Y}}/{\mathcal{O}}_{K}})^{\Gamma}$

###### Proof.

For each $n$ the quotient
${\mathfrak{X}}_{n}:={\mathfrak{Y}}_{{\mathcal{O}}_{K}/{\mathfrak{m}}^{n}}/\Gamma$
exists by [SGA71] Expose 1 Proposition V.1.8 because admissibility of the
action can be checked on the reduced subscheme which is projective in this
case. The closed immersions
${\mathfrak{Y}}_{{\mathcal{O}}_{K}/{\mathfrak{m}}^{n}}\to{\mathfrak{Y}}_{{\mathcal{O}}_{K}/{\mathfrak{m}}^{n+1}}$
induce morphisms ${\mathfrak{X}}_{n}\to{\mathfrak{X}}_{n+1}$. Since the action
is free, by Corollarie V.2.4 loc. cit. the morphisms
$\pi_{n}:{\mathfrak{Y}}_{n}\to{\mathfrak{X}}_{n}$ are etale with Galois group
$\Gamma$ and the canonical maps
${\mathcal{O}}_{{\mathfrak{X}}_{n}}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[\Gamma]\to\pi_{n*}{\mathcal{O}}_{Y_{n}}$
are isomorphisms.

Hence, the maps ${\mathfrak{X}}_{n}\to{\mathfrak{X}}_{n+1}$ are closed
immersions inducing isomorphisms
${\mathfrak{X}}_{n+1}\times_{{\mathcal{O}}_{K}/{\mathfrak{m}}^{n+1}}{\mathcal{O}}_{K}/{\mathfrak{m}}^{n}\simeq{\mathfrak{X}}_{n}$
(this also follows from order of $\Gamma$ being invertible on ${\mathfrak{Y}}$
but is not true in general for a non-free action) and the schemes
${\mathfrak{X}}_{n}$ form the desired quotient formal scheme ${\mathfrak{X}}$.
If $L$ is an ample line bundle on ${\mathfrak{Y}}_{k}$ then
$\bigotimes\limits_{\gamma\in\Gamma}\gamma^{*}(L)$ descends to an ample line
bundle on ${\mathfrak{X}}_{k}$ so ${\mathfrak{X}}_{k}$ is projective over $k$.

By the etaleness of the quotient map $\pi:{\mathfrak{Y}}\to{\mathfrak{X}}$ the
canonical morphisms
$\Omega^{i}_{{\mathfrak{X}}/{\mathcal{O}}_{K}}\to(\pi_{*}\Omega^{i}_{{\mathfrak{Y}}/{\mathcal{O}}_{K}})^{\Gamma}$
are isomorphisms and we get a Hochschild-Serre spectral sequence with the
second page
$E_{2}^{ab}=H^{a}(\Gamma,H^{b}({\mathfrak{Y}},\Omega^{i}_{{\mathfrak{Y}}/{\mathcal{O}}_{K}}))$
converging to
$H^{a+b}({\mathfrak{X}},\Omega^{i}_{{\mathfrak{X}}/{\mathcal{O}}_{K}})$. Since
higher cohomology of $\Gamma$ with coefficients in ${\mathcal{O}}_{K}$-modules
vanishes, we get the desired isomorphism. ∎

###### Proof of Theorem 4.1.

We first treat the case of $i+j=3$ and the general case follows from the Lemma
4.3 below. Let ${\mathfrak{Z}}$ be a formal abelian scheme with an action of
$G$ over the ring of integers ${\mathcal{O}}_{L}=W(k^{\prime})$ in some
unramified extension $L\supset{\mathbb{Q}}_{p}$ provided by Proposition 3.1.
We would like to construct ${\mathfrak{X}}$ by taking the quotient of
${\mathfrak{Z}}$ by $G$ but first we need to make the action free in a way
that does not spoil the discrepancy between ranks of invariant submodules. By
[Ray79] Proposition 4.2.3 or [BMS18] Section 2.2 there exists a smooth
projective complete intersection ${\mathfrak{Y}}$ of dimension $\geq 4$ in
projective space over ${\mathcal{O}}_{L}$ equipped with a free action of $G$.
We equip the product ${\mathfrak{Z}}\times_{{\mathcal{O}}_{L}}{\mathfrak{Y}}$
with the diagonal action of $G$ which is free because the action on the second
factor is free. Note that
$H^{i}({\mathfrak{Z}}\times{\mathfrak{Y}},{\mathcal{O}})\simeq
H^{i}({\mathfrak{Z}},{\mathcal{O}})$ and
$H^{0}({\mathfrak{Z}}\times{\mathfrak{Y}},\Omega^{i}_{{\mathfrak{Z}}\times{\mathfrak{Y}}/{\mathcal{O}}_{L}})\simeq
H^{0}({\mathfrak{Z}},\Omega^{i}_{{\mathfrak{Z}}/{\mathcal{O}}_{L}})$ for
$i\leq 3$ by Künneth formula and Lefschetz hyperplane theorem (Proposition 5.3
of [ABM19]) applied to ${\mathfrak{Y}}$ .

First, we construct the scheme with desired property over the unramified
extension ${\mathcal{O}}_{L}$ as the quotient
${\mathfrak{X}}^{\prime}:=({\mathfrak{Z}}\times{\mathfrak{Y}})/G$ provided by
the Lemma 4.2. We have
$H^{3}({\mathfrak{X}}^{\prime},{\mathcal{O}})=H^{3}({\mathfrak{Z}}\times{\mathfrak{Y}},{\mathcal{O}})^{G}=H^{3}({\mathfrak{Z}},{\mathcal{O}})^{G}$
and
$H^{0}({\mathfrak{X}}^{\prime},\Omega^{3}_{{\mathfrak{X}}^{\prime}/{\mathcal{O}}_{L}})^{G}=H^{0}({\mathfrak{Z}},\Omega^{3}_{{\mathfrak{Z}}/{\mathcal{O}}_{L}})^{G}$.
Proposition 3.1 supplied ${\mathfrak{Z}}$ such that the ranks of invariant
modules $H^{3}({\mathfrak{Z}},{\mathcal{O}})^{G}$ and
$H^{0}({\mathfrak{Z}},\Omega^{3}_{{\mathfrak{Z}}/{\mathcal{O}}_{L}})^{G}$ are
different, so $H^{3}({\mathfrak{X}}^{\prime},{\mathcal{O}})$ and
$H^{0}({\mathfrak{X}}^{\prime},\Omega_{{\mathfrak{X}}^{\prime}/{\mathcal{O}}_{L}}^{3})$
are indeed of different rank. By Proposition 2.1 we have
$3h^{3,0}({\mathfrak{X}}^{\prime})+h^{2,1}({\mathfrak{X}}^{\prime})=3h^{0,3}({\mathfrak{X}}^{\prime})+h^{1,2}({\mathfrak{X}}^{\prime})$
so the ranks of
$H^{2}({\mathfrak{X}}^{\prime},\Omega^{1}_{{\mathfrak{X}}^{\prime}/{\mathcal{O}}_{L}})$
and
$H^{1}({\mathfrak{X}}^{\prime},\Omega^{2}_{{\mathfrak{X}}^{\prime}/{\mathcal{O}}_{L}})$
are forced to be different as well.

Finally, the example over ${\mathbb{Z}}_{p}$ is obtained by Weil restriction
of scalars
${\mathfrak{X}}:=\operatorname{{Res}}^{{\mathcal{O}}_{L}}_{{\mathbb{Z}}_{p}}{\mathfrak{X}}^{\prime}$.
Since ${\mathcal{O}}_{L}$ is finite etale over ${\mathbb{Z}}_{p}$ this is a
smooth proper formal scheme over ${\mathbb{Z}}_{p}$ with projective special
fiber
${\mathfrak{X}}_{{\mathbb{F}}_{p}}=\operatorname{{Res}}^{k^{\prime}}_{{\mathbb{F}}_{p}}{\mathfrak{X}}_{k^{\prime}}$.
Since ${\mathfrak{X}}\times_{{\mathbb{Z}}_{p}}{\mathcal{O}}_{L}$ is isomorphic
to $({\mathfrak{X}}^{\prime})^{\times d}$ where $d$ is the degree of
$k^{\prime}$ over ${\mathbb{F}}_{p}$, the Hodge cohomology modules of
${\mathfrak{X}}$ are free and
$h^{3,0}({\mathfrak{X}})-h^{0,3}({\mathfrak{X}})=d(h^{3,0}({\mathfrak{X}}^{\prime})-h^{0,3}({\mathfrak{X}}^{\prime}))$
because the symmetry on cohomology of ${\mathfrak{X}}^{\prime}$ in degrees $1$
and $2$ holds by Corollary 2.2.

######  Lemma 4.3.

Let ${\mathfrak{X}}$ be a smooth proper formal scheme over ${\mathcal{O}}_{K}$
such that $h^{3,0}({\mathfrak{X}})\neq h^{0,3}({\mathfrak{X}})$. Then for any
$i,j\geq 0$ with $i\neq j$ and $i+j>3$ there exists a smooth projective scheme
${\mathfrak{Y}}$ over ${\mathcal{O}}_{K}$ such that
$h^{i,j}({\mathfrak{X}}\times{\mathfrak{Y}})\neq
h^{j,i}({\mathfrak{X}}\times{\mathfrak{Y}})$.

###### Proof.

Recall that for a formal scheme ${\mathfrak{T}}$ the number
$\delta^{i,j}({\mathfrak{T}})$ is defined as the difference
$h^{i,j}({\mathfrak{T}})-h^{j,i}({\mathfrak{T}})$. Note that if
${\mathfrak{Y}}$ satisfies Hodge symmetry then

(4.1)
$\delta^{i,j}({\mathfrak{T}}\times{\mathfrak{Y}})=\displaystyle\sum\limits_{\begin{subarray}{c}i_{1}+i_{2}=i\\\
j_{1}+j_{2}=j\end{subarray}}\delta^{i_{1},j_{1}}({\mathfrak{T}})h^{i_{2},j_{2}}({\mathfrak{Y}})$

By Proposition 2.1 we have
$\delta^{2,1}({\mathfrak{X}})=-3\delta^{3,0}({\mathfrak{X}})$ and
$\delta^{1,0}({\mathfrak{X}})=\delta^{2,0}({\mathfrak{X}})=0$. Let us assume
that $i>j$. For a scheme ${\mathfrak{Y}}$ denote by
$H_{{\mathfrak{Y}}}(x,y):=\sum\limits_{i,j}h^{i,j}({\mathfrak{Y}})x^{i}y^{j}$
its Hodge polynomial. If ${\mathfrak{Y}}_{1}\subset{\mathfrak{Y}}_{2}$ is a
smooth closed subscheme of codimension $r+1$ in a smooth proper scheme then
the Hodge polynomial of the blow-up is given by

$H_{Bl_{{\mathfrak{Y}}_{1}}{\mathfrak{Y}}_{2}}(x,y)=H_{{\mathfrak{Y}}_{2}}(x,y)+H_{{\mathfrak{Y}}_{1}}(x,y)\cdot(xy+(xy)^{2}+\dots+(xy)^{r})$

Let ${\mathfrak{T}}_{d,n}$ be a smooth degree $d$ hypersurface in
${\mathbb{P}}^{n+1}_{{\mathcal{O}}_{K}}$ where $d$ and $n$ are numbers that we
will choose later. Construct a sequence of smooth projective schemes starting
with ${\mathfrak{Y}}_{0}={\mathfrak{T}}_{d,n}$ and for $s\geq 0$ defining the
scheme ${\mathfrak{Y}}_{s+1}$ as the blow-up of some projective space
${\mathbb{P}}^{N_{s}}_{{\mathcal{O}}_{K}}$ along some embedding
${\mathfrak{Y}}_{s}\subset{\mathbb{P}}^{N_{s}}_{{\mathcal{O}}_{K}}$ of
codimension $\geq 2$ (the numbers
$N_{s}>\dim_{{\mathcal{O}}_{K}}{\mathfrak{Y}}_{s}+1$ can be chosen
arbitrarily). By the above formula we have
$H_{{\mathfrak{Y}}_{s}}(x,y)=F(xy)+H_{{\mathfrak{T}}_{d,n}}(x,y)\cdot(xy)^{s}\cdot
G(xy)$ where $F(t),G(t)\in 1+t{\mathbb{Z}}[t]$ are some polynomials.

Assume first that $i>j+3$. Then take $n=i-3-j,s=j$ and consider
${\mathfrak{Y}}:={\mathfrak{Y}}_{s}$ obtained by the above inductive procedure
from ${\mathfrak{T}}_{d,n}$. We then have
$h^{i-2,j-1}({\mathfrak{Y}})=h^{i-1,j-2}({\mathfrak{Y}})=h^{i,j-3}({\mathfrak{Y}})=0$
and $h^{i-3,j}({\mathfrak{Y}})=h^{n,0}({\mathfrak{T}}_{d,n})$. This turns the
formula (4.1) into

$\delta^{i,j}({\mathfrak{X}}\times{\mathfrak{Y}})=\delta^{3,0}({\mathfrak{X}})\cdot
h^{i-3,j}({\mathfrak{Y}})+\sum\limits_{i_{2}+j_{2}<i+j-3}\delta^{i-i_{2},j-j_{2}}({\mathfrak{X}})\cdot
h^{i_{2},j_{2}}({\mathfrak{Y}})$

By [SGA73] XI.Theoreme 1.5 the Hodge numbers $h^{a,b}({\mathfrak{T}}_{d,n})$
of a hypersurface are zero unless $a+b=n$ or $a=b$ and
$h^{a,a}({\mathfrak{T}}_{d,n})=1$ for $0\leq a\leq n,2a\neq n$. It follows
that the Hodge polynomial $H_{{\mathfrak{Y}}}(x,y)$ is congruent modulo the
ideal $(x,y)^{n+s}$ to a polynomial of the form $K(xy)$ with coefficients not
depending on the degree $d$ of the hypersurface. In particular,
$\delta^{i,j}({\mathfrak{X}}\times{\mathfrak{Y}})$ is the sum of
$\delta^{3,0}({\mathfrak{X}})\cdot
h^{i-3,j}({\mathfrak{Y}})=\delta^{3,0}({\mathfrak{X}})\cdot
h^{n,0}({\mathfrak{T}}_{d,n})$ and a number that does not depend on $d$. By
loc. cit. Corollaire 2.4 the number is $h^{n,0}({\mathfrak{T}}_{d,n})$ is
equal to $\binom{d-1}{n+1}$ so for large enough $d$ this sum will be non-zero.

If $i=j+2$ then take ${\mathfrak{Y}}={\mathfrak{Y}}_{s}$ with $n=1,s=j-1$. An
analogous computation gives that
$\delta^{i,j}({\mathfrak{X}}\times{\mathfrak{Y}})$ is the sum of a number that
does not depend on $d$ and the expression $\delta^{2,1}({\mathfrak{X}})\cdot
h^{1,0}({\mathfrak{T}}_{d,1})+\delta^{3,0}({\mathfrak{X}})\cdot
h^{0,1}({\mathfrak{T}}_{d,1})=\delta^{3,0}({\mathfrak{X}})(h^{0,1}({\mathfrak{T}}_{d,1})-3h^{1,0}({\mathfrak{T}}_{d,1}))=-2\delta^{3,0}({\mathfrak{X}})\binom{d-1}{2}$
which is non-zero for $d>2$.

Finally, if $i=j+3$ or $i=j+1$ we will be able to move asymmetry into higher
cohomology just via Tate twists. Consider the scheme
${\mathfrak{Y}}=({\mathbb{P}}^{1}_{{\mathcal{O}}_{K}})^{d}$ where we will
again choose $d$ at the end. We have

$\delta^{i,j}({\mathfrak{X}}\times{\mathfrak{Y}})=\sum\limits_{r\leq\frac{i+j-3}{2}}\delta^{i-r,j-r}({\mathfrak{X}})\cdot\binom{d}{r}$

This is a polynomial in $d$ with leading coefficient
$\delta^{i-r,j-r}({\mathfrak{X}})$ with $r=\frac{i+j-3}{2}$ which is nothing
but $\pm\delta^{3,0}({\mathfrak{X}})$ or $\pm\delta^{2,1}({\mathfrak{X}})$ so
this expression is non-zero for a large enough value of $d$. ∎

This finishes the proof of Theorem 4.1. ∎

######  Corollary 4.4.

For every pair of positive numbers $i\neq j$ with $i+j\geq 3$ there exists a
smooth proper rigid-analytic variety $X$ over ${\mathbb{Q}}_{p}$ admitting a
smooth formal model ${\mathfrak{X}}$ over
$\operatorname{{Spf}}{\mathbb{Z}}_{p}$ with projective special fiber
${\mathfrak{X}}_{{\mathbb{F}}_{p}}$ such that

$h^{i,j}(X)\neq h^{j,i}(X).$

###### Proof.

Take $X$ to be the generic fiber of a formal scheme ${\mathfrak{X}}$ provided
by Theorem 4.1. Applying Lemma 4.1 gives the result. ∎

##  5\. Embellishing the main example and a question

We end by noting that the examples provided by Theorem 4.1 can be modified to
avoid some of the characteristic $p$ “pathologies”.

In all of the examples constructed above the Hodge cohomology groups
$H^{i}({\mathfrak{X}},\Omega^{j}_{{\mathfrak{X}}/{\mathbb{Z}}_{p}})$ are free
modules over ${\mathbb{Z}}_{p}$ so the Hodge numbers of the special and
generic fibers of ${\mathfrak{X}}$ coincide. In particular, the Hodge symmetry
for the special fiber ${\mathfrak{X}}_{{\mathbb{F}}_{p}}$ fails as well. The
next lemma shows that symmetry for the generic fiber cannot be salvaged by
requiring that for the special fiber: there are enough projective schemes
failing symmetry for the special fiber to cancel out the asymmetry on the
special fiber by taking products. We only treat here one particular pair of
degrees.

######  Lemma 5.1.

There exists a smooth proper formal scheme ${\mathfrak{X}}^{\prime}$ over
$\operatorname{{Spf}}{\mathbb{Z}}_{p}$ with projective special fiber such that
$h^{3,0}({\mathfrak{X}}^{\prime})\neq h^{0,3}({\mathfrak{X}}^{\prime})$ while
$h^{3,0}({\mathfrak{X}}^{\prime}_{{\mathbb{F}}_{p}})=h^{0,3}({\mathfrak{X}}^{\prime}_{{\mathbb{F}}_{p}})$.

###### Proof.

The proof of Theorem 4.1 provides us with a smooth proper scheme
${\mathfrak{X}}$ over ${\mathbb{Z}}_{p}$ with projective special fiber such
that all Hodge groups
$H^{i}({\mathfrak{X}},\Omega^{j}_{{\mathfrak{X}}/{\mathbb{Z}}_{p}})$ are free
over ${\mathbb{Z}}_{p}$ and $h^{3,0}({\mathfrak{X}})<h^{0,3}({\mathfrak{X}})$
while $h^{1,0}({\mathfrak{X}})=h^{0,1}({\mathfrak{X}})=0$ (the Hodge number in
degree $1$ vanish because representations $U,V$ used in proof of Proposition
3.1 have trivial invariants). Since Hodge cohomology modules are free, the
Hodge numbers of the special fiber ${\mathfrak{X}}_{{\mathbb{F}}_{p}}$ are
equal to those of ${\mathfrak{X}}$.

We will now take the product of ${\mathfrak{X}}$ with an auxiliary projective
scheme that will be constructed by approximating the stack
$B(\mu_{p}\times{\mathbb{Z}}/p)$. By Theorem 1.2 of [ABM19] applied over
${\mathbb{Z}}_{p}$ to the group scheme $\mu_{p}\times{\mathbb{Z}}/p$ with
$d=3$ there exists a smooth projective scheme ${\mathfrak{Y}}^{\prime}$ such
that
$H^{j}({\mathfrak{Y}}^{\prime}_{{\mathbb{F}}_{p}},\Omega^{i}_{{\mathfrak{Y}}^{\prime}_{{\mathbb{F}}_{p}}})=H^{j}(B(\mu_{p}\times{\mathbb{Z}}/p)_{{\mathbb{F}}_{p}},\Omega^{i})$
and
$H^{j}({\mathfrak{Y}}^{\prime},\Omega^{i}_{{\mathfrak{Y}}^{\prime}/{\mathbb{Z}}_{p}})=H^{j}(B(\mu_{p}\times{\mathbb{Z}}/p)_{{\mathbb{Z}}_{p}},\Omega^{i})$
for pairs $i,j$ with $i+j\geq 3$ and $i=0$ or $j=0$. The Hodge polynomial
$H_{B(\mu_{p}\times{\mathbb{Z}}/p)_{{\mathbb{F}}_{p}}}(x,y):=\sum\limits_{i,j}h^{i,j}(B(\mu_{p}\times{\mathbb{Z}}/p)_{{\mathbb{F}}_{p}})x^{i}y^{j}$
of this stack is given by the product
$H_{B\mu_{p,{\mathbb{F}}_{p}}}(x,y)H_{B{\mathbb{Z}}/p_{{\mathbb{F}}_{p}}}(x,y)$.
By [Tot18] Proposition 11.1 the first multiple $H_{B\mu_{p,{\mathbb{F}}_{p}}}$
is equal to $\frac{1+x}{1-xy}$ and $H_{B({\mathbb{Z}}/p)_{{\mathbb{F}}_{p}}}$
is equal to $\frac{1}{1-y}$ because the only non-zero Hodge cohomology groups
of the classifying stack of a finite discrete group $\Gamma$ are given by
group cohomology $H^{i}(B\Gamma_{{\mathbb{F}}_{p}},{\mathcal{O}})\simeq
H^{i}(\Gamma,{\mathbb{F}}_{p})$. Take
${\mathfrak{Y}}:={\mathfrak{Y}}^{\prime}\times{\mathfrak{E}}^{l}$ where
${\mathfrak{E}}$ is an elliptic curve over ${\mathbb{Z}}_{p}$ and $l$ is a
number that we will choose at the end.

By the above computation, $H_{{\mathfrak{Y}}_{{\mathbb{F}}_{p}}}(x,y)$ is
given by $\frac{1}{1-y}\cdot\frac{1+x}{1-xy}\cdot(1+x+y+xy)^{l}$ modulo an
element of the ideal $(x^{4},xy,y^{4})$. For the purpose of computing
$h^{3,0}(({\mathfrak{X}}\times{\mathfrak{Y}})_{{\mathbb{F}}_{p}})$ and
$h^{0,3}(({\mathfrak{X}}\times{\mathfrak{Y}})_{{\mathbb{F}}_{p}})$ we only
care about this polynomial modulo $xy$ and we have
$H_{({\mathfrak{X}}\times{\mathfrak{Y}})_{{\mathbb{F}}_{p}}}(x,y)\equiv(1+h^{2,0}({\mathfrak{X}}_{{\mathbb{F}}_{p}})x^{2}+h^{3,0}({\mathfrak{X}}_{{\mathbb{F}}_{p}})x^{3}+h^{0,2}({\mathfrak{X}}_{{\mathbb{F}}_{p}})y^{2}+h^{0,3}({\mathfrak{X}}_{{\mathbb{F}}_{p}})y^{3})(1+(l+1)x+(\binom{l}{2}+l)x^{2}+(\binom{l}{3}+\binom{l}{2})x^{3}+(l+1)y+(\binom{l}{2}+l+1)y^{2}+(\binom{l}{3}+\binom{l}{2}+l+1)y^{3})$
modulo the ideal $(x^{4},xy,y^{4})$. The Hodge numbers of the special fiber
${\mathfrak{X}}_{{\mathbb{F}}_{p}}\times{\mathfrak{Y}}_{{\mathbb{F}}_{p}}$ are
thus given by
$h^{3,0}(({\mathfrak{X}}\times{\mathfrak{Y}})_{{\mathbb{F}}_{p}})=\binom{l}{3}+\binom{l}{2}+h^{3,0}({\mathfrak{X}}_{{\mathbb{F}}_{p}})+(l+1)h^{2,0}({\mathfrak{X}}_{{\mathbb{F}}_{p}})$,
$h^{0,3}(({\mathfrak{X}}\times{\mathfrak{Y}})_{{\mathbb{F}}_{p}})=\binom{l}{3}+\binom{l}{2}+l+1+h^{0,3}({\mathfrak{X}}_{{\mathbb{F}}_{p}})+(l+1)h^{0,2}({\mathfrak{X}}_{{\mathbb{F}}_{p}})$.
Hence,
$\delta^{3,0}(({\mathfrak{X}}\times{\mathfrak{Y}})_{{\mathbb{F}}_{p}})=\delta^{3,0}({\mathfrak{X}}_{{\mathbb{F}}_{p}})+l+1$.
Since $\delta^{3,0}({\mathfrak{X}}_{{\mathbb{F}}_{p}})<0$ there exists $l$
that makes this difference equal to zero.

The product ${\mathfrak{X}}\times{\mathfrak{Y}}$ is our desired scheme
${\mathfrak{X}}^{\prime}$. Note that the Hodge numbers
$h^{i,0}({\mathfrak{Y}}_{{\mathbb{Q}}_{p}})$ of the generic fiber
${\mathfrak{Y}}_{{\mathbb{Q}}_{p}}$ vanish for $i\leq 3$ so the asymmetry
$h^{3,0}({\mathfrak{X}}^{\prime})=h^{3,0}({\mathfrak{X}})\neq
h^{0,3}({\mathfrak{X}})=h^{0,3}({\mathfrak{X}}^{\prime})$ persists. ∎

Another feature of the counterexamples to Hodge symmetry obtained by taking
quotients of formal abelian schemes by finite groups is the absence on the
special fiber of an ample line bundle of degree prime to $p$. Indeed, if the
$d$-dimensional special fiber of ${\mathfrak{X}}$ (in the notation of the
proof of Theorem 4.1) admits a line bundle $L$ such that $L^{d}\not\equiv
0\pmod{p}$ then it induces a $G$-equivariant ample line bundle $L^{\prime}$ on
${\mathfrak{Z}}_{k^{\prime}}$ with
$(L^{\prime})^{\dim{\mathfrak{Z}}_{k^{\prime}}}\not\equiv 0\pmod{p}$ as well
(because the order of $G$ is prime to $p$). It induces a separable
$G$-equivariant polarization and hence and isomorphism
$H^{0}({\mathfrak{Z}},\Omega^{1}_{{\mathfrak{Z}}_{k}^{\prime}/k^{\prime}})\simeq
H^{1}({\mathfrak{Z}}_{k^{\prime}},{\mathcal{O}})^{\vee}$ of
$G$-representations. Hence, for all $i$ and $j$ the representations
$H^{i}({\mathfrak{Z}}_{k^{\prime}},\Omega^{j}_{{\mathfrak{Z}}_{k^{\prime}}/k^{\prime}})$
and
$H^{j}({\mathfrak{Z}}_{k^{\prime}},\Omega^{i}_{{\mathfrak{Z}}_{k^{\prime}}/{k^{\prime}}})$
would be dual to each other and Hodge symmetry for
${\mathfrak{X}}_{k^{\prime}}$ and, hence, for ${\mathfrak{X}}$ would have to
hold.

We note, however, that blowing up a point provides the special fiber with a
prime to $p$ polarization without spoiling Hodge asymmetry

######  Lemma 5.2.

For every pair of distinct positive integers $i,j$ with $i+j\geq 3$ there
exists a smooth proper formal scheme ${\mathfrak{X}}^{\prime}$ over
${\mathbb{Z}}_{p}$ such that $h^{i,j}({\mathfrak{X}}^{\prime})\neq
h^{j,i}({\mathfrak{X}}^{\prime})$ and the special fiber carries an ample
divisor $L$ such that its top self-intersection is prime to $p$:
$L^{\dim({\mathfrak{X}}^{\prime}_{{\mathbb{F}}_{p}})}\not\equiv 0\pmod{p}$.

###### Proof.

Let ${\mathfrak{X}}$ be a formal scheme provided by Theorem 4.1. Choose a
point $x:\operatorname{{Spf}}{\mathbb{Z}}_{p}\to{\mathfrak{X}}$ (one sees
easily that the proof of Theorem 4.1 can be modified to make sure that at
least one point exists). Let $f:{\mathfrak{X}}^{\prime}\to{\mathfrak{X}}$ be
the blow-up of ${\mathfrak{X}}$ at the closed subscheme given by $x$. Note
that $h^{i,j}({\mathfrak{X}}^{\prime})=h^{i,j}({\mathfrak{X}})$ for $i\neq j$.

The special fiber ${\mathfrak{X}}^{\prime}_{{\mathbb{F}}_{p}}$ is the blow-up
of ${\mathfrak{X}}_{{\mathbb{F}}_{p}}$ at a point, denote by
$E\subset{\mathfrak{X}}^{\prime}_{{\mathbb{F}}_{p}}$ the exceptional divisor.
Let $H$ be any ample divisor on ${\mathfrak{X}}^{\prime}_{{\mathbb{F}}_{p}}$.
We will find a linear combination $H+nE$ with $n\geq 0$ such that the top
self-intersection $(H+nE)^{d}$ is prime to $p$. Since the sum of an ample
divisor with an effective divisor is ample, this will provide an ample divisor
with the desired property.

Suppose that $H|_{E}$ has degree $k$ in
$\operatorname{{Pic}}(E)={\mathbb{Z}}$. Then $H^{d-i}\cdot
E^{i}=k^{d-i}\cdot(-1)^{i}$ for $i>0$ by Example 8.3.9. in [Ful98] and the
projection formula, so

(5.1)
$(H+nE)^{d}=H^{d}+\sum\limits_{i=1}^{d}\binom{d}{i}n^{i}k^{d-i}(-1)^{i-1}=H^{d}-(k-n)^{d}+k^{d}$

For varying $n$ the residue of $(k-n)^{d}$ modulo $p$ takes at least two
different values, so we can find $n$ such that the expression (5.1) is prime
to $p$. ∎

Apart from satisfying Hodge symmetry, compact Kähler manifolds must have
nonzero middle Hodge numbers $h^{i,i}(X)\neq 0$ for $i\leq\dim X$ because
$H^{i}(X,\Omega^{i}_{X/{\mathbb{C}}})$ contains the nonzero $i$-th power of
the class of a Kähler form. It would be interesting to find out whether this
is the case for rigid-analytic varieties with projective reduction:

######  Question 1.

Does there exist a smooth proper rigid-analytic variety $X$ of $\dim X>0$
admitting a formal model with projective special fiber such that
$H^{1}(X,\Omega^{1}_{X/K})=0$?

Note that $X$ must have non-zero second de Rham cohomology
$H^{2}_{\mathrm{dR}}(X/K)$. If $X$ is algebraic projective, then a non-zero
class is given by the first Chern class of an ample line bundle. If, on the
contrary, $X$ is not projective, then no ample line bundle on the special
fiber ${\mathfrak{X}}_{k}$ can lift to ${\mathfrak{X}}$. The obstruction to
lifting a line bundle gives a non-zero class in second cohomology as follows.
Consider the long exact sequence

$\dots\to\operatorname{{Pic}}({\mathfrak{X}})\to\operatorname{{Pic}}({\mathfrak{X}}_{k})\xrightarrow{\delta}H^{2}({\mathfrak{X}},(1+{\mathfrak{m}}{\mathcal{O}}_{{\mathfrak{X}}})^{\times})\to\dots$

obtained from the short exact sequence
$1\to(1+{\mathfrak{m}}{\mathcal{O}}_{{\mathfrak{X}}})^{\times}\to{\mathcal{O}}_{{\mathfrak{X}}}^{\times}\to{\mathcal{O}}_{{\mathfrak{X}}_{k}}^{\times}\to
1$ of sheaves of abelian groups on the Zariski site of ${\mathfrak{X}}$. For
any ample line bundle $L\in\operatorname{{Pic}}({\mathfrak{X}}_{k})$ the image
$\delta(L)\in
H^{2}({\mathfrak{X}},(1+{\mathfrak{m}}{\mathcal{O}}_{{\mathfrak{X}}})^{\times})$
is non-zero and, since no power of $L$ lifts to ${\mathfrak{X}}$, is a non-
torsion element. Applying to $\delta(L)$ the isomorphism given by the
logarithm
$\log:(1+{\mathfrak{m}}{\mathcal{O}}_{{\mathfrak{X}}})^{\times}[\frac{1}{p}]\simeq{\mathcal{O}}_{{\mathfrak{X}}}[\frac{1}{p}]$
gives a non-zero class in
$H^{2}({\mathfrak{X}},{\mathcal{O}}_{{\mathfrak{X}}}[\frac{1}{p}])=H^{2}(X,{\mathcal{O}})$.

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