# Arithmetic Deformation of Line Bundles

David Urbanik and Ziquan Yang

###### Abstract

We introduce a new method to study mixed characteristic deformation of line
bundles. In particular, for sufficiently large smooth projective families
$f:\mathcal{X}\to\mathcal{S}$ defined over the ring of $N$-integers
$\mathcal{O}_{L}[1/N]$ of a number field $L$, we produce a proper closed
subscheme $\mathcal{E}\subsetneq\mathcal{S}$ outside of which all line bundles
appearing in positive characteristic fibres of $f$ admit characteristic zero
lifts. This in particular applies to elliptic surfaces over $\mathbb{P}^{1}$
and projective hypersurfaces in $\mathbb{P}^{3}$ of degree $d\geq 5$. We also
study the locus in $\mathcal{E}$ in more detail in the $h^{0,2}=2$ case.

###### Contents

  1. 1 Introduction
    1. 1.1 Main Results
    2. 1.2 Applications
    3. 1.3 Methods
    4. 1.4 Notations and Conventions
  2. 2 Deformation Theory
    1. 2.1 Generalities on Picard Schemes
    2. 2.2 Local Deformation Spaces
  3. 3 Jets and Period Maps
    1. 3.1 Jet Spaces
    2. 3.2 Period Jets
  4. 4 Proofs of Theorems
    1. 4.1 General Setup
    2. 4.2 Construction of the Atypical Locus
    3. 4.3 The Power-primitive Case
    4. 4.4 Proofs of Main Theorems
  5. 5 The $h^{0,2}=2$ Case
    1. 5.1 Theorem of the Fixed Part in a $p$-adic Setting
    2. 5.2 Non-liftable Line Bundles
  6. 6 Applications and Examples
    1. 6.1 Elliptic Surfaces
    2. 6.2 Complete Intersection Surfaces
    3. 6.3 Non-examples
  7. A A Lemma on Flat Frames

##  1 Introduction

### 1.1 Main Results

In [Del81], Deligne proved that every line bundle on a K3 surface in positive
characteristic lifts to characteristic $0$ along with the surface. Deligne’s
theorem and its extensions have historically played a pivotal role in
addressing the Tate conjecture for K3 surfaces and varieties alike in positive
characteristic, because a mainstream method relies on considering
characteristic $0$ liftings and connecting the Tate conjecture to its Hodge-
theoretic analogue, i.e., the Lefschetz $(1,1)$-theorem (e.g., [Nyg83, NO85,
Mad15, HYZ23]). To explore the potential for future advancements in this
direction, we pose the following question: In an arithmetic family, can we
generate all line bundles on the characteristic $p$ fibers by specializing
line bundles on the characteristic $0$ fibers?

To set up a context for the question, let $L\subseteq\mathbb{C}$ be a number
field and $\mathcal{O}_{L}$ be its ring of integers. Suppose that
$\mathcal{S}$ is a smooth quasi-projective $\mathcal{O}_{L}[1/N]$-scheme for
some $N\in\mathbb{N}_{>0}$ with $\mathcal{S}_{\mathbb{C}}$ connected, and
$f:\mathcal{X}\to\mathcal{S}$ is a polarized smooth projective family with
geometrically connected fibers. Then we provide the following answer to
address the above question.

######  Theorem (1.1.1).

Suppose that $f_{\mathbb{C}}$ has large monodromy and sufficiently large
period image. Then there exists a proper closed subscheme
$\mathcal{E}\subsetneq\mathcal{S}$ such that for every algebraically closed
field $k$ of positive characteristic and
$s\in\mathcal{S}(k)\setminus\mathcal{E}(k)$, every line bundle $\xi$ on the
fibre $\mathcal{X}_{s}$ lifts to characteristic $0$.

By “$\xi$ lifts to characteristic $0$”, we mean that there exists a finite
extension $K$ of $W(k)[1/p]$ and a point
$\widetilde{s}\in\mathcal{S}(\mathcal{O}_{K})$ lifting $s$, such that $\xi$
lies in the image of the specialization map
$\mathrm{Pic}(\mathcal{X}_{\widetilde{s}})\to\mathrm{Pic}(\mathcal{X}_{s})$.
We now define the conditions in the hypothesis. Their necessity will be
discussed in §6.3.

######  Definition (1.1.2).

Let $S$ be a connected $\mathbb{C}$-variety and $X\to S$ be a smooth
projective morphism with connected fibers.

  * •

We say that $X\to S$ has _large monodromy_ if for some $\mathbb{C}$-point
$s\in S$ not lying on the Noether-Lefschetz locus, the Zariski closure of the
image of the monodromy representation

$\pi_{1}(S,s)\to\mathrm{O}(\mathrm{H}^{2}(X_{s},\mathbb{Q}))$

contains $\mathrm{SO}(\mathrm{T}^{2}(X_{s},\mathbb{Q}))$ as the identity
component. Here we are equipping $\mathrm{H}^{2}(X_{s},\mathbb{Q})$ with the
symmetric bilinear form induced by the chosen polarization, and taking
$\mathrm{T}^{2}(X_{s},\mathbb{Q})$ to be the orthogonal component of the
subspace of $(1,1)$-classes.

  * •

We say that $X\to S$ has _sufficiently large period image_ if for some (and
hence a general) $\mathbb{C}$-point $s\in S$, the Kodaira-Spencer map

$T_{s}S\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega^{1}_{X_{s}}),\mathrm{H}^{2}(\mathcal{O}_{X_{s}}))$
(1)

has image of dimension at least
$h^{0,2}=\dim_{\mathbb{C}}\mathrm{H}^{2}(\mathcal{O}_{X_{s}})$.

The proof of Thm (1.1.1) yields structural theorems for the arithmetic
Noether-Lefschetz (NL) locus, and the relative Picard scheme
$\mathscr{P}_{\mathcal{X}/\mathcal{S}}$. We say that a proper closed reduced
subscheme $Z\subseteq\mathcal{S}$ is a component of the arithmetic NL locus if
it is the scheme-theoretic image of an irreducible component of the reduced
subscheme of $\mathscr{P}_{\mathcal{X}/\mathcal{S}}$ which does not dominant
$\mathcal{S}$. Then the arithmetic NL locus, which we shall denote by
$\mathcal{N}$, is the collection of all such (typically infinitely many)
components. It is characterized by the property that a geometric point
$s\to\mathcal{S}$ factors through $\mathcal{N}$ if and only if
$\mathrm{rank\,}\mathrm{NS}(\mathcal{X}_{s})>\mathrm{rank\,}\mathrm{NS}(\mathcal{X}_{\bar{\eta}})$,
where $\bar{\eta}$ is a geometric generic point of $\mathcal{S}$. Then we have
the following theorem.

######  Theorem (1.1.3).

In the situation of Thm (1.1.1), the restriction of
$\mathscr{P}_{\mathcal{X}/\mathcal{S}}$ over
$\mathcal{S}\backslash\mathcal{E}$ is syntomic (i.e., flat and lci) over
$\mathcal{O}_{L}[1/N]$, and the restriction of every component of
$\mathcal{N}$ over $\mathcal{S}\backslash\mathcal{E}$ is flat over
$\mathcal{O}_{L}[1/N]$ and has codimension exactly
$h^{0,2}(\mathcal{X}_{\bar{\eta}})$ in $\mathcal{S}\backslash\mathcal{E}$.

In particular, the above theorem implies that when restricted to
$\mathcal{S}\backslash\mathcal{E}$ the arithmetic NL locus $\mathcal{N}$ can
be recovered from the variation of Hodge structures (VHS) on
$R^{2}f_{\mathbb{C}*}\mathbb{Q}$, as every component of $\mathcal{N}$ is the
Zariski closure of some component of the usual (Hodge-theoretic) NL locus over
$\mathcal{S}(\mathbb{C})$. We also remark that our method will give an
explicit construction of $\mathcal{E}$: Away from small primes, $\mathcal{E}$
is the Zariski closure of a so-called “atypical Hodge locus” on
$\mathcal{S}_{\mathbb{C}}$ ([BKU22]) and the locus where the Kodaira-Spencer
map has rank $<h^{0,2}$. The issue of determining $\mathcal{E}$ effectively is
discussed at the end of §1.3.

To illustrate what might happen on $\mathcal{E}$, we give a thorough analysis
in the case when $h^{0,2}=2$. See §5.2. One conclusion we can draw is the
following (cf. (5.2.4)):

######  Theorem (1.1.4).

In the situation of (1.1.1), suppose in addition for every
$s\in\mathcal{S}(\mathbb{C})$, $h^{0,2}(\mathcal{X}_{s})=2$ and the Kodaira-
Spencer map (1) has rank $\geq 2$.

Then for $p\gg 0$, the following holds: Let $k$ be an algebraically closed
field of characteristic $p$ and take any $s\in S(k)$ and
$\xi\in\mathrm{Pic}(\mathcal{X}_{s})$. Up to equi-characteristic deformations
of the pair $(s,\xi)$, $\xi^{m}$ is a sum of liftable line bundles for some
$m\in\mathbb{N}_{>0}$.

The upshot of the above conclusion is that, when $p\gg 0$ and $s$ lies on
$\mathcal{E}$ even if $\xi$ is non-liftable, it still comes from specializing
and deforming line bundles in characteristic $0$, albeit in a more indirect
way. However, in general we expect that the complexity of possible behaviours
that occur over the locus $\mathcal{E}$ to grow quickly as $h^{0,2}$
increases.

### 1.2 Applications

We will give applications of our results to general type and elliptic
surfaces. Let $k$ be an algebraically closed field of characteristic $p$.

######  Theorem (1.2.1).

Let $d\geq 5$ be a number. If $p\gg 0$, then for a general degree $d$
hypersurface $Y$ in $\mathbb{P}^{3}_{k}$, every $\xi\in\mathrm{Pic}(Y)$ is
liftable.

Here the word “general” means that $Y$ comes from a Zariski open dense
subscheme of a natural parameter space. Note that the above surfaces are of
general type.

Let $\pi:X\to\mathbb{P}^{1}$ be an elliptic surface (with a zero section) of
height $h:=\mathrm{deg\,}(R^{1}\pi_{*}\mathcal{O}_{X})^{\vee}$. We say that
$X$ is admissible if all geometric fibers of $\pi$ are irreducible. This
condition is generically satisfied by elliptic surfaces over $\mathbb{P}^{1}$
with a fixed $h$.

######  Theorem (1.2.2).

If $p\gg 0$, then for a general admissible $X$ over $k$, every
$\xi\in\mathrm{Pic}(X)$ is liftable.

When $h=3$, (1.1.4) allows us to remove the word “general” at the expense of a
weaker conclusion:

######  Theorem (1.2.3).

If $p\gg 0$, $h=3$ and $X$ is admissible, then for every
$\xi\in\mathrm{Pic}(X)$, we may equi-characteristically deform $(X,\xi)$ to
another pair $(X^{\prime},\xi^{\prime})$, such that some positive power of
$\xi^{\prime}$ is a sum of liftable line bundles.

The precise forms of the above theorems are given in §6.

Finally we remark that elliptic surfaces are of particular interest because
their line bundles give rise to rational points on their generic fibers, which
are elliptic curves over function fields. Hence lifting results in the style
of (1.2.2) suggest a way of constructing all rational points on an elliptic
curve over a global function field by eventually appealing to the Lefschetz
$(1,1)$-theorem. This idea was indeed used before, in an earlier work of the
second author [HYZ23] joint with Hamacher and Zhao, to establish a new case of
the function field BSD conjecture. It is our hope that (1.2.2) is a start for
future progress.

### 1.3 Methods

The key new idea in our work is that one can apply the theory of Hodge-
theoretic unlikely intersections over $\mathbb{C}$ to understand the
liftability of line bundles in positive characteristic. This is noteworthy
because the former is highly transcendental whereas the latter is completely
algebraic. To explain how this works, let us first give an informal overview
of how line bundles deform, starting with the $\mathbb{C}$-fibre
$\mathcal{S}_{\mathbb{C}}$. If $Z\subset\mathcal{S}_{\mathbb{C}}$ is a
component of the Noether-Lefschetz locus coming from deforming a line bundle
$\xi$ on $\mathcal{X}_{s}$ for some $s\in\mathcal{S}(\mathbb{C})$, then
analytically locally at $s$ one can view $Z$ as the set of points $t$ where
parallel transport of the Betti class $c_{1}(\xi)$ to $t$ remains in
$\mathrm{Fil}^{1}\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}_{t}/\mathbb{C})$,
the middle piece of the Hodge filtration. Then it is not hard to see that $Z$
is locally cut out at $s$ by $h^{0,2}(\mathcal{X}_{s})$ equations.

Now, by generalizing a result of Deligne [Del81, Prop. 1.5], we explain in
(2.2.9) that under some mild assumptions the same statement also applies on
the integral level. Suppose that $k$ is an algebraically closed field of
characteristic $p>0$, $s$ is a $k$-point on $\mathcal{S}$ and
$\xi\in\mathrm{Pic}(\mathcal{X}_{s})$, and assume for simplicity that
$\mathcal{O}_{L}[1/N]$ is unramified over $\mathbb{Z}[1/N]$. Then there is
formal subscheme $\mathrm{Def}(\xi)$ inside the formal completion of
$\mathcal{S}{\otimes}_{\mathcal{O}_{L}[1/N]}W(k)$ at $s$ such that
$\mathrm{Def}(\xi)$ is the maximal locus to which $\xi$ deforms and is cut out
by $h^{0,2}(\mathcal{X}_{s})$ equations. The analogue of the locus $Z$ in the
preceding paragraph is the Zariski closure of $\mathrm{Def}(\xi)$ in
$\mathcal{S}$, which we shall denote by $\mathcal{Z}$. But — and this where
the connection to unlikely intersections appears — passing to the integral
level gains us a dimension. More specifically, if $\xi$ does not deform to
characteristic zero, which is to say that $\mathrm{Def}(\xi)$ has no
characteristic zero fiber, then the codimension of $\mathcal{Z}_{k}$ inside
$\mathcal{S}_{k}$ must be _at most_ $h^{0,2}(\mathcal{X}_{s})-1$.

Let us first explain why it would be unusual to see such a phenomenon in
characteristic zero. Indeed, if we consider again the locus
$Z\subset\mathcal{S}_{\mathbb{C}}$ in the first paragraph, then we can
alternatively define the germ of $Z$ at $s$ in the following way: consider a
Hodge-theoretic period map $\psi:B\to\widecheck{D}$ on a small ball
$B\subset\mathcal{S}(\mathbb{C})$ around $s$, and define $Z\cap B$ as
$\psi^{-1}(\widecheck{D}_{c_{1}(\xi)})$, where
$\widecheck{D}_{c_{1}(\xi)}\subset\widecheck{D}$ is a flag subvariety of
codimension $h^{0,2}$. Then if
$\Gamma_{\psi}\subset\mathcal{S}_{\mathbb{C}}\times\widecheck{D}$ is the graph
of $\psi$, the intersection between $\Gamma_{\psi}$ and
$\mathcal{S}_{\mathbb{C}}\times\widecheck{D}_{c^{1}(\xi_{s})}$ is now
unlikely: it has codimension at most $\dim\widecheck{D}+(h^{0,2}-1)$, smaller
than the “expected” codimension of $\dim\widecheck{D}+h^{0,2}$.

Let us call such a $Z$ giving rise to an unlikely intersection of this type an
“atypical Hodge locus”.111Strictly speaking, to be in agreement with [BKU22],
$\widecheck{D}$ should be an orbit of the generic Mumford-Tate group. But
under the large monodromy assumption we may simply take $\widecheck{D}$ to be
the variety of all polarized Hodge flags.  Hodge locus which are atypical in
this sense are studied in the recent paper [BKU22], where the authors prove
the following “Geometric Zilber-Pink” statement: the atypical Hodge locus of
positive period dimension associated to an integral variation of Hodge
structure $\mathbb{V}$ on $S$ with $\mathbb{Q}$-simple monodromy lie in
finitely many strict subvarieties of $S$. Thus the phenomenon just described
can only occur for $Z$ lying inside one of these subvarieties; it in
particular does not occur for $Z$ intersecting a Zariski open subset of $S$.

At this point one can observe that what is needed for our main theorems is a
kind of analogue of this “Geometric Zilber-Pink” result at the integral level.
Tools which will make this possible were developed in recent work [Urb22] by
the first author. More specifically, the paper [Urb22] gives a purely algebro-
geometric way of obtaining the results of [BKU22] which also allows one to
study “integral” unlikely intersections. The idea is that one studies
infinitesimal period maps — formal objects giving maps
$\psi:\mathbb{D}_{r}\to\widecheck{D}$ where $\mathbb{D}_{r}\subset\mathcal{S}$
is an infinitesimal disk of order $r$ — and looks at the locus
$\mathcal{A}_{r}$ in a jet space $J^{d}_{r}\mathcal{S}$ of disks
$\mathbb{D}^{d}_{r}\subset\mathcal{S}$ of dimension
$d=\dim\mathcal{S}_{\mathbb{C}}-(h^{0,2}-1)$ lying in the inverse image under
some such $\psi$ of carefully chosen flag subschemes.222In the proof of
(4.2.3) where this argument takes place we will find it convenient not to
construct $\mathcal{A}_{r}$ directly, but instead understand it through a
larger locus $\mathcal{I}_{r}$ lying in the product of a bundle over
$J^{d}_{r}\mathcal{S}$ with the jet space of a flag variety. Similar, but more
algebro-geometric, reasoning to [BKU22] — which ultimately reduces as in
[BKU22] to an application of the Ax-Schanuel Theorem [BT19] — shows that
$\mathcal{A}_{r,\mathbb{C}}$ projects to a proper closed subvariety of
$\mathcal{S}_{\mathbb{C}}$ for $r\gg 0$. But then the Noetherianity of
$\mathcal{A}_{r}$ implies the same fact at the integral level, and shows that
“unlikely intersections” of positive dimension are constrained by
intersections of the same type in characteristic zero for $p\gg 0$.

Let us make two further comments about our large monodromy and period image
hypotheses. The methods of this paper should apply even when $f$ does not have
large monodromy, although the reasoning becomes more complicated. In order to
apply the Ax-Schanuel theorem, the flag variety $\widecheck{D}$ must be taken
to be an orbit of the algebraic monodromy group. If this group shrinks, so
does $\widecheck{D}$, which means that the expected codimension of $Z$ becomes
smaller. One then has to take this into account at the crystalline level,
which to our knowledge requires knowing that the global cycles defining the
monodromy group satisfy absolute-Hodge-like properties. We plan future work to
deal with this more refined situation.

Secondly, the “sufficiently large period image” assumption is necessary to
ensure that if $Z$ is atypical, it is also positive-dimensional (in the sense
of period dimension). As we explain in §6.3 below, our results are false
without this assumption, and the fact that geometric Zilber-Pink reasoning
requires the intersections to be positive dimensional allows one to see the
necessity of this assumption in a different way.

##### Comment on Effectivity

If the family in (1.1.1) or (1.1.3) is known to satisfy some stronger
conditions (take $R=\mathcal{O}_{L}[1/N]$ case in §4.1), then the locus
$\mathcal{E}$ appearing in the statements of the theorems and various loci
which appeared in the sketch above can in principle be computed by an
algorithm run on a sufficiently powerful computer, given explicit projective
equations for the family as input. In the case of the generic fibres of these
loci (i.e., if we were to compute them over a characteristic zero field), the
necessary theory for writing down these algorithms is developed in [Urb23]. In
particular, [Urb23] gives explicit algorithms for computing the Hodge bundle,
Hodge filtration, and Gauss-Manin connection from a set of projective
equations as well as for computing all of the jet-related constructions we
will use. Subject to limitations in the computational literature concerning
algebro-geometric constructions over a more general base ring
$R=\mathcal{O}_{L}[1/N]$ — for instance, we are not aware of a detailed
algorithmic approach to Chevalley’s Theorem on images of constructible sets in
this context — this could be generalized to compute the locus in our paper as
well.

### 1.4 Notations and Conventions

Let $S$ be a scheme. For a point or geometric point $s$ on $S$, we write
$k(s)$ for the field of definition for $s$. An alg-geometric point on $S$ is a
geometric point $s\to S$ such that $k(s)$ is algebraically (as opposed to just
separably) closed. For a geometric point $s\to S$, we denote its image on the
underlying topological space of $S$ by $s^{\flat}$. We will often consider
smooth projective morphisms $\mathcal{X}\to S$ such that
$h^{i,j}(\mathcal{X}_{s}):=\dim_{k(s)}\mathrm{H}^{j}(\Omega^{i}_{\mathcal{X}_{s}})$
is independent of $s\in S$. In this case, we will simply write this number as
$h^{i,j}$ when the family $\mathcal{X}/S$ is understood.

Suppose that $R^{\prime}\to R$ is a morphism between complete local rings.
Then we define the scheme theoretic image of the induced morphism
$\mathrm{Spf}(R)\to\mathrm{Spf}(R^{\prime})$ to be the formal subscheme
$\mathrm{Spf}(R^{\prime\prime})\subseteq\mathrm{Spf}(R^{\prime})$, where
$R^{\prime\prime}=R^{\prime}/(\ker(R^{\prime}\to R))$. Note that
$\mathrm{Spec}(R^{\prime\prime})\subseteq\mathrm{Spec}(R^{\prime})$ is the
scheme theoretic image of $\mathrm{Spec}(R)\to\mathrm{Spec}(R^{\prime})$ in
the usual sense [Sta22, 01R5]. We abbreviate “local complete intersection” to
“lci”.

When $S$ is a smooth complex variety, by a variation of Hodge structures
$\mathbb{V}$ on $S$ we shall mean a $\mathbb{Q}$-VHS unless otherwise noted.
Moreover, we write the $\mathbb{Q}$-local system as $\mathbb{V}_{B}$ and the
filtered flat vector bundle as $\mathbb{V}_{\mathrm{dR}}$, so that
$\mathbb{V}$ is given by the pair $(\mathbb{V}_{B},\mathbb{V}_{\mathrm{dR}})$.

When $S$ is a scheme and $\mathcal{X}\to S$ is a smooth projective morphism
with geometrically connected fibers, we use $\mathscr{P}_{\mathcal{X}/S}$ to
denote the relative Picard scheme (see §2.1 below). If $k$ is a field and $X$
is a geometrically connected smooth projective variety over $k$, then we let
$\mathscr{P}^{0}_{X/k}$ denote the identity component of $\mathscr{P}_{X/k}$.
The Néron-Severi group $\mathrm{NS}(X)$ of $X/k$ is defined to be the group of
connected components $\mathscr{P}_{X/k}/\mathscr{P}^{0}_{X/k}$. For two line
bundles $\xi,\xi^{\prime}\in\mathrm{Pic}(X)$, we say that they are
algebraically equivalent (denoted by $\xi\sim\xi^{\prime}$) if they give the
same class in $\mathrm{NS}(X)$; similarly, we write
$\xi\sim_{\mathbb{Q}}\xi^{\prime}$ if their classes in
$\mathrm{NS}(X)_{\mathbb{Q}}$ differ by a $\mathbb{Q}^{\times}$-multiple.

##### Acknowledgments

This collaboration started at a conference at the Simons-Laufer Mathematical
Sciences Institute (SLMath, previously known as MSRI), and the authors would
like to thank SLMath for organizing the event and their hospitality. The first
author additionally thanks SLMath for hosting him during the spring of 2023,
and the Institut des Hautes Études Scientifiques (IHES) for hosting him and
supporting his work during the remainder of the year. The second author would
like to thank Ananth Shankar and Mark Kisin for their interest in the work.

##  2 Deformation Theory

### 2.1 Generalities on Picard Schemes

We briefly recall some basic facts about the (relative) Picard scheme and set
up some preliminary lemmas. In this section, we shall always assume that $S$
is a base scheme and $f:\mathcal{X}\to S$ is a smooth and projective morphism
with connected geometric fibers.

By a fundamental theorem of Artin and Grothendieck ([BLR90, §8.4 Thm 3], see
also [Kle05, Thm 4.8]), the sheafification of the presheaf on the fppf site of
$S$ defined by

$T\mapsto\mathrm{Pic}(\mathcal{X}_{T})/\mathrm{Pic}(T)$

is representable by a separated scheme $\mathscr{P}_{\mathcal{X}/S}$ over $S$
which is locally of finite type; moreover, every closed subscheme of
$\mathscr{P}_{\mathcal{X}/S}$ which is of finite type over $S$ is projective
over $S$. We often implicitly use the fact that the formation of
$\mathscr{P}_{\mathcal{X}/S}$ commutes with base change ([Kle05, Ex. 4.4]).
Another handy fact we shall often use is that if $f$ admits a section, then
$\mathscr{P}_{\mathcal{X}/S}(S)=\mathrm{Pic}(\mathcal{X}_{S})/\mathrm{Pic}(S)$
([BLR90, §8.1 Prop. 4]), i.e., sheafification does not add new global
sections.

The following fact is a special case of [Kle05, Prop. 5.20].

######  Proposition (2.1.1).

Assume that for every geometric point $s\to S$, the identity component
$\mathscr{P}^{0}_{\mathcal{X}_{s}/s}$ of $\mathscr{P}_{\mathcal{X}_{s}/s}$ is
smooth of a fixed dimension. Then $\mathscr{P}_{\mathcal{X}/S}$ has an open
and closed subscheme $\mathscr{P}^{0}_{\mathcal{X}/S}$ which is of finite type
over $S$ and whose fiber above a geometric point $s\to S$ is
$\mathscr{P}^{0}_{\mathcal{X}_{s}/s}$. Morevoer, if $S$ is reduced, then
$\mathscr{P}^{0}_{\mathcal{X}/S}$ is smooth over $S$.

######  Definition (2.1.2).

We will often consider pairs of the form $(s,\xi)$ where $s\to S$ is a alg-
geometric point and $\xi\in\mathrm{Pic}(\mathcal{X}_{s})$ is a line bundle on
$\mathcal{X}_{s}$. We refer to such a pair as a LB-pair. Note that a LB-pair
corresponds bijectively to an alg-geometric point on
$\mathscr{P}_{\mathcal{X}/S}$.

######  Definition (2.1.3).

Let $\mathscr{P}(S,s,\xi)$ be the localization of
$\mathscr{P}_{\mathcal{X}/S}$ at $(s,\xi)^{\flat}$ (i.e., the spectrum of the
local ring of $\mathscr{P}_{\mathcal{X}/S}$ at $(s,\xi)^{\flat}$), and
$Z(S,s,\xi)\subseteq S$ be the scheme-theoretic image of
$\mathscr{P}(S,s,\xi)$. We say that $(s,\xi)$ is liftable (resp. strongly
liftable) if $\mathscr{P}(S,s,\xi)$ has a non-empty (resp. topologically
dense) characteristic $0$ fiber. When $S$ is an integral scheme, we say that
$(s,\xi)$ is global (on $S$) if $\mathscr{P}(S,s,\xi)$ dominates $S$.

Below we give a few lemmas about the above definition.

######  Lemma (2.1.4).

Let $(V,(\pi))$ be a DVR with fraction field $F$ and $S$ be a $V$-scheme. Let
$Y$ be an $F$-scheme admitting a quasi-compact $V$-morphism to $S$. Then the
scheme-theoretic image $Z$ of $Y$ is flat over $V$.

###### Proof.

We may assume that $S=\mathrm{Spec}(A)$ and $Y=\mathrm{Spec}(B)$ are affine.
Suppose that $Z_{F}\to S$ is given by a morphism of $V$-algebras $\varphi:A\to
B$. Then $Z$ is given by $A/\ker(\varphi)$ ([Sta22, 01R8]). The flatness of
$Z$ is equivalent to the $\pi$-torsion-freeness of $A/\ker(\varphi)$, i.e., if
$a\not\in\ker(\varphi)$, then $\pi a\not\in\ker(\varphi)$. But this is clear
because $\varphi(\pi)$ is a unit in $B$ by assumption. ∎

######  Lemma (2.1.5).

Let $(V,(\pi))$ be a complete DVR with a characteristic $0$ fraction field $F$
and $R$ be a Noetherian complete local $V$-algebra. Then there exists a DVR
$V^{\prime}$ which is finite and flat over $V$ admitting a local morphism to
$R$.

###### Proof.

One can dig out the following argument from [Del81, Thm 1.6]: Let
$(\pi,x_{1},\cdots,x_{d})$ be a system of parameters for $R$. Set
$R^{\prime}=R/(x_{1},\cdots,x_{d})$ ([Sta22, 0BWY]). By [Sta22, 00LJ], $\pi$
is not a zero divisor in $R^{\prime}$, so that $R^{\prime}$ is flat over $V$.
As $R^{\prime}$ has zero-dimensional fibers over $V$ it is quasi-finite, and
hence finite over $V$ by [Sta22, 04GG(13)]. Therefore, we can find a morphism
$R^{\prime}\to F^{\prime}$ for some finite extension $F^{\prime}$ of $F$. Then
the morphism $R^{\prime}\to F^{\prime}$ necessarily factors through the
integral closure $V^{\prime}$ of $V$ in $F^{\prime}$. ∎

######  Proposition (2.1.6).

A LB-pair $(s,\xi)$ is liftable if and only if there exists a finite extension
$K$ of $K_{0}:=W(k(s))[1/p]$ such that there exists a morphism
$\widetilde{s}:\mathrm{Spec}(\mathcal{O}_{K})\to S$ extending $s$ and a line
bundle $\widetilde{\xi}$ on $\mathcal{X}_{\widetilde{s}}$ whose special fiber
is $\xi$.

###### Proof.

The “if” part follows from definition: Up to replacing $K$ by a finite
extension, $\widetilde{s}$ lifts to a morphism
$\mathrm{Spec}(\mathcal{O}_{K})\to\mathscr{P}_{\mathcal{X}/S}$ which maps the
special point to $(s,\xi)$.

Conversely, if $\mathscr{P}(S,s,\xi)$ contains a characteristic $0$ point,
then by taking its scheme-theoretic image we obtain a closed subscheme
$B\subseteq\mathscr{P}_{\mathcal{X}/S}$ which contains $(s,\xi)^{\flat}$ and
is flat over $\mathbb{Z}_{(p)}$ ($p=\mathrm{char\,}k(s)$) by (2.1.4). Now
$(s,\xi)$ is a closed point on $B_{W}$ for $W:=W(k(s))$, which is flat over
$W$. By (2.1.5), the formal completion of $B_{W}$ at $s$, and hence $B_{W}$
itself, admits some $\mathcal{O}_{K}$-point which lifts $(s,\xi)$ for some
finite extension $K/K_{0}$. This translates to the desired statement. ∎

######  Remark (2.1.7).

  1. (a)

Roughly speaking, being “strongly liftable” means a LB-pair $(s,\xi)$ remains
liftable after small pertubation: there exists an open neighborhood $U$ of
$(s,\xi)^{\flat}$ in $\mathscr{P}_{\mathcal{X}/S}$ such that every LB-pair
$(t,\zeta)$ with $(t,\zeta)^{\flat}\in U$ is liftable. Clearly, if
$\mathscr{P}(S,s,\xi)$ is flat over $\mathbb{Z}_{(p)}$ for
$p=\mathrm{char\,}k(s)$, then $(s,\xi)$ is strongly liftable.

  2. (b)

Being liftable or strongly liftable is clearly a closed property: Suppose that
$(s,\xi)$ and $(t,\zeta)$ are two LB-pairs and $(t,\zeta)^{\flat}$ specializes
to $(s,\xi)^{\flat}$ on $\mathscr{P}_{\mathcal{X}/S}$. If $(t,\zeta)$ is
liftable, then so is $(s,\xi)$.

######  Definition (2.1.8).

Let $T$ be a connected smooth $\mathbb{C}$-variety and $t\in T$ be a point. If
$\mathbb{V}_{B}$ is a $\mathbb{Q}$-local system, let
$\mathrm{Mon}(\mathbb{V}_{B},t)$ be the Zariski closure of $\pi_{1}(T,t)$ in
$\mathrm{GL}(\mathbb{V}_{B,t})$ and let
$\mathrm{Mon}^{\circ}(\mathbb{V}_{B},t)$ denote the connected component. We
call the isomorphism class of $\mathrm{Mon}^{\circ}(\mathbb{V}_{B},t)$, which
is independent of the choice of $t$, the algebraic monodromy group. When $T$
is irreducible but not necessarily smooth, the algebraic monodromy group is
understood to be that of $\mathbb{V}_{B}$ restricted to the smooth locus of
$T$.

Similarly, if $T$ is a general connected normal Noetherian scheme and $t$ is a
geometric point, and $\mathcal{H}_{\ell}$ is an $\mathbb{Q}_{\ell}$-local
system for a prime $\ell$ invertible in $\mathcal{O}_{T}$, define
$\mathrm{Mon}(\mathcal{H}_{\ell},t)$ to be the Zariski closure of
$\pi_{1}^{\textrm{{\'{e}}t}}(T,t)$ in $\mathrm{GL}(\mathcal{H}_{\ell,t})$ and
let $\mathrm{Mon}^{\circ}(\mathcal{H}_{\ell},t)$ be its identity component.

#### (2.1.9)

Below we shall assume that $S$ is a notherian integral scheme with generic
point $\eta$, and give some remarks on specializing line bundles. Let
$\bar{\eta}$ be a geometric point over $\eta$ and $s$ be any other geometric
point. Then we can choose a morphism $\gamma:\mathrm{Spec}(R)\to S$ where $R$
is a strictly Henselian DVR, such that $\gamma$ sends its generic point
$\eta_{0}$ to $\eta$ and special point $s_{0}$ to $s^{\flat}\in S$. Let
$\bar{\eta}_{0}$ and $\bar{s}_{0}$ be geometric points over $\eta_{0}$ and
$s_{0}$ such that their morphisms to $S$ factor through $\bar{\eta}$ and $s$
respectively. By [MP12, Prop. 3.1], there are natural identifications
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})=\mathrm{NS}(\mathcal{X}_{\bar{\eta}_{0}})$
and
$\mathrm{NS}(\mathcal{X}_{\bar{s}})=\mathrm{NS}(\mathcal{X}_{\bar{s}_{0}})$.
This induces a specialization map
$\mathrm{sp}_{\gamma}:\mathrm{NS}(\mathcal{X}_{\bar{\eta}})\to\mathrm{NS}(\mathcal{X}_{\bar{s}})$
(cf. [MP12, Prop. 3.6] and its proof). Let us also write $\gamma$ for the
triple $(\gamma,\bar{\eta}_{0},s_{0})$. We call such a triple a path from
$\bar{\eta}$ to $s$ as it induces an étale path on $S$ from $\bar{\eta}$ to
$s$ in the sense of [Sta22, Tag 03VD], which we also denote as $\gamma$. Using
[Sta22, 054F], one deduces that the LB-pair $(s,\xi)$ is global on $S$ if and
only if $\xi$ lies in the image of $\mathrm{sp}_{\gamma}$ for some choice of
$\gamma$.

######  Lemma (2.1.10).

If $S$ is in addition normal, then we have the following.

  1. (a)

The natural sequence
$0\to\mathrm{Pic}(S)\to\mathrm{Pic}(\mathcal{X})\to\mathrm{Pic}(\mathcal{X}_{\eta})\to
0$ is exact.

  2. (b)

The image of $\mathrm{sp}_{\gamma}{\otimes}\mathbb{Q}$ is independent of the
choice of $\gamma$.

  3. (c)

If moreover $\mathrm{NS}(\mathcal{X}_{\bar{\eta}})$ is
$\mathrm{char\,}k(\eta)$-torsion-free, then (b) is true for the image of
$\mathrm{sp}_{\gamma}$ integrally.

  4. (d)

If $(s,\xi)$ a global LB-pair on $S$ and
$\mathrm{Mon}(R^{2}f_{\textrm{{\'{e}}t}*}\mathbb{Q}_{\ell},s)$ is connected
for some prime $\ell$ invertible in $\mathcal{O}_{S}$, then there exists a
relative line bundle
$\boldsymbol{\xi}\in\mathrm{Pic}(\mathcal{X})/\mathrm{Pic}(S)$ such that
$\boldsymbol{\xi}_{s}\sim_{\mathbb{Q}}\xi$.

We remark that by (a), for any field-valued point $t$ on $S$, there are well
defined specialization maps
$\mathrm{Pic}(\mathcal{X}_{\eta})\to\mathrm{Pic}(\mathcal{X}_{t})$ and
$\mathrm{NS}(\mathcal{X}_{\eta})\to\mathrm{NS}(\mathcal{X}_{t})$.

###### Proof.

(a) follows from [Gro67, $\mathrm{Err}_{\mathrm{IV}}$ Cor. 21.4.13]. For (b)
and (c), note that the following diagram commutes.

${{\mathrm{NS}(\mathcal{X}_{\bar{\eta}})}}$${{\mathrm{NS}(\mathcal{X}_{\bar{s}})}}$${{\mathrm{H}^{2}_{\textrm{{\'{e}}t}}(\mathcal{X}_{\bar{\eta}},\widehat{\mathbb{Z}}^{p})}}$${{\mathrm{H}^{2}_{\textrm{{\'{e}}t}}(\mathcal{X}_{\bar{s}},\widehat{\mathbb{Z}}^{p})}}$$\scriptstyle{\mathrm{sp}_{\gamma}}$$\scriptstyle{c_{1}}$$\scriptstyle{c_{1}}$$\scriptstyle{\gamma}$

Now if $\gamma^{\prime}$ are a different choice, then the composition of étale
paths $\gamma^{\prime}\circ\gamma^{-1}$ defines an element of
$\pi_{1}^{\textrm{{\'{e}}t}}(S,\bar{\eta})$, whose action on
$\mathrm{H}^{2}_{\textrm{{\'{e}}t}}(\mathcal{X}_{\bar{\eta}},\widehat{\mathbb{Z}}^{p})$
preserves $\mathrm{NS}(\mathcal{X}_{\bar{\eta}})$. Indeed, this follows from
the fact that the $\mathrm{Gal}(\bar{\eta}/\eta)$-action preserves
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})$ and factors through the natural map
$\mathrm{Gal}(\bar{\eta}/\eta)\to\pi_{1}^{\textrm{{\'{e}}t}}(S,\bar{\eta})$,
which is a surjection by the normality of $S$. The conclusion easily follows.

(d) As the action of $\mathrm{Gal}(\bar{\eta}/\eta)$ on
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathbb{Q}}$ factors through
$\pi_{1}^{\textrm{{\'{e}}t}}(S,\bar{\eta})$ and a finite quotient, the
connectedness of $\mathrm{Mon}(R^{2}f_{*}\mathbb{Q}_{\ell},s)$ implies that
this action is in fact trivial, so that the natural map
$\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}\to\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathbb{Q}}$
is an isomorphism. Suppose that $\xi=\mathrm{sp}_{\gamma}(\beta)$ for some
path $\gamma$ and $\beta\in\mathrm{Pic}(\mathcal{X}_{\bar{\eta}})$. Then the
class of $\beta$ in $\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathbb{Q}}$
descends to a class $\boldsymbol{\xi}$ in
$\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}$. Choose a representative of
$\boldsymbol{\xi}$ in $\mathrm{Pic}(\mathcal{X}_{\eta})$ and apply (a). ∎

######  Remark (2.1.11).

In general, for (d) we cannot ask in the above proposition to take
$\boldsymbol{\xi}$ such that $\boldsymbol{\xi}_{s}=\xi$ as elements of
$\mathrm{Pic}(\mathcal{X}_{s})$. Even when $S=\mathrm{Spec}(k)$ is a point,
this cannot be guaranteed if $k$ is not algebraically closed: the morphism
$\mathrm{Pic}(\mathcal{X})\hookrightarrow\mathrm{Pic}(\mathcal{X}_{\bar{k}})^{\mathrm{Gal}_{k}}$
is in general not surjective —the cokernel is a subgroup of the Brauer group
$\mathrm{Br}(k)$ and does not always vanish.

### 2.2 Local Deformation Spaces

#### (2.2.1)

Let $\kappa$ be a field of characteristic $0$, $R$ be a complete local
$\kappa$-algebra with residue field $\kappa$. Let $S:=\mathrm{Spf}(R)$ and $s$
be the special point. Suppose that $f:\mathcal{X}\to S$ is a smooth projective
family with geometrically connected fibers. Let
$\xi\in\mathrm{Pic}(\mathcal{X}_{s})$ be any element. Then there exists a
closed formal subscheme $\mathrm{Def}(\xi)\subseteq S$ characterized by the
property that for every Artin $\kappa$-algebra $A$, an element $\widetilde{s}$
in $S(A)$ lies in $\mathrm{Def}(\xi)(A)$ if and only if $\xi$ deforms,
possibly non-uniquely, to $\mathcal{X}_{\widetilde{s}}$. It is not hard to
construct $\mathrm{Def}(\xi)$ explicitly using the standard deformation theory
of line bundles: Let $(\mathcal{H},\nabla,\mathrm{Fil}^{\bullet}\mathcal{H})$
be the filtered flat vector bundle
$R^{2}f_{*}\Omega^{\bullet}_{\mathcal{X}/S}$. Then $c_{1,\mathrm{dR}}(\xi)$
extends uniquely to a horizontal section $\sigma$ of $\mathcal{H}$, as
$\mathcal{H}$ admits a flat trivialization
$(\mathcal{H},\nabla)\simeq\mathcal{H}_{s}{\otimes}(R,d)$ (see for example the
proof of [BO78, Prop. 2.16]). Then $\mathrm{Def}(\xi)=\mathrm{Spf}(R/I)$ where
$I$ is the minimal ideal such that the image of $\sigma$ vanishes in
$(\mathcal{H}/\mathrm{Fil}^{1}\mathcal{H}){\otimes}_{R}(R/I)$. In particular,
it is clear that $\mathrm{Def}(\xi)$ is generated by
$h^{0,2}:=\mathrm{rank\,}\mathcal{H}/\mathrm{Fil}^{1}\mathcal{H}$ elements.

Our goal in this section is to prove the analogue of (2.2.1) in positive and
mixed characteristic. The difficulty to overcome in this case is that in
general $R^{2}f_{*}\Omega^{\bullet}_{\mathcal{X}/S}$ does not admit a flat
trivialization.

######  Definition (2.2.2).

Let $S$ be a noetherian (possibly formal) scheme and $f:\mathcal{X}\to S$ be a
smooth projective morphism with geometrically connected fibers. We say that
$\mathcal{X}/S$ is an admissible family if the following conditions hold:

  1. (i)

the Hodge-de Rham spectral sequence degenerates at $E_{1}$ page;

  2. (ii)

the Hodge cohomology
$\mathrm{H}^{j}(\Omega^{i}_{\mathcal{X}/S}):=R^{i}f_{*}\Omega_{\mathcal{X}/S}^{j}$
is locally free for every $(i,j)$.

  3. (iii)

for every prime $p$, the crystal of coherent sheaves
$R^{r}f_{\mathbb{F}_{p},\mathrm{cris}*}\mathcal{O}_{\mathcal{X}_{\mathbb{F}_{p}}/\mathbb{Z}_{p}}$
on $\mathrm{Cris}(S_{\mathbb{F}_{p}}/\mathbb{Z}_{p})$ is locally free for
every $r$.

When $S$ is connected, we shall denote
$\mathrm{rank\,}\mathrm{H}^{j}(\Omega^{i}_{\mathcal{X}/S})$ by $h^{i,j}$.

######  Remark (2.2.3).

The above assumptions are used to make sure that various cohomology modules
commute with base change: (i) and (ii) easily imply that for each $r$,
$\mathrm{H}^{r}_{\mathrm{dR}}(\mathcal{X}/S):=R^{r}f_{*}\Omega^{\bullet}_{\mathcal{X}/S}$
is locally free. Moreover, using [Har77, III Cor. 12.7], one quickly checks by
induction that for any morphism $T\to S$ between noetherian (formal) schemes,
the natural base change morphisms
$\mathrm{H}^{j}(\Omega^{i}_{\mathcal{X}/S}){\otimes}_{\mathcal{O}_{S}}\mathcal{O}_{T}\to\mathrm{H}^{j}(\Omega^{i}_{\mathcal{X}_{T}/T})$
and hence
$\mathrm{H}^{r}_{\mathrm{dR}}(\mathcal{X}/S){\otimes}_{\mathcal{O}_{S}}\mathcal{O}_{T}\to\mathrm{H}^{r}_{\mathrm{dR}}(\mathcal{X}_{T}/T)$
are isomorphisms for all $i,j,r$. Similarly, suppose that $T$ is a noetherian
$\mathbb{F}_{p}$-scheme over $S_{\mathbb{F}_{p}}$ and there is PD morphism
$T^{\prime}\to S^{\prime}$ between PD thickenings $T^{\prime}$ of $T$ and
$S^{\prime}$ of $S_{\mathbb{F}_{p}}$. Then the natural morphism

$(R^{r}f_{\mathbb{F}_{p},\mathrm{cris}*}\mathcal{O}_{\mathcal{X}_{\mathbb{F}_{p}}/\mathbb{Z}_{p}})_{S^{\prime}}{\otimes}_{\mathcal{O}_{S^{\prime}}}\mathcal{O}_{T^{\prime}}\to(R^{r}f_{T,\mathrm{cris}*}\mathcal{O}_{\mathcal{X}_{T}/\mathbb{Z}_{p}})_{T^{\prime}}$

is an isomorphism. One may deduce this from the commutivity with base change
on the level of derived categories [BO83, Thm 7.8] and the universal
coefficient theorem.

The following lemma follows from a famous theorem of Deligne-Illusie [DI87],
standard base change theorem for coherent cohomology, and crystalline de-Rham
comparison isomorphisms.

######  Lemma (2.2.4).

Let $L$ be a number field, $\mathcal{S}$ be a connected smooth scheme over
$\mathcal{O}_{L}[1/N]$ of finite type, and $f:\mathcal{X}\to\mathcal{S}$ be a
smooth projective morphism with geometrically connected fibers. Then up to
increasing $N$, $\mathcal{X}$ is admissible.

For the content below, let $k$ be an algebraically closed field of
characteristic $p>0$ and let $W=W(k)$.

######  Definition (2.2.5).

Let $S$ be the $p$-adic completion of a noetherian $W$-scheme. A filtered
F-crystal on $S$ is a crystal $\mathbf{H}$ of vector bundles on
$\mathrm{Cris}(S_{k}/W)$ equipped with a decreasing filtration
$\mathrm{Fil}^{\bullet}$ on $\mathbf{H}_{S}$ by direct summands and a
Frobenius morphism $F:F^{*}_{S_{k}/W}\mathbf{H}\to\mathbf{H}$, where
$F_{S_{k}/W}$ is the morphism on the topos $(S/W)_{\mathrm{cris}}$ induced by
the absolute Frobenius morphism $F_{S_{k}}$ on $S_{k}$.

Here $\mathbf{H}_{S}$ denotes the $\mathcal{O}_{S}$-module obtained by
evaluating $\mathbf{H}$ on $S$, which is viewed as a (pro-)object of
$\mathrm{Cris}(S_{k}/W)$.

######  Remark (2.2.6).

We remark that, if
$\mathbf{H}=R^{r}f_{\mathrm{cris}*}\mathcal{O}_{\mathcal{X}/W}$ for an
admissible family $f:\mathcal{X}\to S_{\mathbb{F}_{p}}$, then the filtration
on $\mathbf{H}_{S}{\otimes}\mathbb{F}_{p}$ and the Frobenius action satisfy a
compatibility condition ([BO83, Thm 8.26]). We will only consider situations
in which this condition is satisfied, but we will not make use of it, so we do
not incorporate it into the definition above.

######  Proposition (2.2.7).

Let $R$ be a northerian complete local $k$-algebra. Set $S=\mathrm{Spf}(R)$
and let $s$ be the closed point. Let $\mathbf{H}$ be an F-crystal over
$\mathrm{Cris}(S/W)$. If $R$ is lci, then for any $m\in\mathbb{N}_{>0}$ the
map

$\Gamma((S/W)_{\mathrm{cris}},\mathbf{H})^{F=p^{m}}:=\\{\alpha\in\Gamma((S/W)_{\mathrm{cris}},\mathbf{H})\mid
F(\alpha{\otimes}1)=p^{m}\alpha\\}\to\Gamma((s/W)_{\mathrm{cris}},\mathbf{H}|_{s})$
(2)

is injective.

The proof below is essentially a version of “Dwork trick”, which was used in
[BO83].

###### Proof.

Let $\mathfrak{m}$ be the maximal ideal and $d$ be the Krull dimension of $R$
and choose a system of parameters $\\{x_{1},x_{2},\cdots,x_{d}\\}$. Let
$\alpha=(\alpha_{1},\cdots,\alpha_{d})\in\mathbb{N}_{>0}^{d}$ be a multi-index
equipped with a partial order such that $\alpha\leq\alpha^{\prime}$ if and
only if $\alpha_{i}\leq\alpha^{\prime}_{i}$ for every $i$. Let
$I_{\alpha}:=(x_{1}^{\alpha_{1}},\cdots,x_{d}^{\alpha_{d}})$ so that
$R=\varprojlim R/I_{\alpha}$. Note that each $R/I_{\alpha}$ is Artinian and
lci. It suffices to show that the statement is true for each $R/I_{\alpha}$,
so that we reduce to the case when $R$ itself is Artinian, i.e.,
$\mathfrak{m}$ is nilpotent.

Choose a smooth $W$-scheme $\widetilde{S}$ such that $S$ admits a closed
embedding into $\widetilde{S}$ and let $D$ be the PD envelope of $S$ in
$\widetilde{S}$. Since $S$ is lci, $D$ (and hence $\mathbf{H}_{D}$) has no
$p$-torsion (cf. [BO83, Rmk 1.10(i)]). This implies that
$\Gamma((S/W)_{\mathrm{cris}},\mathbf{H})$ has no $p$-torsion. On the other
hand, we note that for $n\gg 0$, $F_{S}^{n}$ factors as $S\to
s\stackrel{{\scriptstyle j}}{{\hookrightarrow}}S$ for some morphism $S\to s$,
where $j$ is the embedding and $F_{S}$ is the absolute Frobenius morphism.
Suppose now that $\gamma\in\Gamma((S/W)_{\mathrm{cris}},\mathbf{H})^{F=p^{m}}$
is an element which lies in the kernel of (2), i.e., $j^{*}(\gamma)=0$. Since
$F_{S}^{n}$ factors through $j$, this implies that
$F^{n}(\gamma{\otimes}1)=p^{mn}\gamma=0$, so that $\gamma=0$. ∎

######  Example (2.2.8).

The assumption on $R$ being lci is necessary: Let
$R=k[\\![x,y]\\!]/(x^{2},xy,y^{2})$. Set $S=\mathrm{Spec}(R)$ and let $s$ be
the closed point. Let $\mathcal{E}$ be a constant elliptic curve over $R$
whose fiber over the closed point is ordinary. Set
$\mathcal{A}=\mathcal{E}\times_{S}\mathcal{E}$. Since
$\mathrm{End}(\mathbf{H}^{1}_{\mathrm{cris}}(\mathcal{E}/S))(-1)$ is a direct
summand of $\mathbf{H}:=\mathbf{H}^{2}_{\mathrm{cris}}(\mathcal{A}/S)$, the
map
$\Gamma((S/W)_{\mathrm{cris}},\mathbf{H})^{F=p}\to\Gamma((s/W)_{\mathrm{cris}},\mathbf{H}|_{s})$
is not injective ([BO83, (A.4)]). The problem is that the PD envelope of $S$
in a smooth $W$-scheme indeed has $p$-torsion.

######  Theorem (2.2.9).

Let $R$ be a complete local $W$-algebra with residue field $k$ such that
$R_{k}=R{\otimes}k$ is reduced and lci. Let $S=\mathrm{Spec}(R)$,
$\hat{S}:=\mathrm{Spf}(R)$ and $s$ be the special point. Suppose that
$f:\mathcal{X}\to S$ is an admissibe family. Let
$\xi\in\mathrm{Pic}(\mathcal{X}_{s})=\mathscr{P}_{\mathcal{X}/S}(k)$ be any
element.

Let $\hat{\mathscr{P}}^{\xi}$ be the formal completion of
$\mathscr{P}_{\mathcal{X}/S}$ at the closed point $\xi$ and let
$\mathrm{Def}(\xi)\subseteq\hat{S}$ be its scheme-theoretic image. Then
$\hat{\mathscr{P}}^{\xi}$ is formally smooth over $\mathrm{Def}(\xi)$ and the
ideal $J\subseteq R$ defining $\mathrm{Def}(\xi)$ is generated by at most
$h:=\dim_{k}\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}})$ elements; moreover
$\mathrm{Def}(\xi)$ only depends on the class of $\xi$ in
$\mathrm{NS}(\mathcal{X}_{s})$.

Note that the fact the formal smoothness of $\hat{\mathscr{P}}^{\xi}$ over
$\mathrm{Def}(\xi)$ implies that the closed formal subscheme
$\mathrm{Def}(\xi)\subseteq\hat{S}$ is indeed characterized by the property
that for any $R\in\mathsf{Art}_{W}$ (the category of Artin $W$-algebras) and
$\widetilde{s}\in\hat{\mathcal{S}}(R)$, $\widetilde{s}$ factors through
$\mathrm{Def}(\xi)$ if and only if $\xi$ deforms to
$\mathcal{X}_{\widetilde{s}}$. This justifies the notation. The functor of
points of $\mathrm{Def}(\xi)$ is denoted by $\Phi^{\xi}$ in the proof below.

###### Proof.

For every $A\in\mathsf{Art}_{W}$, note that a point in
$\mathscr{P}_{\mathcal{X}/S}(A)$ is of the form $(z,\zeta)$, where
$z\in\hat{S}(A)$ and $\zeta\in\mathrm{Pic}(\mathcal{X}_{z})$ because
$\mathcal{X}\to S$ admits a section. Let
$\Phi^{\xi}:\mathsf{Art}_{W}\to\mathsf{Set}$ be the functor defined by sending
$A\in\mathsf{Art}_{W}$ to the set of pairs $(z,\alpha)$, where
$z\in\hat{S}(A)$ and
$\alpha=c_{1,\mathrm{cris}}(\zeta)\in\Gamma(\mathrm{Cris}(A/W),\mathbf{H}|_{z})$
for some $\zeta\in\mathrm{Pic}(\mathcal{X}_{z})$ such that $\zeta_{s}\sim\xi$.
Equivalently, the pair $(z,\alpha)$ could be thought of as an equivalence
class of pairs $(z,\zeta)$, where $\zeta\in\mathrm{Pic}(\mathcal{X}_{z})$ is
an element with $\zeta_{s}\sim\xi$ and $(z,\zeta)$ is equivalent to
$(z,\zeta^{\prime})$ if and only if
$c_{1,\mathrm{cris}}(\zeta)=c_{1,\mathrm{cris}}(\zeta^{\prime})$.

We start by proving two lemmas:

######  Lemma (2.2.10).

For every $A\in\mathsf{Art}_{W}$, the projection $\Phi^{\xi}(A)\to\hat{S}(A)$
is injective.

###### Proof.

It suffices to show that for each $z\in\hat{S}(A)$, given
$\zeta,\zeta^{\prime}\in\mathrm{Pic}(\mathcal{X}_{z})$ such that
$\zeta_{s}\sim\zeta_{s}^{\prime}$, we have
$c_{1,\mathrm{cris}}(\zeta)=c_{1,\mathrm{cris}}(\zeta^{\prime})$ in
$\Gamma((A/W)_{\mathrm{cris}},\mathbf{H}|_{z})$. Let
$\delta:=\zeta^{\prime}{\otimes}\zeta^{\vee}$. Then
$\delta_{k}\in\mathscr{P}^{0}_{\mathcal{X}_{s}/k}(k)$ and it suffices to show
that $c_{1,\mathrm{cris}}(\delta)=0$. Recall that for every alg-geometric
point $t\to S$, $\mathrm{H}^{2}_{\mathrm{cris}}(\mathcal{X}_{t}/W(k(t)))$ is
torsion-free, which implies that the fiber
$\mathscr{P}^{0}_{\mathcal{X}/S}|_{t}$ is reduced (or smooth) by [Ill79, II
Prop. 5.16]; moreover, the dimension of $\mathscr{P}^{0}_{\mathcal{X}/S}|_{t}$
is $\dim\mathrm{H}^{1}(\mathcal{O}_{\mathcal{X}_{t}})$, which by assumption is
independent of $t$. Since $S$ is reduced, by (2.1.1)
$\mathscr{P}^{0}_{\mathcal{X}/S}$ is smooth, so that there exists a
deformation
$\widetilde{\delta}\in\mathrm{Pic}(\mathcal{X})=\mathscr{P}_{\mathcal{X}/S}(S)$
such that $\widetilde{\delta}_{z}=\delta$. Then by the functoriality the Chern
class map (see e.g., [BI70, Thm 2.4(i)]) $c_{1,\mathrm{cris}}(\delta)$ is the
image of $c_{1,\mathrm{cris}}(\widetilde{\delta})$ under the natural
restriction map
$\Gamma((S/W)_{\mathrm{cris}},\mathbf{H})\to\Gamma((A/W)_{\mathrm{cris}},\mathbf{H}|_{z})$.
Therefore, it suffices to show that
$c_{1,\mathrm{cris}}(\widetilde{\delta})=0$. Now we note that
$c_{1,\mathrm{cris}}(\widetilde{\delta})$ lies in the kernel of the map
$\Gamma((S/W)_{\mathrm{cris}},\mathbf{H})^{F=p}\to\Gamma((s/W)_{\mathrm{cris}},\mathbf{H}|_{s})$,
which is trivial by (2.2.7). ∎

######  Lemma (2.2.11).

The natural projection $\hat{\mathscr{P}}^{\xi}\to\Phi^{\xi}$ is formally
smooth. That is, given any surjection $A\to A^{\prime}$ whose kernel is square
zero, the natural morphism

$\hat{\mathscr{P}}^{\xi}(A)\to\hat{\mathscr{P}}^{\xi}(A^{\prime})\times_{\Phi^{\xi}(A^{\prime})}\Phi^{\xi}(A)$

is surjective.

###### Proof.

Concretely, the above amounts to saying that given $z\in\hat{S}(A)$ and line
bundles $\zeta_{1},\zeta_{2}\in\mathrm{Pic}(\mathcal{X}_{z^{\prime}})$, where
$z^{\prime}\in\hat{S}(A^{\prime})$ is the image of $z$, then if $\zeta_{1}$
extends to $\mathcal{X}_{z}$, so does $\zeta_{2}$. By [Ogu79, Prop. 1.12],
$\zeta_{i}$ extends to $\mathcal{X}_{z}$ if and only if
$c_{1,\mathrm{cris}}(\zeta_{i})_{z}\in(\mathbf{H}|_{z^{\prime}})_{A}\,{\cong}\,\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}_{z}/A)$
lies in the $\mathrm{Fil}^{1}$ part of Hodge filtration. However, (2.2.10)
implies that $c_{1,\mathrm{cris}}(\zeta_{1})=c_{1,\mathrm{cris}}(\zeta_{2})$,
so the conclusion follows. ∎

Back to the proof of (2.2.9), given morphisms $A^{\prime}\to A$ and
$A^{\prime\prime}\to A$ in $\mathsf{Art}_{W}$, we have a commutative diagram

${{\hat{\mathscr{P}}^{\xi}(A^{\prime}\times_{A}A^{\prime\prime})}}$${{\hat{\mathscr{P}}^{\xi}(A^{\prime})\times_{\hat{\mathscr{P}}(A)}\hat{\mathscr{P}}(A^{\prime\prime})}}$${{\Phi^{\xi}(A^{\prime}\times_{A}A^{\prime\prime})}}$${{\Phi^{\xi}(A^{\prime})\times_{\Phi^{\xi}(A)}\Phi^{\xi}(A^{\prime\prime})}}$${{\hat{S}(A^{\prime}\times_{A}A^{\prime\prime})}}$${{\hat{S}(A^{\prime})\times_{\hat{S}(A)}\hat{S}(A^{\prime\prime})}}$$\scriptstyle{\sim}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\sim}$$\scriptstyle{\iota_{2}}$

The top and bottom horizontal arrows are isomorphisms because
$\hat{\mathscr{P}}^{\xi}$ and $\hat{S}$ are pro-representable by assumption.
The surjectivity of $\pi_{1}$ is clear by definition, and the injectivity of
$\iota_{i}$’s follow from (2.2.10). To see the surjectivity of $\pi_{2}$, it
suffices to see that if
$(z,\zeta),(z^{\prime},\zeta^{\prime}),(z^{\prime\prime},\zeta^{\prime\prime})$
are $A,A^{\prime},A^{\prime\prime}$ points on $\hat{\mathscr{P}}^{\xi}$ such
that $z^{\prime}$ and $z^{\prime\prime}$ are both deformations of $z$, and
$c_{1,\mathrm{cris}}(\zeta^{\prime}|_{z})=c_{1,\mathrm{cris}}(\zeta^{\prime\prime}|_{z})=c_{1,\mathrm{cris}}(\zeta)$,
then $\zeta$ can be deformed to $z^{\prime}$ and $z^{\prime\prime}$. This
follows easily from the argument of (2.2.11) by iteratively considering square
zero extensions.

Now it is easy to infer that the middle horizontal arrow is an isomorphism.
Note that the tangent space $\Phi^{\xi}(k[\epsilon]/(\epsilon^{2}))$ is finite
dimensional as it admits a surjection from
$\hat{\mathscr{P}}^{\xi}(k[\epsilon]/(\epsilon^{2}))$. Schlessinger’s
criterion [Sch68, Thm 2.11] implies that $\Phi^{\xi}$ is pro-representable by
a formal scheme $\hat{Z}:=\mathrm{Spf}(R^{\prime})$. By [DH22, Lem. 2.1.7],
the natural morphism $\hat{Z}\to\hat{S}$ is a closed immersion, and by
(2.2.11) the morphism $\hat{\mathscr{P}}^{\xi}\to\hat{Z}$ is formally smooth.
This implies that $\hat{Z}$ is indeed the scheme-theoretic image of
$\hat{\mathscr{P}}^{\xi}$, i.e., $\hat{Z}=\mathrm{Def}(\xi)$. Moreover, it is
clear from the definition of the functor $\Phi^{\xi}$ that
$\Phi^{\xi}=\Phi^{\xi^{\prime}}$ if $\xi\sim\xi^{\prime}$, so that
$\mathrm{Def}(\xi)$ only depends on the class of $\xi$ in
$\mathrm{NS}(\mathcal{X}_{s})$.

It remains to show that the ideal $J=\ker(R\to R^{\prime})$ defining
$\hat{Z}\subseteq\hat{S}$ is generated by $h$ elements, which follows from a
simple adaptation of Delgine’s arguments for [Del81, Prop. 1.5]. By choosing a
section of $\hat{\mathscr{P}}^{\xi}\to\hat{Z}$ we obtain a line bundle
$\widetilde{\xi}\in\mathrm{Pic}(\mathcal{X}_{\hat{Z}})$ which deforms $\xi$.
Let $\mathfrak{m}\subseteq R$ be the maximal ideal. The obstruction to
deforming $\widetilde{\xi}$ to
$\hat{Z}^{\prime\prime}:=\mathrm{Spf}(R/\mathfrak{m}J)$ is given by an element
$\mathfrak{o}\in\mathrm{H}^{2}(X_{s},J/\mathfrak{m}J)\,{\cong}\,\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}}){\otimes}J/\mathfrak{m}J$.
By choosing a basis for $\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}})$,
$\mathfrak{o}$ gives us $h$ elements in $J/\mathfrak{m}J$. Choose liftings of
the elements $f_{1},\cdots,f_{h}\in J$. Let $\Sigma\subseteq S$ be defined by
the ideal $\mathfrak{m}J+(f_{1},\cdots,f_{h})$. Then by construction
$\hat{Z}\subseteq\Sigma\subseteq\hat{Z}^{\prime\prime}$. Since
$\widetilde{\xi}$ deforms to $\Sigma$, there is a section
$\Sigma\to\hat{\mathscr{P}}^{\xi}$. As $\hat{Z}$ is the scheme-theoretic
image, $\Sigma\subseteq\hat{Z}$. Therefore, we must have $\hat{Z}=\Sigma$ and
hence $\mathfrak{m}J+(f_{1},\cdots,f_{h})=J$. By Nakayama’s lemma
$J=(f_{1},\cdots,f_{h})$. ∎

Below are some simple observations for future reference.

######  Lemma (2.2.12).

In (2.2.9), for any $m\in\mathbb{N}$ with $p\nmid m$,
$\mathrm{Def}(\xi)=\mathrm{Def}(m\xi)$.

###### Proof.

We run a simple induction. Suppose that $R^{\prime}\to R$ is a surjection in
$\mathsf{Art}_{W}$ with square zero kernel and
$\mathrm{Def}(\xi)(R)=\mathrm{Def}(m\xi)(R)\subseteq\hat{S}(R)$. We just need
to show that $\mathrm{Def}(\xi)(R^{\prime})=\mathrm{Def}(m\xi)(R^{\prime})$.
That $\mathrm{Def}(\xi)(R^{\prime})\subseteq\mathrm{Def}(m\xi)(R^{\prime})$ is
automatic, so we only need to show the reverse inclusion.

Take $z^{\prime}\in\mathrm{Def}(m\xi)(R^{\prime})$ and a deformation
$\xi^{\prime}_{z^{\prime}}$ of $m\xi$ to $z^{\prime}$. Let
$z:=z^{\prime}{\otimes}_{R^{\prime}}R$. Then by assumption
$z\in\mathrm{Def}(\xi)(R)$ so there exists a deformation $\xi_{z}$ of $\xi$
over $z$. Setting
$\xi^{\prime}_{z}:=\xi^{\prime}_{z^{\prime}}{\otimes}_{R^{\prime}}z$. Then
$m\xi_{z}\sim\xi^{\prime}_{z}$. In particular, by (2.2.10),
$c_{1,\mathrm{cris}}(m\xi_{z})=c_{1,\mathrm{cris}}(\xi^{\prime}_{z})$.
Therefore, by [Ogu79, Prop. 1.12], the fact that $\xi^{\prime}_{z}$ deforms
over $z^{\prime}$ implies that $m\xi_{z}$ deforms to $R^{\prime}$ as well.
Applying this proposition again, as
$c_{1,\mathrm{cris}}(m\xi_{z})=mc_{1,\mathrm{cris}}(\xi_{z})$ and $m$ is a
unit, $\xi_{z}$ also deforms to $z^{\prime}$, i.e.,
$z^{\prime}\in\mathrm{Def}(\xi)(R^{\prime})$. ∎

######  Lemma (2.2.13).

Let $\kappa$ be a perfect field and $X$ be an admissible smooth proper variety
over $\kappa$. Then
$\mathrm{NS}(X)_{\mathrm{tf}}\to(\mathrm{NS}(X_{\bar{\kappa}})_{\mathrm{tf}})^{\mathrm{Gal}_{\kappa}}$
is an isomorphism. If $\mathrm{char\,}\kappa=p>0$, then $\mathrm{NS}(X)$ has
no $p$-torsion.

###### Proof.

It is well known that the statement is true for $\mathrm{NS}(X)_{\mathbb{Q}}$,
i.e., the natural map
$\mathrm{NS}(X)_{\mathbb{Q}}\to(\mathrm{NS}(X_{\bar{\kappa}})_{\mathbb{Q}})^{\mathrm{Gal}_{\kappa}}$
is an isomorphism. For every $\ell\neq\mathrm{char\,}k$,
$\mathrm{NS}(X){\otimes}\mathbb{Z}_{\ell}\to\mathrm{H}^{2}_{\textrm{{\'{e}}t}}(X_{\bar{\kappa}},\mathbb{Z}_{\ell}(1))$
is injective and has torsion-free cokernel. Similarly, if
$\mathrm{char\,}k=p>0$, as $\mathrm{H}^{2}_{\mathrm{cris}}(X/W(\kappa))$ is
$p$-torsion free, $\mathrm{NS}(X)$ has no $p$-torsion and the map
$\mathrm{NS}(X){\otimes}\mathbb{Z}_{p}\to\mathrm{H}^{2}_{\mathrm{cris}}(X/W(\kappa))^{F=p}$
is injective and torsion-free. Hence the statement for
$\mathrm{NS}(X)_{\mathbb{Q}}$ implies the statement for
$\mathrm{NS}(X)_{\mathrm{tf}}$. ∎

######  Lemma (2.2.14).

Suppose that $S$ is a connected smooth scheme over $W$ with generic point
$\eta$, $s\in S(k)$ is a closed point, and $\mathscr{P}(S,s,\xi)$ dominates
$S$. Then $\xi$ deforms to the formal completion of $S$ at $s$.

###### Proof.

Since $\mathscr{P}(S,s,\xi)$ dominates $S$, $\xi$ is obtained by specializing
a line bundle on $\mathcal{X}_{\bar{\eta}}$. Up to replacing $S$ by a
connected étale cover and $s$ by a lift, we may assume that
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathrm{tf}}=\mathrm{NS}(\mathcal{X}_{\eta})_{\mathrm{tf}}$.
As $\mathrm{NS}(\mathcal{X}_{s})$ has no $p$-torsion, for some
$m\in\mathbb{N}$ with $p\nmid m$, $m\xi$ lies in the image of
$\mathrm{NS}(\mathcal{X}_{\eta})\to\mathrm{NS}(\mathcal{X}_{s})$. As every
line bundle on $\mathcal{X}_{\eta}$ extends to a global line bundle over $S$,
the conclusion follows. ∎

##  3 Jets and Period Maps

### 3.1 Jet Spaces

Our arguments will use _jet spaces_ , which we now define. Given a ring $R$,
we set

$A^{d}_{r,R}=R[t_{1},\ldots,t_{d}]/(t_{1},\ldots,t_{d})^{r+1},$

and write $\mathbb{D}^{d}_{r,R}$ for $\mathrm{Spec}A^{d}_{r,R}$. If $S$ is an
$R$-scheme, the _jet space_ $J^{d}_{r}S$ associated to the non-negative
integers $d,r\geq 0$ and $S$ is defined to be the $R$-scheme representing the
functor $\textsf{Sch}_{R}\to\textsf{Set}$ given by

$T\mapsto\mathrm{Hom}_{R}(T\times_{R}\mathbb{D}^{d}_{r,R},S),\hskip
15.00002pt[T\to
T^{\prime}]\mapsto[\mathrm{Hom}_{R}(T^{\prime}\times_{R}\mathbb{D}^{d}_{r,R},S)\to\mathrm{Hom}_{R}(T\times_{R}\mathbb{D}^{d}_{r,R},S)],$

where the natural map
$\mathrm{Hom}_{R}(T^{\prime}\times_{R}\mathbb{D}^{d}_{r,R},S)\to\mathrm{Hom}_{R}(T\times_{R}\mathbb{D}^{d}_{r,R},S)$
obtained by pulling back along $T\times_{R}\mathbb{D}^{d}_{r,R}\to
T^{\prime}\times_{R}\mathbb{D}^{d}_{r,R}$. To check representability of the
functor one may produce as in [Urb21, Prop 2.2] an isomorphism to the Weil
restriction functor
$\textrm{Res}_{\mathbb{D}^{d}_{r,R}/R}(\mathbb{D}^{d}_{r,R}\times_{R}(-))$,
and hence reduce to the corresponding representability questions for Weil
restrictions. For our purposes it will be enough that $J^{d}_{r}S$ exists when
$S$ is a quasi-projective $R$-scheme.

A few basic properties of the functors $J^{d}_{r}$ which are useful and
immediately checked from the definition are as follows:

  * (i)

given a ring map $R\to R^{\prime}$, we have that
$J^{d}_{r}(S_{R^{\prime}})=(J^{d}_{r}S)_{R^{\prime}}$ (c.f. [BHJ22, Prop
6.2.2(2)] for the Weil restriction version);

  * (ii)

each map $\mathbb{D}^{d}_{r,R}\to\mathbb{D}^{d^{\prime}}_{r^{\prime},R}$
between infinitesimal disks induces a morphism of functors
$J^{d^{\prime}}_{r^{\prime}}\to J^{d}_{r}$ by pre-composition; and

  * (iii)

given a further composition
$\mathbb{D}^{d^{\prime}}_{r^{\prime},R}\to\mathbb{D}^{d^{\prime\prime}}_{r^{\prime\prime},R}$
the maps $J^{d^{\prime}}_{r^{\prime}}\to J^{d}_{r}$,
$J^{d^{\prime\prime}}_{r^{\prime\prime}}\to J^{d^{\prime}}_{r^{\prime}}$ and
$J^{d^{\prime\prime}}_{r^{\prime\prime}}\to J^{d}_{r}$ are related by natural
transformations in the obvious way.

Denote by $\pi^{r}_{r-k}$ the natural map $J^{d}_{r}\to J^{d}_{r-1}$ induced
by the natural reduction map $A^{d}_{r,R}\to A^{d}_{r-k,R}$ sending each
$t_{i}$ to $t_{i}$. If $r=k$, we simply write $\pi:J^{d}_{r}S\to J^{d}_{0}S=S$
for $\pi^{r}_{0}$. Given $j\in J^{d}_{r}S(R)$, we also write $j(0)$ for the
image $\pi(j)$ in $S(R)$.

Lastly, we write $J^{d}_{r,nd}S\subset J^{d}_{r}S$ for the open subscheme of
_non-degenerate_ jets, and $J^{d}_{r,c}S\subset J^{d}_{r}S$ for the closed
subscheme of constant jets; see [Urb22, Def. 3.4] and [Urb22, §3] for precise
definitions. The field-valued points of $J^{d}_{r,nd}S$ can be identified with
maps from $\mathbb{D}^{d}_{r}$ to $S$ which induce embeddings on the level of
tangent spaces, and the field-valued points of $J^{d}_{r,c}S$ can be
identified with maps from $\mathbb{D}^{r}_{d}$ to $S$ which are constant. The
construction of both $J^{d}_{r,nd}S$ and $J^{d}_{r,c}S$ is compatible with
base-change in the sense of (i) above.

### 3.2 Period Jets

In this section we fix an integral domain $R$, a smooth $R$-scheme $S$ and the
data $(\mathcal{H},\mathrm{Fil}^{\bullet},\nabla)$ where $\mathcal{H}$ is a
vector bundle, $\mathrm{Fil}^{\bullet}$ is a decreasing filtration on
$\mathcal{H}$ and an $R$-linear integrable connection
$\nabla:\mathcal{H}\to\mathcal{H}{\otimes}\Omega_{S/R}$. We additionally write
$\widecheck{L}$ for the $R$-scheme parameterizing flags on $R^{m}$ (graded
locally free modules) with graded quotients of the same dimension as
$\mathrm{Fil}^{\bullet}$ (c.f. [Urb22, §1.2]). Below we shall fix some
$r\in\mathbb{N}$ and assume that $r!$ is invertibe in $R$.

######  Definition (3.2.1).

Let $k$ be a field over $R$ and $s\in S(k)$. Let
$\Delta^{r}_{s}:=\mathrm{Spec}(\mathcal{O}_{S_{k},s}/\mathfrak{m}^{r+1})$,
where $\mathfrak{m}$ is the maximal ideal of the local ring
$\mathcal{O}_{S_{k},s}$. Given a frame (i.e., a basis)
$b^{\bullet}=\\{b^{1},\cdots,b^{m}\\}$ of $\mathcal{H}_{s}$, there is a unique
way of extending $b^{\bullet}$ to a set of horizontal generators of
$\mathcal{H}|_{\Delta^{r}_{s}}$, which induces a horizontal trivialization
$(\mathcal{H},\nabla)|_{\Delta^{r}_{s}}\,{\cong}\,(\mathcal{O}_{\Delta^{r}_{s}},d)^{\oplus
m}$.333This fact can be dug out from [Urb22, Lem. 2.6]. A more direct
alternative argument is given in the appendix. Then the filtration
$\mathrm{Fil}^{\bullet}\mathcal{H}$ restricts to a filtration on
$(\mathcal{O}_{\Delta^{r}_{s}})^{\oplus m}$, which gives rise to a morphism
$\psi_{b^{\bullet}}:\Delta^{r}_{s}\to\widecheck{L}_{k}$. We call
$\psi_{b^{\bullet}}$ the $r$th order inifinitesimal period morphism ($r$-limp)
associated to the frame $b^{\bullet}$.

#### (3.2.2)

We will study $r$-limps through the maps of jets they induce. For any
$d\in\mathbb{N}$, consider the $\mathrm{GL}_{m,R}$-torsor
$\gamma^{d}_{r}:\mathcal{P}^{d}_{r}\to J^{d}_{r}S$ obtained as the base-change
along $J^{d}_{r}S\to S$ of the frame bundle of $\mathcal{H}$. Then for every
field $k$, there is a natural bijection between $\mathcal{P}^{d}_{r}(k)$ and
pairs of the form $(j,b^{\bullet})$, where $j$ is a morphism
$\mathbb{D}^{d}_{r,k}\to S_{k}$ and $b^{\bullet}$ is a frame on
$\mathcal{H}_{s}$ for $s:=j(0)$. Note that $j:\mathbb{D}^{d}_{r,k}\to S_{k}$
uniquely factors through $\mathbb{D}^{d}_{r,k}\to\Delta^{r}_{s}\subseteq
S_{k}$. We can then define a natural map
$\alpha^{d}_{r}(k):\mathcal{P}^{d}_{r}(k)\to J^{d}_{r}\widecheck{L}(k)$ by
sending each $(j,b^{\bullet})$ to the composition

$\mathbb{D}^{d}_{r,k}\stackrel{{\scriptstyle
j}}{{\to}}\Delta^{r}_{s}\stackrel{{\scriptstyle\psi_{b^{\bullet}}}}{{\to}}\widecheck{L}_{k}.$

The following result says that the maps $\alpha^{d}_{r}(k)$ “glue” in some
natural way, and can be obtained by unpacking [Urb22, Thm 1.14] and its proof:

######  Theorem (3.2.3).

Associated to the triple $(\mathcal{H},\mathrm{Fil}^{\bullet},\nabla)$, there
exists a $\mathrm{GL}_{m,R}$-equivariant morphism
$\alpha^{d}_{r}:\mathcal{P}^{d}_{r}\to J^{d}_{r}\widecheck{L}$ whose induced
map on $k$-points for each algebraically closed field $k$ is
$\alpha^{d}_{r}(k)$ as above. The resulting morphism

$\eta^{d}_{r}:J^{d}_{r}S\to\mathrm{GL}_{m,R}\backslash J^{d}_{r}\widecheck{L}$

of Artin stacks over $R$ is compatible with

  * -

base-change along maps $R\to R^{\prime}$ of $\mathbb{Z}[1/r!]$-algebras;

  * -

pullback along maps $g:S^{\prime}\to S$ of smooth $R$-schemes, in the sense
that $\eta^{d}_{r}\circ J^{d}_{r}g$ is the map associated to
$(g^{*}\mathcal{H},g^{*}\mathrm{Fil}^{\bullet},g^{*}\nabla)$; and

  * -

maps of functors $J^{d}_{r}\to J^{d^{\prime}}_{r^{\prime}}$ for
$r^{\prime}\leq r$ induced by maps
$\mathbb{D}^{d^{\prime}}_{r^{\prime},R}\to\mathbb{D}^{d}_{r,R}$, in the sense
that the natural square involving
$J^{d}_{r}S,J^{d^{\prime}}_{r^{\prime}}S,\mathrm{GL}_{m}\backslash
J^{d}_{r}\widecheck{L}$ and $\mathrm{GL}_{m}\backslash
J^{d^{\prime}}_{r^{\prime}}\widecheck{L}$ commutes.

##  4 Proofs of Theorems

### 4.1 General Setup

Let $R\subseteq\mathbb{C}$ be a sub-$\mathbb{Z}[1/N]$-algebra for some
$N\in\mathbb{N}_{>0}$ and let $F\subseteq\mathbb{C}$ be its fraction field.
Suppose that $\mathcal{S}$ is a smooth quasi-projective $R$-scheme such that
$\mathcal{S}_{\mathbb{C}}$ is connected and $f:\mathcal{X}\to\mathcal{S}$ be
an admissible family of relative dimension $d_{0}$. Choose a polarization
(i.e., a relatively ample line bundle) $\boldsymbol{\xi}_{0}$ on
$\mathcal{X}/\mathcal{S}$. Let $\mathcal{H}$ be the filtered flat vector
bundle $R^{2}f_{*}\Omega^{\bullet}_{\mathcal{X}/\mathcal{S}}$ on
$\mathcal{S}$, equipped with a Hodge filtration $\mathrm{Fil}^{\bullet}$ and
Gauss-Manin connection $\nabla$. We let $\eta$ be the generic point of
$\mathcal{S}$ and $\bar{\eta}$ is a geometric point over $\eta$.

Let $\mathbb{V}=(\mathbb{V}_{B},\mathbb{V}_{\mathrm{dR}})$ be the VHS given by
on $\mathbb{V}_{B}:=R^{2}f_{\mathbb{C}*}\mathbb{Q}$ and
$\mathbb{V}_{\mathrm{dR}}:=\mathcal{H}_{\mathrm{dR}}|_{\mathcal{S}_{\mathbb{C}}}$.
Note that $\mathbb{V}_{B}$ has a natural integral structure given by
$(R^{2}f_{\mathbb{C}*}\mathbb{Z})_{\mathrm{tf}}$, where “tf” standards for the
torsion-free quotient. Using $\boldsymbol{\xi}_{0}$, one defines a symmetric
bilinear pairing on $\mathbb{V}$ via the usual formula $\langle
x,y\rangle=x\cup y\cup c_{1}(\boldsymbol{\xi}_{0,\mathbb{C}})^{d_{0}-2}$,
where $x,y$ are local sections. As $\boldsymbol{\xi}_{0}$ is defined over $S$,
using the same formula one defines a symmetric bilinear pairing on
$\mathcal{H}$. We impose the following assumptions and notations.

  1. (i)

$f_{\mathbb{C}}$ has large monodromy and sufficiently large period image.

  2. (ii)

$\mathrm{Mon}(\mathbb{V}_{B})$ is connected.

  3. (iii)

The pairing on $\mathcal{H}_{\mathrm{dR}}$ induced by $\boldsymbol{\xi}_{0}$
is self-dual.

  4. (iv)

Every prime dividing the order of
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathrm{tor}}$ is a prime factor of
$N$, and $\mathrm{NS}(\mathcal{X}_{\bar{\eta}})[N^{-1}]$ equipped with the
pairing induced by $\boldsymbol{\xi}_{0}$ is a self-dual as a quadratic
$\mathbb{Z}[1/N]$-lattice.

  5. (v)

The natural maps
$\mathrm{NS}(\mathcal{X}_{\eta})_{\mathrm{tf}}\to\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathrm{tf}}$
and
$\mathrm{NS}(\mathcal{X}_{\eta})[N^{-1}]\to\mathrm{NS}(\mathcal{X}_{\bar{\eta}})[N^{-1}]$
are isomorphisms.

  6. (vi)

The relative de Rham Chern class map sends444Choose a section of
$\mathrm{NS}(\mathcal{X}_{\eta})\to\mathrm{Pic}(\mathcal{X}_{\eta})=\mathrm{Pic}(\mathcal{X})/\mathrm{Pic}(\mathcal{S})$.
The choice is unimportant.
$\mathrm{NS}(\mathcal{X}_{\eta}){\otimes}_{\mathbb{Z}}\mathcal{O}_{\mathcal{S}}$
isomorphically onto an orthogonal direct summand
$\mathcal{H}^{0}_{\mathrm{dR}}$ of $\mathcal{H}_{\mathrm{dR}}$.

Note that $\mathcal{H}^{0}$ is a filtered flat sub-bundle of $\mathcal{H}$. In
fact $\mathcal{H}^{0}\subseteq\mathrm{Fil}^{2}\mathcal{H}$. Let
$\eta_{\mathbb{C}}$ be the generic point of $\mathcal{S}_{\mathbb{C}}$. Then
the relative Chern class of
$\mathrm{NS}(\mathcal{X}_{\eta_{\mathbb{C}}})_{\mathrm{tf}}$ gives rise to a
sub-VHS $\mathbb{V}^{0}=(\mathbb{V}^{0}_{B},\mathbb{V}^{0}_{\mathrm{dR}})$ of
$\mathbb{V}$. Condition (iii) implies that the natural map
$\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}\to\mathrm{NS}(\mathcal{X}_{\eta_{\mathbb{C}}})_{\mathbb{Q}}$
is also an isomorphism, so that
$\mathrm{NS}(\mathcal{X}_{\eta})_{\mathrm{tf}},\mathrm{NS}(\mathcal{X}_{\eta_{\mathbb{C}}})_{\mathrm{tf}}$
and $\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathrm{tf}}$ can all be
identified. This implies that
$\mathbb{V}^{0}_{\mathrm{dR}}=\mathcal{H}^{0}|_{\mathcal{S}_{\mathbb{C}}}$. We
denote by $\mathbb{V}^{2}=(\mathbb{V}^{2}_{B},\mathbb{V}^{2}_{\mathrm{dR}})$
and $\mathcal{H}^{2}$ the orthogonal complements of $\mathbb{V}^{0}$ and
$\mathcal{H}^{0}$ in $\mathbb{V}$ and $\mathcal{H}$ respectively, and write
the natural projections $\mathbb{V}\to\mathbb{V}^{2}$ and
$\mathcal{H}\to\mathcal{H}^{2}$ both as $\pi_{2}$. We write $h^{0,2}$ for
$h^{0,2}(\mathbb{V})=h^{0,2}(\mathbb{V}^{2})$. Note that as $\mathbb{V}^{0}$
is a trivial VHS,
$\mathrm{Mon}(\mathbb{V}_{B},s)=\mathrm{Mon}(\mathbb{V}^{2}_{B},s)$ for every
$s\in\mathcal{S}(\mathbb{C})$.

######  Definition (4.1.1).

Let $s\to\mathcal{S}$ be a geometric point and let $p:=\mathrm{char\,}k(s)$.
Here we allow $p=0$, in which case $\mathbb{Z}_{(p)}=\mathbb{Q}$. Recall that
there is a well defined specialization map
$\mathrm{NS}(\mathcal{X}_{\eta})\to\mathrm{NS}(\mathcal{X}_{s})$ which is
injective after inverting $p$. Denote the image by
$\mathrm{NS}(\mathcal{X}_{\eta})_{s}$. Then since by assumption (iv)
$\mathrm{NS}(\mathcal{X}_{\eta}){\otimes}\mathbb{Z}_{(p)}$ is a torsion-free
self-dual quadratic lattice over $\mathbb{Z}_{(p)}$,
$\mathrm{NS}(\mathcal{X}_{\eta}){\otimes}\mathbb{Z}_{(p)}$ maps isomorphically
onto $\mathrm{NS}(\mathcal{X}_{\eta})_{s}{\otimes}\mathbb{Z}_{(p)}$, and there
is an orthogonal decomposition

$\mathrm{NS}(\mathcal{X}_{s}){\otimes}\mathbb{Z}_{(p)}=[\mathrm{NS}(\mathcal{X}_{\eta})_{s}{\otimes}\mathbb{Z}_{(p)}]\oplus[\mathrm{NS}(\mathcal{X}_{\eta})_{s}{\otimes}\mathbb{Z}_{(p)}]^{\perp}.$
(3)

For $\xi\in\mathrm{Pic}(\mathcal{X}_{s})$, we say that $\xi$ is power-
primitive if its class lies in
$[\mathrm{NS}(\mathcal{X}_{\eta})_{s}{\otimes}\mathbb{Z}_{(p)}]^{\perp}$ and
is nonzero mod $p$.

######  Lemma (4.1.2).

Consider the setting in (4.1.1) and write $k$ for $k(s)$.

  1. (a)

The map
$c_{1,\mathrm{dR}}{\otimes}k:\mathrm{NS}(\mathcal{X}_{s}){\otimes}_{\mathbb{Z}}k\to\mathcal{H}_{s}=\mathrm{H}^{2}_{\mathrm{dR}}(X/k)$
sends $\mathrm{NS}(\mathcal{X}_{\eta})_{s}{\otimes}_{\mathbb{Z}}k$
isomorphically onto $\mathcal{H}^{0}_{s}\subseteq\mathcal{H}_{s}$.

  2. (b)

If $\xi\in\mathrm{Pic}(\mathcal{X}_{s})$ is power-primitive, then
$c_{1,\mathrm{dR}}(\xi)\in\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}_{s}/k(s))=\mathcal{H}_{s}$
is nonzero and lies in $\mathcal{H}^{2}_{s}\subseteq\mathcal{H}_{s}$.

  3. (c)

For every $\xi\in\mathrm{NS}(\mathcal{X}_{s})$, there exists some
$m\in\mathbb{N}\backslash p\mathbb{N}$ such that
$m\xi=\xi_{0}+p^{h}\xi^{\prime}$ for some $\xi_{0}$ whose class lies in
$\mathrm{NS}(\mathcal{X}_{\eta})_{s}$, $h\in\mathbb{N}$ and a power-primitive
$\xi^{\prime}$.

###### Proof.

(a) follows from the definition of $\mathcal{H}^{0}$ and the commutativity of
taking de Rham Chern class with pullback. (b) As we assumed that
$\mathcal{X}\to\mathcal{S}$ is admissible, the Hodge-de Rham spectral sequence
of $\mathcal{X}_{s}$ degenerates at $E_{1}$-page and
$\mathrm{H}^{*}_{\mathrm{cris}}(\mathcal{X}_{s}/W(k)$ is torsion-free. Then by
[Del81, Rmk 3.5] (see also [Ill79, Cor. 5.18] and [Ogu79, Cor. 1.4]), the map
$c_{1,\mathrm{dR}}:\mathrm{NS}(\mathcal{X}_{s}){\otimes}\mathbb{F}_{p}\to\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}_{s}/k)$
is injective. This implies that $c_{1,\mathrm{dR}}(\xi)$ is nonzero. Moreover,
from (a) it is clear that $c_{1,\mathrm{dR}}(\xi)$ is orthogonal to
$\mathcal{H}^{0}_{s}$ and hence lies in $\mathcal{H}^{2}_{s}$. (c) is a direct
consequence of (3). ∎

We set $\widecheck{L}$ to be $R$-scheme parameterizing flags on $R^{m}$ with
the same Hodge numbers as $\mathbb{V}^{2}$. Let $s$ be a point on
$\mathcal{S}$ valued over a field $k$. Then for a chosen basis $b^{\bullet}$
of $\mathcal{H}^{2}_{s}$, we have a $r$-limp map
$\psi_{b^{\bullet}}:\Delta^{r}_{s}\to\check{L}_{k}$ for any $r>1$ such that
$r!$ is invertible in $k$ (see (3.2.1)). Below we shall relate this map to the
Kodaira-Spencer map.

#### (4.1.3)

The map
$T_{s}\mathcal{S}_{k(s)}\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega_{\mathcal{X}_{s}}),\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}}))$
that we call the Kodaira-Spencer map is given by the composition

$T_{s}\mathcal{S}_{k(s)}\to\mathrm{H}^{1}(T_{\mathcal{X}_{s}})\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega_{\mathcal{X}_{s}}),\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}}))$
(4)

where the first map exists because $H^{1}(X_{s},T_{X_{s}})$ classifies
deformations of $\mathcal{X}_{s}$ over the dual numbers
$k(s)[\varepsilon]/(\varepsilon^{2})$ and the second map is given by the cup
product pairing. In fact, using cup product again we can extend to a map

$T_{s}\mathcal{S}_{k(s)}\to\mathrm{H}^{1}(T_{\mathcal{X}_{s}})\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega_{\mathcal{X}_{s}}),\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}}))\oplus\mathrm{Hom}(\mathrm{H}^{0}(\Omega_{\mathcal{X}_{s}}^{2}),\mathrm{H}^{1}(\Omega^{1}_{\mathcal{X}_{s}})).$
(5)

We now relate the above notion of Kodaira-Spencer mapping to its Hodge-
theoretic variant. Given a vector space $V$ over a field and a flag variety
$\mathcal{F}$ parameterizing two-step flags $F^{\bullet}=\\{V=F^{0}V\supseteq
F^{1}V\supseteq F^{2}V\\}$ on $V$, its tangent space
$T_{F^{\bullet}}\mathcal{F}$ at $F^{\bullet}$ is naturally identified with

$(\mathrm{Hom}(F^{1}/F^{2},F^{0}/F^{1})\oplus\mathrm{Hom}(F^{2},F^{1}/F^{2}))\oplus\mathrm{Hom}(F^{2},F^{0}/F^{1})$

(c.f. [GGK12, Chap II]). When $V$ is taken to be
$\mathcal{H}_{s}=\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}_{s}/k(s))$, and
$F^{\bullet}$ is taken to be the Hodge filtration $\mathrm{Fil}^{\bullet}$,
the variation of Hodge filtrations on $\mathcal{H}$ at $s$ gives us a map
$T_{s}\mathcal{S}_{k(s)}\to T_{F^{\bullet}}\mathcal{F}$, which is also called
the Kodaira-Spencer map. Griffiths transversality guarantees that this map
factors through the first two summands, so we obtain a map

$T_{s}\mathcal{S}_{k(s)}\to\mathrm{Hom}(\mathrm{Fil}^{1}\mathcal{H}_{s}/\mathrm{Fil}^{2}\mathcal{H}_{s},\mathcal{H}_{s}/\mathrm{Fil}^{1}\mathcal{H}_{s})\oplus\mathrm{Hom}(\mathrm{Fil}^{2}\mathcal{H}_{s},\mathrm{Fil}^{1}\mathcal{H}_{s}/\mathrm{Fil}^{2}\mathcal{H}_{s}).$
(6)

In fact, this agrees with (5) via the natural identifications
$\mathrm{Fil}^{2}\mathcal{H}_{s}=\mathrm{H}^{0}(\Omega^{2}_{\mathcal{X}_{s}})$,
$\mathrm{Fil}^{1}\mathcal{H}_{s}/\mathrm{Fil}^{2}\mathcal{H}_{s}=\mathrm{H}^{1}(\Omega^{1}_{\mathcal{X}_{s}})$,
and
$\mathcal{H}_{s}/\mathrm{Fil}^{1}\mathcal{H}_{s}=\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}})$
provided by the degeneration of Hodge-de Rham spectral sequence of
$\mathcal{X}_{s}$ at $E_{1}$ page. Deligne explained this in [Del81, (2.3.12)]
in the context of deforming K3 surfaces but the argument easily generalizes.
Moreover, since the pairing on $\mathcal{H}$ induced by the polarization
$\boldsymbol{\xi}_{0}$ is horizontal, and under this pairing
$\mathrm{H}^{1}(\Omega_{\mathcal{X}_{s}}^{1})$ is self-dual and
$\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}})$ is dual to
$\mathrm{H}^{0}(\Omega_{\mathcal{X}_{s}}^{2})$, the two factors
$T_{s}\mathcal{S}_{k(s)}\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega_{\mathcal{X}_{s}}),\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}}))$
and
$T_{s}\mathcal{S}_{k(s)}\to\mathrm{Hom}(\mathrm{H}^{0}(\Omega_{\mathcal{X}_{s}}^{2}),\mathrm{H}^{1}(\Omega^{1}_{\mathcal{X}_{s}}))$
of (5) are dual to each other up to a sign (cf. [Del81, (2.3.14)]). Therefore,
(5) is remembered by (4).

Suppose now we take $V$ to be $\mathcal{H}^{2}_{s}$ and $F^{\bullet}$ to be
the filtration on $\mathcal{H}^{2}_{s}$, which is obtained by restricting that
on $\mathcal{H}_{s}$. Since the orthogonal complement $\mathcal{H}^{0}$ of
$\mathcal{H}^{2}$ in $\mathcal{H}$ lies completely in
$\mathrm{Fil}^{1}\mathcal{H}$,
$\mathrm{Fil}^{2}\mathcal{H}^{2}=\mathrm{Fil}^{2}\mathcal{H}$ and
$\mathcal{H}^{2}/\mathrm{Fil}^{1}\mathcal{H}^{2}=\mathcal{H}/\mathrm{Fil}^{1}\mathcal{H}$.
Moreover, as
$\mathcal{H}^{0}=\mathrm{NS}(\mathcal{X}_{\eta}){\otimes}_{\mathbb{Z}}\mathcal{O}_{\mathcal{S}}$
is generated by horizontal global sections, one easily checks that the map (6)
factors through

$T_{s}\mathcal{S}_{k(s)}\to\mathrm{Hom}(\mathrm{Fil}^{1}\mathcal{H}^{2}_{s}/\mathrm{Fil}^{2}\mathcal{H}^{2}_{s},\mathcal{H}^{2}_{s}/\mathrm{Fil}^{1}\mathcal{H}^{2}_{s})\oplus\mathrm{Hom}(\mathrm{Fil}^{2}\mathcal{H}^{2}_{s},\mathrm{Fil}^{1}\mathcal{H}^{2}_{s}/\mathrm{Fil}^{2}\mathcal{H}^{2}_{s})\subseteq
T_{F^{\bullet}}\mathcal{F},$ (7)

where the right hand side above is naturally viewed as a submodule of the
right hand side of (6). The chosen basis $b^{\bullet}$ of
$\mathcal{H}^{2}_{s}$ gives us an identification
$\mathcal{F}\stackrel{{\scriptstyle b^{\bullet}}}{{\simeq}}\widecheck{L}_{k}$,
and there is a commutative diagram

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### 4.2 Construction of the Atypical Locus

Consider the set-up in §4.1. Let
$d=\dim_{R}\mathcal{S}-(h^{0,2}(\mathbb{V}^{2})-1)$. Write
$\mathcal{P}^{d}_{r,nd}\subset\mathcal{P}^{d}_{r}$ for the fibre of
$\mathcal{P}^{d}_{r}\to J^{d}_{r}\mathcal{S}$ above $J^{d}_{r,nd}\mathcal{S}$.
Let $\widecheck{L}_{v}\subset\widecheck{L}$ denote the subscheme of flags
whose $\mathrm{Fil}^{1}$ part contains the first standard basis vector $v$ of
$R^{m}$.

Let $\Phi^{d}_{r}$ be the graph of the map $\alpha^{d}_{r}$ in (3.2.3)
corresponding to $(\mathcal{H}^{2},\mathrm{Fil}^{\bullet},\nabla)$. Define the
intersections of schemes over $R[1/r!]$

$\displaystyle\mathcal{I}_{0}$
$\displaystyle:=\Phi^{d}_{0}\cap\left(\mathcal{P}^{d}_{r}\times\widecheck{L}_{v}\right)=\Phi^{d}_{0}\cap\left(\mathcal{P}^{d}_{0}\times\widecheck{L}_{v}\right)\text{
for $r=0$}$ $\displaystyle\mathcal{I}_{r}$
$\displaystyle:=\Phi^{d}_{r}\cap\left(\mathcal{P}^{d}_{r,nd}\times(J^{d}_{r}\widecheck{L}_{v}\setminus(\pi^{r}_{1})^{-1}(J^{d}_{1,c}\widecheck{L}_{v}))\right)\text{
for $r>0$}.$

######  Remark (4.2.1).

For readers’ convenience, let us unpack the definitions and give a description
of $\mathcal{I}_{0}$ and $\mathcal{I}_{r}$ on field-valued points. Let $k$ be
a field. Recall the description of $\mathcal{P}^{d}_{r}(k)$ in (3.2.2). Then
$\mathcal{I}_{0}(k)$ corresponds to pairs
$(p_{0},\sigma_{0})\in\mathcal{P}^{d}_{0}(k)\times\widecheck{L}(k)$ such that
$p_{0}$ is given by a pair $(s,b^{\bullet})$ where $s\in S(k)$ and
$b^{\bullet}$ is a frame of $\mathcal{H}_{s}$,
$\psi_{b^{\bullet}}^{-1}(v)\in\mathrm{Fil}^{1}\mathcal{H}_{s}$, and the flag
$\sigma_{0}$ is given by the filtration on $\mathcal{H}_{s}$ via the choice of
frame $b^{\bullet}$. An element of $\mathcal{I}_{r}(k)$ is given by a pair
$(p_{r},\sigma_{r})\in\mathcal{P}^{d}_{r}(k)\times J^{d}_{r}\widecheck{L}(k)$,
such that $p_{r}$ is given by a triple
$(s,b^{\bullet},j_{r}:\mathbb{D}^{d}_{r,k}\to S_{k})$ such that

  * •

$j$ is injective on tangent spaces,

  * •

the composition $\mathbb{D}^{d}_{r,k}\stackrel{{\scriptstyle
j_{r}}}{{\to}}\Delta^{r}_{s}\stackrel{{\scriptstyle\psi_{b^{\bullet}}}}{{\to}}\widecheck{L}_{k}$
is nonconstant when restricted to $\mathbb{D}^{d}_{1,k}$, and

  * •

$(\psi_{b^{\bullet}}\circ j_{r})^{-1}(v)$ as a global section of
$j^{*}\mathcal{H}$ over $\mathbb{D}^{d}_{r,k}$ lies in the $\mathrm{Fil}^{1}$
part.

and $\sigma_{r}=\alpha^{d}_{r}(p_{r})=\psi_{b^{\bullet}}\circ j_{r}$. For
$r\geq r^{\prime}$, there is a natural projection
$\mathcal{I}_{r}\to\mathcal{I}_{r^{\prime}}$ such that on $k$-points it is
given by sending $(p_{r}:=(s,b^{\bullet},j_{r}),\alpha_{r}^{d}(p_{r}))$ to
$(p_{r^{\prime}}:=(s,b^{\bullet},j_{r^{\prime}}),\alpha^{d}_{r^{\prime}}(p_{r^{\prime}}))$,
where $j_{r^{\prime}}=\pi^{r}_{r^{\prime}}(j_{r})$.

######  Remark (4.2.2).

Thanks to the compatibility-with-base-change statements in (3.2.3), the
formation of the tuple of $\mathcal{S}$-schemes
$\mathcal{T}:=(\Phi^{d}_{r},\mathcal{I}_{0},\mathcal{I}_{r})$ commutes with
extension of scalars within $\mathbb{C}$: If $R^{\prime}\subseteq\mathbb{C}$
is a sub-$R$-algebra, then the tuple $\mathcal{T}^{\prime}$ of
$\mathcal{S}{\otimes}_{R}R^{\prime}$-schemes formed by applying the above
constructions to the family
$(f:\mathcal{X}\to\mathcal{S}){\otimes}_{R}R^{\prime}$ is naturally identified
with $\mathcal{T}{\otimes}_{R}R^{\prime}$ in the obvious sense.

We first prove the following, which is a sharpening of the more abstract
statement [Urb22, Prop. 5.8] to the situation which will be of interest to us.

######  Proposition (4.2.3).

There exists $r_{0}$ such that

  * (i)

for every $r\geq r_{0}$, every irreducible component of the scheme
$\mathcal{I}_{r}$ has non-empty generic fiber over $\mathrm{Spec}(R[1/r!])$,
and non-Zariski dense image in $\mathcal{S}_{R[1/r!]}$;

  * (ii)

if $E\subset\mathcal{S}$ is the scheme-theoretic image of
$\mathcal{I}_{r_{0}}\to\mathcal{S}$, then every irreducible component
$E_{0}\subset E_{\mathbb{C}}$ has dimension at least $d$ and the algebraic
monodromy group of
${\left.\kern-1.2pt\mathbb{V}\vphantom{\big{|}}\right|_{E_{0}}}$ is properly
contained in that of $\mathbb{V}$; and

  * (iii)

for every $r\geq r_{0}$ the map $\mathcal{I}_{r,\mathbb{C}}\to E_{\mathbb{C}}$
factors through $J^{d}_{r}E_{\mathbb{C}}$.

The proof requires two notions from Hodge theory, which we now recall.

######  Definition (4.2.4).

Given a variation of Hodge structure $\mathbb{V}$ on a smooth complex
algebraic variety $S$, we say an irreducible complex algebraic subvariety
$Z\subset S$ is weakly special if it is maximal for its algebraic monodromy
group. We say it is special if it is maximal for its Mumford-Tate group (the
Mumford-Tate group of $\mathbb{V}_{s}$ for a very general point $s\in
Z(\mathbb{C})$).

######  Remark (4.2.5).

Special subvarieties correspond to Hodge locus for variations of Hodge
structures in the tensor category generated by $\mathbb{V}$. Since they are
defined by rational tensors, there are countably many. On the other hand there
are in general uncountably many weakly special subvarieties; for instance, if
$\mathbb{V}$ induces a quasi-finite period map, for every point $s\in
S(\mathbb{C})$ the variety $\\{s\\}$ is weakly special.

###### Proof.

If $d\leq 0$ then $\mathcal{I}_{r}$ is empty and there is nothing to show, so
we may assume $d>0$.

The case $R=\mathbb{C}$: We first prove the statement when $R=\mathbb{C}$. Let
$E_{r}$ be the scheme-theoretic image of $\mathcal{I}_{r}\to\mathcal{S}$, set
$E=\bigcap_{r}E_{r}$ and choose $r_{0}$ such that $E=E_{r_{0}}$. Fix some
$t\geq r_{0}$ and consider the intersection of constructible sets ([Sta22,
054J])

$\mathcal{J}_{t}=\bigcap_{r\geq
t}\textrm{im}\left[\mathcal{I}_{r}\to\mathcal{P}^{d}_{t}\times
J^{d}_{t}\widecheck{L}\right].$

Let $(p_{t},\sigma_{t})$ be a $\mathbb{C}$-point in $\mathcal{J}_{t}$.
Applying [Urb21, Lem. 5.3] to the sequence $\\{\mathcal{I}_{r}\\}_{r\geq t}$
at the point $(p_{t},\sigma_{t})$ we obtain a sequence of pairs
$\\{(p_{r},\sigma_{r})\\}_{r\geq t}$, compatible with the projections
$\mathcal{I}_{r}\to\mathcal{I}_{r-1}$ induced by the maps
$\mathcal{P}^{d}_{r}\times
J^{d}_{r}\widecheck{L}\to\mathcal{P}^{d}_{r-1}\times
J^{d}_{r-1}\widecheck{L}$, such that $(p_{r},\sigma_{r})\in\mathcal{I}_{r}$
for all $r\geq t$. (Note that the letter “$E$” in [Urb21, Lem. 5.3] differs
from our $E$.) Let
$\mathbb{D}^{d}_{\infty,\mathbb{C}}:=\varinjlim_{r}\mathbb{D}^{d}_{r}$, which
is (abstractly) isomorphic to
$\mathrm{Spf}(\mathbb{C}[\\![x_{1},\cdots,x_{d}]\\!])$. The sequence
$\\{(p_{r},\sigma_{r})\\}_{r\geq t}$ defines formal objects
$\sigma_{\infty}:\mathbb{D}^{d}_{r,\mathbb{C}}\to\widecheck{L}_{v,\mathbb{C}}$,
$\psi_{\infty}:\hat{\mathcal{S}}_{s}\to\widecheck{L}_{\mathbb{C}}$ and
$j_{\infty}:\mathbb{D}^{d}_{\infty,\mathbb{C}}\to\hat{\mathcal{S}}_{s}$
satisfying $\sigma_{\infty}=\psi_{\infty}\circ j_{\infty}$. By (4.2.1),
$j_{\infty}$ is injective on tangent spaces, which implies that the map
$j_{\infty}$ is an embedding, and $\sigma_{\infty}$ is non-constant.

Write
$(\hat{\mathcal{H}}^{2}_{s},\hat{\mathrm{Fil}}^{\bullet}_{s},\hat{\nabla}_{s})$
for the restriction of the data
$(\mathcal{H}^{2},\mathrm{Fil}^{\bullet},\nabla)$ to $\hat{\mathcal{S}}_{s}$.
Let $b^{1},\ldots,b^{m}\in\mathcal{H}_{s}$ be vectors in $b^{\bullet}$, which
we shall extend to flat sections which trivialize $\hat{\mathcal{H}}^{2}_{s}$.
Choose $A\in\mathrm{GL}_{m}(\mathbb{C})$ so that the $A$-translated basis
$\\{b^{i}_{A}:=A\cdot b^{i}\\}$ is identified under the Betti-de Rham
comparison with an integral basis for $\mathbb{V}^{2}$, and hence
$\psi_{A}:=A\cdot\psi$ is an honest Hodge-theoretic local period map (to be
explained in more detail in the paragraph below). In particular, transferring
the polarization on $\mathbb{V}^{2}$ along the isomorphism
$i_{s}:\mathbb{V}^{2}_{s}\cong\mathbb{Q}^{m}$ induced by
$b^{1}_{A},\ldots,b^{m}_{A}$ to obtain a polarization
$Q:\mathbb{Q}^{m}\otimes\mathbb{Q}^{m}\to\mathbb{Q}$, we obtain a subvariety
$\widecheck{D}\subset\widecheck{L}$ through which $\psi_{A}$ factors
consisting of those Hodge flags satisfying the first Hodge-Riemann bilinear
relation, and by the large monodromy assumption and elementary linear algebra
(c.f. [GGK12, Ch.2, pg.48]) this subvariety is exactly the orbit of the
algebraic monodromy group of $\mathbb{V}$ regarded inside
$\mathrm{GL}_{m}(\mathbb{C})$ using $i_{s}$. (Note here that $\widecheck{L}$
and $\widecheck{D}$ are flag varieties parameterizing flags with the same
Hodge numbers as $\mathbb{V}^{2}$ rather than $\mathbb{V}$, otherwise we would
have to include conditions coming from global flat sections as well.)

We now relate our construction to a Hodge-theoretic situation so that we may
apply the Ax-Schanuel Theorem [BT19] of Bakker-Tsimerman. Below we shall view
$\mathcal{S}(\mathbb{C})$ as a complex analytic space. The formal map
$\psi_{A}$ is in fact convergent on a sufficiently small ball
$B\subset\mathcal{S}(\mathbb{C})$ containing $s$. Here we are identifying the
completion of the ring of analytic functions on $B$ at $s$ with
$\hat{\mathcal{S}}_{s}$. The fact that the $b^{1}_{A},\ldots,b^{m}_{A}$ are
sections of a local system guarantees that they can be represented by power
series which converge on $B$. We denote this convergent morphism
$B\to\widecheck{L}_{\mathbb{C}}$ of complex-analytic spaces still by
$\psi_{A}$, and define
$\widecheck{D}_{v}=\widecheck{D}\cap(A\cdot\widecheck{L}_{v})$. Then if we
consider the intersection of $U=(S\times\widecheck{D}_{v})\cap\Gamma_{A}$
inside $S\times\widecheck{D}$, where $\Gamma_{A}$ denotes the graph of
$\psi_{A}$, the Ax-Schanuel Theorem [BT19] tells us one of the following two
situations happens:

  * (a)

we either have that

$\displaystyle\mathrm{codim}_{\mathcal{S}\times\widecheck{D}}\,U$
$\displaystyle\geq\mathrm{codim}_{\mathcal{S}\times\widecheck{D}}(\mathcal{S}\times\widecheck{D}_{v})+\mathrm{codim}_{\mathcal{S}\times\widecheck{D}}\,\Gamma_{A}$

or equivalently that

$\displaystyle\mathrm{codim}_{\mathcal{S}}\,\psi_{A}^{-1}(\widecheck{D}_{v})$
$\displaystyle\geq\mathrm{codim}_{\widecheck{D}}\widecheck{D}_{v};$

or

  * (b)

the projection of $U$ to $\mathcal{S}$ (and hence the image of $j_{\infty}$)
lies inside a strict (i.e., properly contained in $\mathcal{S}$) weakly
special subvariety of $\mathcal{S}$ (c.f. [KO21, Cor. 4.14] for an equivalence
with the notion of weakly special used in [BT19]).

Now by Lemma (4.2.6) proven below, one has
$\mathrm{codim}_{\widecheck{D}}\widecheck{D}_{v}=h^{2,0}$, and because
$j_{\infty}$ embeds a formal disk of dimension
$d=\dim_{\mathbb{C}}\mathcal{S}-(h^{2,0}-1)$ into
$\psi^{-1}_{A}(\widecheck{D}_{v})$ we must necessarily be in case (b).
Moreover, because $\psi_{A}\circ j_{\infty}$ is non-constant by construction,
the restriction of $\psi_{A}$ to the formal neighbourhood of $s$ in $Z$ is
non-constant. This implies that the restriction
${\left.\kern-1.2pt\psi_{A}\vphantom{\big{|}}\right|_{B_{Z}}}$ of $\psi_{A}$
to some (complex-analytic) open neighbourhood $B_{Z}$ of $s$ in $Z$ is non-
constant everywhere on $B_{Z}$. Let $Z^{\textrm{sm}}$ be the smooth locus $Z$.
As $B_{Z}\cap Z^{\mathrm{sm}}\neq\emptyset$, it follows that $Z^{\textrm{sm}}$
admits a non-constant period map. Hence we have shown:

> Claim: Any point
> $j_{t}\in\mathcal{K}_{t}:=\textrm{im}\left[\mathcal{J}_{t}\to
> J^{d}_{t}\mathcal{S}\right]$ factors through a strict weakly special
> subvariety $Z\subset\mathcal{S}$ whose smooth locus $Z^{\textrm{sm}}$ admits
> a non-constant period map.

Before continuing further, let us pause to note that we also learn from this
argument that $j_{t}$ lies inside $J^{d}_{t}E$. Indeed, consider the closed
analytic locus $C=\psi^{-1}_{A}(\widecheck{D}_{v})$ inside $B$, let
$C_{0}\subset C$ be a component through which $j_{\infty}$ factors, and write
$C^{\textrm{sm}}_{0}$ for its smooth locus. Since $C$ factors $j_{\infty}$ it
is necessarily of dimension $\geq d$, hence so is $C^{\textrm{sm}}_{0}$.
Moreover, each point of $C^{\textrm{sm}}_{0}$ admits a local period map
mapping to $\widecheck{D}_{v}$, and hence admits some $\infty$-limp mapping to
$\widecheck{L}_{v}$. Thus each point in $C^{\textrm{sm}}_{0}$ where
${\left.\kern-1.2pt\psi_{A}\vphantom{\big{|}}\right|_{C^{\textrm{sm}}_{0}}}$
is non-constant to first order — for instance, on the dense locus of
$C^{\textrm{sm}}_{0}$ where it is a submersion onto its image — lies below
$\mathcal{I}_{t}$ and therefore inside $E$. Since $E$ is closed (in the
complex-analytic topology) this proves that $C_{0}\subset E$, and hence $j\in
J^{d}_{t}C_{0}\subset J^{d}_{t}E$. Thus we have shown (iii), and similarly one
sees the dimension claim in (ii) since $C_{0}$ has dimension $\geq d$.

It remains to show (ii). It is a consequence of the Theorem of the Fixed part
that the period map on $Z^{\textrm{sm}}$ being non-constant implies the
algebraic monodromy group of
${\left.\kern-1.2pt\mathbb{V}\vphantom{\big{|}}\right|_{Z^{\textrm{sm}}}}$,
which agrees with the algebraic monodromy group of
${\left.\kern-1.2pt\mathbb{V}\vphantom{\big{|}}\right|_{Z}}$, is non-trivial.
It now follows from the $\mathbb{Q}$-simplicity of the monodromy on $S$ that
every such $Z$ is contained in a strict weakly non-factor weakly special
subvariety $Z^{\prime}$ in the sense of [KOU23, Def. 1.13]. Because the
monodromy group is $\mathbb{Q}$-simple, by [KOU23, Lem 2.5] $Z^{\prime}$ is
special. Since there are countably many special subvarieties, we conclude
there is a countable collection $\\{Z_{i}\\}_{i\in I}$ of strict closed
subvarieties $Z_{i}\subset\mathcal{S}$ such that any strict weakly special
subvariety with non-trivial algebraic monodromy group lies inside one of the
$Z_{i}$. In particular, $\mathcal{K}_{t}\subset\bigcup_{i\in I}J^{d}_{t}Z_{i}$
as a consequence of the claim. The set $\mathcal{K}_{t}$ is a countable
intersection of constructible sets, so its complement $\mathcal{K}^{c}_{t}$ is
a countable increasing union of constructible sets, call them
$\\{T_{j}\\}_{j\in J}$. Then
$J^{d}_{r}\mathcal{S}_{\mathbb{C}}=\left(\bigcup_{j\in
J}T_{j}\right)\cup\left(\bigcup_{i\in I}J^{d}_{t}Z_{i}\right)$. Since the
complex analytic space $J^{d}_{r}\mathcal{S}_{\mathbb{C}}$ is not a countable
union of meagre sets, it follows that $T_{j}$ is not meagre for some $j$. But
then $\mathcal{K}_{t}$ must lie in a proper Zariski closed subset
$\mathcal{Z}\subset J^{d}_{r}\mathcal{S}$. Replacing $\mathcal{K}_{t}$ and all
of $\\{J^{d}_{t}Z_{i}\\}_{i\in I}$ with their intersections with $\mathcal{Z}$
and iterating this argument, we learn that $\mathcal{K}_{t}$ must lie in
finitely many of the sets $\\{J^{d}_{t}Z_{i}\\}_{i\in I}$, say
$J^{d}_{r}Z_{1},\ldots,J^{d}_{r}Z_{j}$.

General $R$: We now return to our original problem. Again let $E_{r}$ be the
scheme-theoretic image of $\mathcal{I}_{r}\to\mathcal{S}$, set
$E=\bigcap_{r}E_{r}$ and choose $r_{0}$ such that $E=E_{r_{0}}$. By (4.2.2)
and the fact that scheme-theoretic image commutes with base change [Sta22, Tag
081I], the base change $E_{\mathbb{C}}$ of $E$ to $\mathbb{C}$ can be viewed
as the “$E$” we discussed above in the $R=\mathbb{C}$ case. Therefore, (ii)
and (iii) hold.

It only remains to check the conclusion that the components of
$\mathcal{I}_{r}$ for $r\geq r_{0}$ with non-empty generic fiber over
$R[1/r!]$, which follows after possibly increasing $r_{0}$. Indeed,
$\mathcal{I}_{r_{0}}$ has only finitely many components with empty generic
fiber over $R[1/r_{0}!]$, and any component of $\mathcal{I}_{r}$ with empty
generic fiber over $R[1/r!]$ must map into such a component of
$\mathcal{I}_{r_{0}}$. ∎

######  Lemma (4.2.6).

Suppose that $\mathbb{Z}^{m}$ is equipped with a pure Hodge structure of
weight $2$ and a polarization $Q$. Let
$\overline{h}=(h^{2,0},h^{1,1},h^{0,2})$ be the Hodge numbers and
$\widecheck{D}$ be the variety of all flags on $\mathbb{C}^{m}$ with Hodge
numbers $\overline{h}$ satisfying the first Hodge-Riemann bilinear relation,
and let $v\in\mathbb{C}^{m}$ be any vector for which there exists
$F^{\bullet}\in\widecheck{D}(\mathbb{C})$ such that $v\in F^{1}$. Then the
locus

$\widecheck{D}_{v}=\\{F^{\bullet}\in\widecheck{D}(\mathbb{C}):v\in F^{1}\\}$

has codimension $h^{2,0}$ in $\widecheck{D}$.

###### Proof.

The complex points of the group $G=\textrm{Aut}(\mathbb{Z}^{m},Q)$ act
transitively on $\widecheck{D}$, so we may reduce to the case where
$v\in\mathbb{Q}^{m}$ is a Hodge vector. The result then follows from [CGG09,
Prop 5.4]. ∎

### 4.3 The Power-primitive Case

We continue to work with the family $f:\mathcal{X}\to\mathcal{S}$ in §4.1, but
with the base ring $R$ set to be $\mathcal{O}_{L}[1/N]$ for some number field
$L\subseteq\mathbb{C}$ and $N\in\mathbb{N}_{>1}$. We shall assume in addition
assume that $\mathcal{O}_{L}[1/N]$ is unramified (or equivalently, étale) over
$\mathbb{Z}[1/N]$. This implies that if $s\to\mathcal{S}$ is an alg-geometric
point and $k:=k(s)$ has characteristic $p>0$, then the morphism
$\mathcal{O}_{L}[1/N]\to k$ determined by $s$ uniquely lifts to a morphism
$\mathcal{O}_{L}[1/N]\to W(k)$, so that we may form
$\mathcal{S}{\otimes}_{\mathcal{O}_{L}[1/N]}W(k)$. Below we shall simply write
it as $\mathcal{S}{\otimes}W(k)$.

######  Lemma (4.3.1).

Let $(s,\xi)$ be a LB-pair for $\mathcal{X}/\mathcal{S}$ such that
$\mathrm{char\,}k=p>0$ for $k:=k(s)$. Then the following statements are
equivalent.

  1. (i)

The class of $\xi$ lies in
$\mathrm{NS}(\mathcal{X}_{s}){\otimes}\mathbb{Z}_{(p)}$ lies in
$\mathrm{NS}(\mathcal{X}_{\eta})_{s}{\otimes}\mathbb{Z}_{(p)}$.

  2. (ii)

Let $\hat{\mathcal{S}}_{s}$ be the formal completion of
$\mathcal{S}{\otimes}W(k)$ at $s$ and
$\mathrm{Def}(\xi)\subseteq\hat{\mathcal{S}}_{s}$ be as defined in (2.2.9) for
the restriction of $\mathcal{X}$ to $\hat{\mathcal{S}}_{s}$. Then
$\mathrm{Def}(\xi)=\hat{\mathcal{S}}_{s}$.

  3. (iii)

$(s,\xi)$ is global on $\mathcal{S}$ in the sense of (2.1.3), i.e.,
$\mathscr{P}(\mathcal{S},s,\xi)$ dominates $\mathcal{S}$.

###### Proof.

(i) $\Rightarrow$ (ii): By (2.2.12) we may replace $\xi$ by a prime-to-$p$
power and assume that the class of $\xi$ lies in
$\mathrm{NS}(\mathcal{X}_{\eta})_{s}$. Moreover, as $\mathrm{Def}(\xi)$
depends only on the class of $\xi$ in $\mathrm{NS}(\mathcal{X}_{s})$, we may
assume that $\xi$ is global on $\mathcal{S}{\otimes}W$ by (2.1.10)(a). Then we
apply (2.2.14).

(ii) $\Rightarrow$ (iii): This is clear, because by (2.2.9), (ii) implies that
the formal completion of $\mathscr{P}_{\mathcal{X}/\mathcal{S}}$ at $\xi$ is
formally smooth over $\hat{\mathcal{S}}_{s}$ and hence
$\mathscr{P}(\mathcal{S},s,\xi)$ dominates $\mathcal{S}$.

(iii) $\Rightarrow$ (i): That $\mathscr{P}(\mathcal{S},s,\xi)$ dominates
$\mathcal{S}$ implies that the class of $\xi$ in
$\mathrm{NS}(\mathcal{X}_{s})$ lies in the image of
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})$ under the specialization map
$\mathrm{sp}_{\gamma}:\mathrm{NS}(\mathcal{X}_{\bar{\eta}})\to\mathrm{NS}(\mathcal{X}_{s})$
for some path $\gamma$ connecting $\bar{\eta}$ and $s$ (2.1.9). However, we
have
$\mathrm{NS}(\mathcal{X}_{\eta}){\otimes}\mathbb{Z}_{(p)}=\mathrm{NS}(\mathcal{X}_{\bar{\eta}}){\otimes}\mathbb{Z}_{(p)}$
by assumption (v) in the setup. ∎

######  Lemma (4.3.2).

Let $E\subset\mathcal{S}$ and $r_{0}\in\mathbb{N}$ be as introduced in (4.2.3)
and assume that $r_{0}!$ is invertible in $R$. Let
$\mathcal{E}\subseteq\mathcal{S}$ be the union of $E$ and the smallest closed
subscheme $E^{\prime}$ containg all points $t\in\mathcal{S}$ such that the
Kodaira-Spencer map has rank
$<h^{0,2}(\mathbb{V})=\mathrm{rank\,}\mathcal{H}/\mathrm{Fil}^{1}\mathcal{H}$.

Let $(s,\xi)$ be an LB-pair over $\mathcal{S}\backslash\mathcal{E}$ and set
$k:=k(s)$. Let $\hat{\mathcal{S}}_{s,k}$ be the formal completion of
$\mathcal{S}_{k}$ at $s$, and let $\mathrm{Def}(\xi)_{k}$ be the scheme-
theoretic image in $\hat{\mathcal{S}}_{s,k}$ of the formal completion
$\hat{\mathscr{P}}^{\xi}_{k}$ of
$\mathscr{P}_{\mathcal{X}/\mathcal{S}}{\otimes}k$ at $(s,\xi)$. If $\xi$ is
power-primitive, then $\mathrm{Def}(\xi)_{k}$ has codimension $h^{0,2}$ in
$\mathcal{S}_{k}$.

###### Proof.

If $\textrm{char}\,k=p>0$ the fact that $\mathrm{Def}(\xi)_{k}$ has at most
codimension $h^{0,2}$ follows from (2.2.9). If $\textrm{char}\,k=0$ the same
fact follows from (2.2.1). We are then reduced to proving the contrapositive
statement which shows that if for an LB-pair $(s,\xi)$ the locus
$\mathrm{Def}(\xi)_{k}$ has codimension less than $h^{0,2}$ in
$\mathcal{S}_{k}$ then $(s,\xi)$ lies over $\mathcal{E}$.

Since the map $\hat{\mathscr{P}}^{\xi}_{k}\to\mathrm{Def}(\xi)_{k}$ is
formally smooth with fiber dimension $h^{0,1}$, the condition that
$\mathrm{Def}(\xi)_{k}$ has codimension less than $h^{0,2}$ is equivalent to
the condition that there is a component $C$ of
$(\mathscr{P}_{\mathcal{X}/\mathcal{S}}{\otimes}k)_{\mathrm{red}}$ containing
$(s,\xi)$ which has dimension greater than $m-h^{0,2}+h^{0,1}$, where $m$ is
the dimension of $\mathcal{S}_{k}$ over $k$. In particular, the scheme-
theoretic image $Z^{\prime}$ of $C$ on $\mathcal{S}_{k}$ has dimension
$>m-h^{0,2}$. To show that $(s,\xi)$ lies over $\mathcal{E}$, it will suffice
to show that $Z^{\prime}$ lies inside $\mathcal{E}$. More concretely, we may
move the pair $(s,\xi)$ along $C$ and assume that

  1. (a)

$s$ lies on the smooth locus of $Z^{\prime}$, and

  2. (b)

$(s,\xi)^{\flat}$ lies on $C$ but not on any other irreducible component of
$\mathscr{P}_{\mathcal{X}/\mathcal{S}}\otimes k$.

Because such $s$ are Zariski dense in $Z^{\prime}$, we may reduce to showing
the claim for LB-pairs $(s,\xi)$ satisfying these conditions. (We note here
that being power-primitive is an open condition.) Note also that
$\mathrm{Def}(\xi)_{k,\mathrm{red}}$ automatically lies inside
$Z^{\prime}=Z^{\prime}_{\mathrm{red}}$, and is therefore isomorphic to
$\mathrm{Spf}(k[\\![x_{1},\ldots,x_{e}]\\!])$ for $e=\dim Z^{\prime}$.

We shall now assume that $s\not\in E^{\prime}_{k}$ and show that $s\in E_{k}$,
which will complete the proof. As $E$ is defined to be the scheme-theoretic
image of $\mathcal{I}_{r_{0}}\to\mathcal{S}$, it suffices to construct a
preimage of $s$ in $\mathcal{I}_{r_{0}}(k)$. Recall the description of the
field-valued points of $\mathcal{I}_{r_{0}}$ in (4.2.1). We may choose an
isomorphism
$\hat{\mathcal{S}}_{s,k}\cong\mathrm{Spf}(k[\\![x_{1},\cdots,x_{m}]\\!])$ such
that the coordinates agree with our trivialization of
$\mathrm{Def}(\xi)_{k,\mathrm{red}}$. Let
$j:\mathbb{D}^{d}_{r_{0},k}\to\mathcal{S}_{k}$ be the jet centered at $s$
defined by $k[\\![x_{1},\cdots,x_{d}]\\!]/(x_{1},\cdots,x_{d})^{r_{0}+1}$ for
these chosen coordinates. In particular, $j$ is clearly injective on tangent
spaces as it is a closed embedding of $\mathbb{D}^{d}_{r_{0},k}$ into
$\mathrm{Def}(\xi)_{k}$ (or $\hat{\mathcal{S}}_{s,k}$). Next, we choose a
basis $b^{\bullet}:=\\{b^{i}\\}$ of $\mathcal{H}^{2}_{s}$ such that
$b^{1}=(\pi_{2}\circ c_{1,\mathrm{dR}})(\xi)$. If $\textrm{char}\,k=p$ this is
allowed because $\xi$ is power-primitive, so that by (4.1.2) $(\pi_{2}\circ
c_{1,\mathrm{dR}})(\xi)\in\mathcal{H}^{2}_{s}$ is nonzero. Now consider the
$r_{0}$-limp $\psi_{b^{\bullet}}:\Delta^{r_{0}}_{s}\to\widecheck{L}_{k}$.

We have a commutative diagram (cf. (8))

${{T_{s}\mathcal{S}_{k}}}$${{\mathrm{Hom}(\mathrm{Fil}^{1}\mathcal{H}^{2}_{s}/\mathrm{Fil}^{2}\mathcal{H}^{2}_{s},\mathcal{H}^{2}_{s}/\mathrm{Fil}^{1}\mathcal{H}^{2}_{s})}}$${{T_{s}\Delta^{r_{0}}_{s}}}$${{T_{\psi_{b^{\bullet}}(s)}\widecheck{L}_{k}}.}$$\scriptstyle{T_{s}(\psi_{b^{\bullet}})}$

By the assumption that $s\not\in E^{\prime}_{k}$,
${T_{s}(\psi_{b^{\bullet}})}$ has rank $\geq h^{0,2}$, and hence cannot vanish
on the tangent space of $\mathrm{Def}(\xi)_{k,\mathrm{red}}$, or that of
$\mathbb{D}^{d}_{r_{0},k}$. This verifies (4.2.1)(ii). Finally, as $\xi$
deforms over $\mathbb{D}^{d}_{r_{0},k}\stackrel{{\scriptstyle
j}}{{\hookrightarrow}}\mathrm{Def}(\xi)_{k}$, $b^{1}=(\pi_{2}\circ
c_{1,\mathrm{dR}})(\xi)$ extends to a flat section of
$\mathrm{Fil}^{1}\mathcal{H}_{k}$ over $\mathbb{D}^{d}_{r_{0},k}$. This
verifies (4.2.1)(iii). Therefore, $((s,b^{\bullet},j),\psi_{b^{\bullet}}\circ
j)$ defines a point in $\mathcal{I}_{r_{0}}(k)$ lying above $s$, as desired. ∎

######  Corollary (4.3.3).

Suppose that in the situation of (4.3.2) $\mathrm{char\,}k=p>0$. Let
$\hat{\mathcal{S}}_{s}$ be the formal completion of
$\mathcal{S}_{W}:=\mathcal{S}{\otimes}W$ at $s$ for $W:=W(k)$, and let
$\mathrm{Def}(\xi)$ be the scheme-theoretic image of $\mathcal{S}_{s}$ of the
formal completion $\hat{\mathscr{P}}^{\xi}$ of
$\mathscr{P}_{\mathcal{X}/\mathcal{S}}{\otimes}W$ at $(s,\xi)$. If $\xi$ is
power-primitive, then $\mathrm{Def}(\xi)$ is flat over $W$.

###### Proof.

By (2.2.9), $\mathrm{Def}(\xi)\subseteq\hat{\mathcal{S}}_{s}$ is cut out by at
most $h^{0,2}$ equations, and by (4.3.2),
$\mathrm{Def}(\xi)_{k}=\mathrm{Def}(\xi){\otimes}k$ has codimension exactly
$h^{0,2}$ in $\hat{\mathcal{S}}_{s}{\otimes}k$. This implies that
$\mathrm{Def}(\xi)$ has codimension exactly $h^{0,2}$ in
$\hat{\mathcal{S}}_{s}$ and hence is Cohen-Macaulay by [Sta22, 02JN].
Moroever, by miracle flatness [Sta22, 00R4] $\mathrm{Def}(\xi)$ is flat. ∎

######  Remark (4.3.4).

It will be clear from (4.4.3) below and the proof therein that a fortiori the
conclusion of (4.3.2) and (4.3.3) continue to hold if the assumption that
$\xi$ is power-primitive is replaced by the more general assumption that
$(s,\xi)$ is not global over $\mathcal{S}$. However, due to the nature of the
jet theoretic arguments, treating the power-primitive case is an important
step in treating this more general case.

### 4.4 Proofs of Main Theorems

Let $\mathcal{S}$ be as in the introduction. Namely, let
$L\subseteq\mathbb{C}$ be a number field, $N\in\mathbb{N}_{\geq 1}$ and
$\mathcal{S}$ be a smooth quasi-projective $\mathcal{O}_{L}[1/N]$-scheme such
that $\mathcal{S}_{\mathbb{C}}$ is connected. Let
$f:\mathcal{X}\to\mathcal{S}$ be a smooth projective morphism with
geometrically connected fibers. Assume that $f_{\mathbb{C}}$ has large
monodromy and sufficiently large period image. Note that to prove (1.1.1) and
(1.1.3), we are allowed to increase $N$, so that we may also assume that the
family is admissible (2.2.4). Moreover, suppose that there is a finite
extension $L^{\prime}$ of $L$ in $\mathbb{C}$, $N^{\prime}\geq N$, and a
smooth scheme $\mathcal{S}^{\prime}$ over
$\mathcal{O}_{L^{\prime}}[1/N^{\prime}]$ such that the generic fiber
$\mathcal{S}^{\prime}_{L^{\prime}}$ remains geometrically connected (as an
$L^{\prime}$-variety), and $\mathcal{S}^{\prime}$ is an étale cover of
$\mathcal{S}{\otimes}_{\mathcal{O}_{L}[1/N]}\mathcal{O}_{L^{\prime}}[1/N^{\prime}]$.
Then to prove the theorems we may replace $(L,N,\mathcal{S})$ by
$(L^{\prime},N^{\prime},\mathcal{S}^{\prime})$ and $\mathcal{X}$ by its
restriction to $\mathcal{S}^{\prime}$.

######  Lemma (4.4.1).

After such a replacement, we may assume that the family
$f:\mathcal{X}\to\mathcal{S}$ satisfies the assumptions in §4.1 with
$R=\mathcal{O}_{L}[1/N]$.

###### Proof.

Only conditions (ii) and (iv) §4.1 need explanation. The other conditions are
automatically satisfied up to increasing $N$. To satisfy (ii), we first find
some connected étale cover $\mathcal{S}^{\prime}_{\mathbb{C}}$ of
$\mathcal{S}_{\mathbb{C}}$ such that
$\mathrm{Mon}(R^{2}f_{\mathbb{C}*}\mathbb{Q}|_{\mathcal{S}^{\prime}_{\mathbb{C}}})$
is connected. Let $\overline{L}\subseteq\mathbb{C}$ be an algebraic closure of
$L$. Since for any $s\in\mathcal{S}(\mathbb{C})$,
$\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{S}_{\overline{L}},s)=\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{S}_{\mathbb{C}},s)$,
such $\mathcal{S}^{\prime}_{\mathbb{C}}$ is defined over $\overline{L}$ and
hence over some finite extension $L^{\prime}$ of $L$ in $\mathbb{C}$. Then by
spreading out, we obtain some $N^{\prime}>N$ and
$\mathcal{O}_{L^{\prime}}[1/N^{\prime}]$-scheme $\mathcal{S}^{\prime}$
admittible an étale $\mathcal{O}_{L}[1/N^{\prime}]$-morphism to $\mathcal{S}$.
Make replacements accordingly. Then to satisfy (iv), we note that the action
of $\mathrm{Gal}_{k(\eta)}=\mathrm{Gal}(\eta/\bar{\eta})$ on
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathrm{tf}}$ factors through a finite
quotient and the surjective open morphism
$\mathrm{Gal}_{k(\eta)}\to\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{S},\bar{\eta})$.
Hence some finite open subgroup of
$\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{S},\bar{\eta})$ acts trivially on
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathrm{tf}}$. Hence by replacing
$\mathcal{S}$ by a connected étale cover, $\mathrm{Gal}_{k(\eta)}$ acts
trivially on $\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathrm{tf}}$. Then we
conclude by (2.2.13). ∎

######  Lemma (4.4.2).

Let $k$ be a field and $S$ be a local $k$-algebra essentially of finite type
with residue field $k$. Then $S$ is a complete intersection (over $k$) in the
sense of [Sta22, 00SD] if and only if its completion $\hat{S}$ is of the form
$k[\\![x_{1},\cdots,x_{n}]\\!]/I$ for some $n$ and ideal $I$ generated by a
regular sequence.

###### Proof.

This can be deduced from [Sta22, 09Q1], since by Cohen structure theorem
[Sta22, 032A] there exists a surjection
$k[\\![x_{1},\cdots,x_{n}]\\!]\twoheadrightarrow S$ for some $n$. ∎

#### (4.4.3)

Now we shall prove (1.1.1) and (1.1.3) under the assumption that
$\mathcal{X}/\mathcal{S}$ satisfies the assumptions in §4.1. Moreover, we may
assume that $\mathcal{O}_{L}[1/N]$ is unramified over $\mathbb{Z}[1/N]$. Let
$k$ be an algebraically closed field of characteristic $p>0$ and write $W$ for
$W(k)$. Let $(s,\xi)$ be an LB-pair defined over $k$ and
$\hat{\mathcal{S}}_{s}$ be the formal completion of
$\mathcal{S}_{W}:=\mathcal{S}{\otimes}W$ at $s$. Let $\mathcal{E}$ be the
proper closed subscheme of $\mathcal{S}$ defined in (4.3.2). Assume that
$s\not\in\mathcal{E}_{k}$. Let $\mathrm{Def}(\xi)$ be the deformation space of
$\xi$ when $\mathcal{X}$ is restricted to $\hat{\mathcal{S}}_{s}$ (cf.
(2.2.9)).

For (1.1.1) what we need to show is that $\mathrm{Def}(\xi)$ admits a point
valued in $\mathcal{O}_{K}$ for some finite extension $K$ over
$K_{0}:=W[1/p]$. By (4.3.1) it suffices to consider the case when the class of
$\xi$ in $\mathrm{NS}(\mathcal{X}_{s}){\otimes}\mathbb{Z}_{(p)}$ does not lie
in $\mathrm{NS}(\mathcal{X}_{\eta})_{s}{\otimes}\mathbb{Z}_{(p)}$, as
otherwise $\mathrm{Def}(\xi)=\hat{\mathcal{S}}_{s}$. As $\mathrm{Def}(\xi)$
depends only on the class of $\xi$ in $\mathrm{NS}(\mathcal{X}_{s})$, using
(4.1.2) we may assume that $m^{\prime}\xi=\xi_{0}+p^{h}\xi^{\prime}$ for some
$\xi_{0}$ whose class lies in $\mathrm{NS}(\mathcal{X}_{\eta})_{s}$, a power-
primitive $\xi^{\prime}$, $h\in\mathbb{N}_{>1}$ and
$m^{\prime}\in\mathbb{N}\backslash p\mathbb{N}$. By (2.2.12) we are free to
replace $\xi$ by a prime-to-$p$ power, so we may assume $m^{\prime}=1$. By
(2.2.14), $\xi_{0}$ deforms to the entire $\hat{\mathcal{S}}_{s}$, so that
$\mathrm{Def}(\xi)=\mathrm{Def}(p^{h}\xi^{\prime})$. Clearly,
$\mathrm{Def}(p^{h}\xi^{\prime})$ contains $\mathrm{Def}(\xi^{\prime})$, so it
suffices to show that $\mathrm{Def}(\xi^{\prime})$ admits some
$\mathcal{O}_{K}$-point for some $K$. Then the conclusion follows from
(4.3.3), which tells us that $\mathrm{Def}(\xi^{\prime})$ is flat over $W$,
and hence we can find an $\mathcal{O}_{K}$-valued point for some finite
extension $K$ of $K_{0}$ by (2.1.5).

Now we use (1.1.1) to prove (1.1.3). Let
$(s,\xi)\to\mathscr{P}_{\mathcal{X}/\mathcal{S}}$ be any LB-pair as in the
first paragraph. Our goal is to show the following two statements.

  1. (a)

The relative Picard scheme $\mathscr{P}_{\mathcal{X}/\mathcal{S}}$ is syntomic
over $\mathcal{O}_{L}[1/N]$ at the point $(s,\xi)^{\flat}$.

  2. (b)

Let $C$ be any irreducible component of
$(\mathscr{P}_{\mathcal{X}/\mathcal{S}})_{\mathrm{red}}$ which does not
dominate $\mathcal{S}$. Then the scheme-theoretic image $Z$ of $C$ on
$\mathcal{S}$ is flat over $\mathcal{O}_{L}[1/N]$ and has codimension exactly
$h^{0,2}:=\mathrm{rank\,}\mathcal{H}/\mathrm{Fil}^{1}\mathcal{H}$.

Again, if the class of $\xi$ in
$\mathrm{NS}(\mathcal{X}_{s}){\otimes}\mathbb{Z}_{(p)}$ lies in
$\mathrm{NS}(\mathcal{X}_{\eta})_{s}{\otimes}\mathbb{Z}_{(p)}$, then by
(4.3.1) $\mathrm{Def}(\xi)=\hat{\mathcal{S}}_{s}$. As
$\hat{\mathscr{P}}^{\xi}$ is formally smooth over $\mathrm{Def}(\xi)$, we have
in this case that $\mathscr{P}_{\mathcal{X}/\mathcal{S}}$ is smooth over
$\mathcal{S}$ at $(s,\xi)^{\flat}$. Therefore, we again restrict to
considering the case when the class of $\xi$ does not lie in
$\mathrm{NS}(\mathcal{X}_{\eta})_{s}{\otimes}\mathbb{Z}_{(p)}$. Note that
(4.3.1) tells us that in this case $(s,\xi)$ is not global over $\mathcal{S}$,
so that none of the irreducible components of
$(\mathscr{P}_{\mathcal{X}/\mathcal{S}})_{\mathrm{red}}$ which contains
$(s,\xi)^{\flat}$ dominates $\mathcal{S}$.

We first argue that, in order to show (a), it suffices to show that
$\mathrm{Def}(\xi){\otimes}k\subseteq\hat{\mathcal{S}}_{s}{\otimes}k$ has
codimension exactly $h^{0,2}$. Recall that we used
$\mathscr{P}(\mathcal{S},s,\xi)$ to denote the localization of
$\mathscr{P}_{\mathcal{X}/\mathcal{S}}$ at $(s,\xi)^{\flat}$ (2.1.3). Let
$\mathfrak{p}\in\mathrm{Spec}(\mathcal{O}_{L}[1/N])$ be the image of
$(s,\xi)^{\flat}\in\mathscr{P}_{\mathcal{X}/\mathcal{S}}$. By [Sta22, 01UE],
to show that $\mathscr{P}_{\mathcal{X}/\mathcal{S}}$ is syntomic over
$\mathrm{Spec}(\mathcal{O}_{L}[1/N])$ at $(s,\xi)^{\flat}$ it suffices to show
the following statements.

  1. (i)

$\mathscr{P}(\mathcal{S},s,\xi)$ is flat over $\mathcal{O}_{L,\mathfrak{p}}$.

  2. (ii)

$\mathscr{P}(\mathcal{S},s,\xi){\otimes}_{\mathcal{O}_{L}}k(\mathfrak{p})$ is
a complete intersection over $k(\mathfrak{p})$.

Note that $s$ induces an embedding $k(\mathfrak{p})\hookrightarrow k$. We
reduce to showing

  1. (iii)

$\hat{\mathscr{P}}^{\xi}$ is flat over $W$, and

  2. (iv)

$\hat{\mathscr{P}}^{\xi}_{k}:=\hat{\mathscr{P}}^{\xi}{\otimes}_{W}k$ is a
complete intersection over $k$.

That (iii) implies (i) is clear, as $\hat{\mathscr{P}}^{\xi}$ is faithfully
flat over $\mathscr{P}(\mathcal{S},s,\xi)$. That (iv) implies (ii) follows
from [Sta22, 00SI] and (4.4.2). Now, using the fact that
$\hat{\mathscr{P}}^{\xi}$ is formally smooth over $\mathrm{Def}(\xi)$, in
(iii) and (iv) we may equivalently replace $\hat{\mathscr{P}}^{\xi}$ by
$\mathrm{Def}(\xi)$. Then both statements would follow if
$\mathrm{Def}(\xi)_{k}=\mathrm{Def}(\xi){\otimes}k$ has codimension exactly
$h^{0,2}$ in $\hat{\mathcal{S}}_{s,k}$, as by (2.2.9)
$\mathrm{Def}(\xi)\subseteq\hat{\mathcal{S}}_{s}$ is cut out by at most
$h^{0,2}$ equations (cf. the argument for (4.3.3)).

Next, we argue that (b) implies (a). Indeed, suppose for the sake of
contradiction that
$\mathrm{Def}(\xi){\otimes}k\subseteq\hat{\mathcal{S}}_{s}{\otimes}k$ has
codimension $<h^{0,2}$. Then we have $\dim\mathrm{Def}(\xi){\otimes}k\geq
m-h^{0,2}+1$. Here $m$ denotes the relative dimension of $\mathcal{S}$ over
$\mathcal{O}_{L}[1/N]$. Recall that $\mathcal{X}\to\mathcal{S}$ is an
admissible family, so that $h^{0,1}$ of its fibers are constant. As
$\hat{\mathscr{P}}^{\xi}$ is formally smooth over $\mathrm{Def}(\xi)$ with
fiber dimension $h^{0,1}$, there exists an irreducible component $C^{\prime}$
of
$(\mathscr{P}_{\mathcal{X}/\mathcal{S}}{\otimes}_{\mathcal{O}_{L}}k(\mathfrak{p}))_{\mathrm{red}}$
which contains $(s,\xi)^{\flat}$ and has dimension $\geq m-h^{0,2}+1+h^{0,1}$.
The scheme-theoretic image of $C^{\prime}$ on $\mathcal{S}_{k(\mathfrak{p})}$
hence as dimension $\geq m-h^{0,2}+1$. Then any irreducible component $C$ of
$(\mathscr{P}_{\mathcal{X}/\mathcal{S}})_{\mathrm{red}}$ which contains
$C^{\prime}$ will not satisfy the condition in (b).

Finally, we prove (b). Note that $Z$ is reduced and irreducible, so that it
has a generic point $\eta_{Z}$. Choose an LB-pair $(t,\zeta)$ such that $t$
lies above $\eta_{Z}$. We first argue that $\mathrm{char\,}\eta_{Z}=0$.
Indeed, otherwise we must have that the pair $(t,\zeta)$ does not lift to
characteristic $0$, which contradicts (1.1.1) that we just proved. Now the
fact that $\mathrm{char\,}\eta_{Z}=0$ automatically implies that $Z$ is flat
over $\mathcal{O}_{L}[1/N]$ by (2.1.4), as $Z$ is the scheme-theoretic image
of the morphism $\eta_{Z}\to\mathcal{S}$. Moreover, we argue that $Z$ has
codimension exactly $h^{0,2}$ in $\mathcal{S}$. We reduce to showing that
$Z_{k(t)}$ has codimension exactly $h^{0,2}$ in $\mathcal{S}_{k(t)}$. This
follows from (4.3.2) because $Z_{k(t)}$ is contained in the Zariksi closure of
$\mathrm{Def}(\zeta)_{k(t)}$ on $\mathcal{S}_{k(t)}$. ∎

######  Remark (4.4.4).

Assuming that $\mathcal{O}_{L}[1/N]$ is unramified over $\mathbb{Z}[1/N]$ in
(4.4.3) is not strictly necessary for the argument if one is willing to keep
track of possibly ramified extensions of $W(k(s))$. However, as we already
have to increase $N$ in many other steps, it is not clear that we gain any
generality in doing so.

##  5 The $h^{0,2}=2$ Case

### 5.1 Theorem of the Fixed Part in a $p$-adic Setting

#### (5.1.1)

We shall define a variant of the general setup in §4.1. The main difference is
that we drop the assumption (i), and we will impose a two-step filtration on
various cohomology sheaves.

Let $L\subseteq\mathbb{C}$ be a number field, $N\in\mathbb{N}_{>1}$ and
$\mathcal{T}$ is a connected smooth $\mathcal{O}_{L}[1/N]$-scheme of finite
type. Let $f:\mathcal{X}\to\mathcal{T}$ be an admissible family. Suppose that
we are given a polarization $\boldsymbol{\xi}_{0}$ for this family. Let
$\mathcal{H}_{\mathrm{dR}}$ be the filtered flat vector bundle
$R^{2}f_{*}\Omega^{\bullet}_{\mathcal{X}/\mathcal{T}}$ on $\mathcal{T}$ and
let $\mathcal{H}_{\ell}$ be the $\mathbb{Q}_{\ell}$-local system
$R^{2}(f|_{\mathcal{T}[1/\ell]})_{\textrm{{\'{e}}t}*}\mathbb{Q}_{\ell}$ over
$\mathcal{T}[1/\ell]$. Let
$\mathbb{V}=(\mathbb{V}_{B},\mathbb{V}_{\mathrm{dR}})$ be the VHS given by on
$\mathbb{V}_{B}:=R^{2}f_{\mathbb{C}*}\mathbb{Q}$ and
$\mathbb{V}_{\mathrm{dR}}:=\mathcal{H}_{\mathrm{dR}}|_{\mathcal{T}_{\mathbb{C}}}$.
As in §4.1, $\boldsymbol{\xi}_{0}$ defines a symmetric bilinear pairing on
$\mathbb{V}$, $\mathcal{H}_{\mathrm{dR}}$ and $\mathcal{H}_{\ell}$. Let $\eta$
be the generic point of $\mathcal{T}$ and $\bar{\eta}$ be the geometric point
defined by a chosen algebraic closure of $k(\eta)$. Assume the following
conditions:

  1. (i)

$\mathcal{O}_{L}$ is unramified at every finite prime
$\mathfrak{p}\in\mathrm{Spec}(\mathcal{O}_{L}[1/N])$.

  2. (ii)

The pairing induced on $\mathcal{H}_{\mathrm{dR}}$ is self-dual.

  3. (iii)

$\mathrm{Mon}(\mathbb{V}_{B},s)$ is connected for some (and hence every)
$s\in\mathcal{T}(\mathbb{C})$.

  4. (iv)

For some (and hence every, by [LP92, Prop. 6.14]) prime $\ell$,
$\mathrm{Mon}(\mathcal{H}_{\ell}|_{\mathcal{T}_{L}},\bar{\eta})$ is connected.

  5. (v)

There exists a section $b:\mathrm{Spec}(\mathcal{O}_{L}[1/N])\to\mathcal{T}$.

  6. (vi)

Every prime dividing the order of
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathrm{tor}}$ is a factor of $N$.

Note that these conditions can always be achieved up to enlarging $N$ and $L$,
and replacing $\mathcal{T}$ by a connected étale cover. For each closed point
$\mathfrak{p}\in\mathrm{Spec}(\mathcal{O}_{L}[1/N])$, we denote by
$f_{\mathfrak{p}}$ the mod $\mathfrak{p}$ reduction of $f$ and
$\hat{\mathcal{O}}_{L,\mathfrak{p}}$ the completion of $\mathcal{O}_{L}$ at
$\mathfrak{p}$. Let
$\mathcal{T}_{\mathfrak{p}}:=\mathcal{T}{\otimes}k(\mathfrak{p})$ and
$\hat{\mathcal{T}}_{\mathfrak{p}}:=\mathcal{T}\times_{\mathcal{O}_{L}[1/N]}\mathrm{Spf}(\hat{\mathcal{O}}_{L,\mathfrak{p}})$.
We identify $\hat{\mathcal{O}}_{L,\mathfrak{p}}$ with the ring of Witt vectors
$W(k(\mathfrak{p}))$ of the residue field $k(\mathfrak{p})$.

Let $\eta_{\mathbb{C}}$ be the generic point of $\mathcal{T}_{\mathbb{C}}$.
Then the relative Chern classes of the elements of
$\mathrm{NS}(\mathcal{X}_{\eta_{\mathbb{C}}})\simeq\mathrm{Pic}(\mathcal{X}_{\mathbb{C}})/\mathrm{Pic}(\mathcal{T}_{\mathbb{C}})$
gives rise to a sub-VHS
$\mathbb{V}^{0}=(\mathbb{V}^{0}_{B},\mathbb{V}^{0}_{\mathrm{dR}})$ of
$\mathbb{V}$. We remark that the assumption (5.1.1)(iv) implies that every
element of $\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathbb{Q}}$ descends to
$\eta$. Moreover, there is a natural identification of
$\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathbb{Q}}$ with
$\mathrm{H}^{0}(\mathbb{V}^{0})$. Let $\mathbb{V}^{f}_{B}$ be the maximal sub-
local system of $\mathbb{V}_{B}$ such that for some (and hence every)
$s\in\mathcal{T}(\mathbb{C})$, $\pi_{1}(\mathcal{T}_{\mathbb{C}},s)$ acts
trivially on $\mathbb{V}^{f}_{B,s}$. By the theorem of the fixed part, there
exists a filtered flat sub-bundle
$\mathbb{V}^{f}_{\mathrm{dR}}\subseteq\mathbb{V}_{\mathrm{dR}}$ such that
$(\mathbb{V}^{f}_{B},\mathbb{V}^{f}_{\mathrm{dR}})$ is a sub-VHS. Moreover, we
have $\mathbb{V}^{0}\subseteq\mathbb{V}^{f}$. We denote the orthogonal
complement of $\mathbb{V}^{0}$ in $\mathbb{V}^{f}$ by $\mathbb{V}^{1}$, and
that of $\mathbb{V}^{f}$ in $\mathbb{V}$ by $\mathbb{V}^{2}$. Note that if
$f_{\mathbb{C}}$ has large monodromy, then $\mathbb{V}^{1}=0$ and
$\mathbb{V}^{0}=\mathbb{V}^{f}$, but we will be primarily interested in the
case when this condition fails.

######  Proposition (5.1.2).

Up to further enlarging $N$, the following conditions hold.

  1. (a)

There exists a decomposition of $\ell$-adic local systems
$\mathcal{H}_{\ell}=\oplus_{i=0}^{2}\mathcal{H}^{i}_{\ell}$ over
$\mathcal{T}[1/\ell]$ whose restriction to $\mathcal{T}_{\mathbb{C}}$ agrees
with
$\mathbb{V}_{B}{\otimes}\mathbb{Q}_{\ell}=\oplus_{i=0}^{2}\mathbb{V}^{i}_{B}{\otimes}\mathbb{Q}_{\ell}$.

  2. (b)

There exists a decomposition
$\mathcal{H}_{\mathrm{dR}}=\oplus_{i=0}^{2}\mathcal{H}^{i}_{\mathrm{dR}}$ of
filtered flat vector bundles whose restriction to $\mathcal{T}_{\mathbb{C}}$
agrees with
$\mathbb{V}_{\mathrm{dR}}=\oplus_{i=0}^{2}\mathbb{V}^{i}_{\mathrm{dR}}$.

  3. (c)

For every finite prime $\mathfrak{p}\subseteq\mathcal{O}_{L}[1/N]$ and
$\mathcal{H}_{\mathfrak{p},\mathrm{cris}}:=R^{2}f_{\mathfrak{p},\mathrm{cris}*}\mathcal{O}_{\mathcal{X}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}}}$,
there exists a decomposition of F-crystals
$\mathcal{H}_{\mathfrak{p},\mathrm{cris}}=\oplus_{i=0}^{2}\mathcal{H}^{i}_{\mathfrak{p},\mathrm{cris}}$
such that under the crystalline-de Rham comparison isomorphism

$\mathcal{H}_{\mathrm{dR}}|_{\hat{\mathcal{T}}_{\mathfrak{p}}}\,{\cong}\,\mathcal{H}_{\mathfrak{p},\mathrm{cris}}(\hat{\mathcal{T}}_{\mathfrak{p}})$
(9)

the component
$\mathcal{H}^{i}_{\mathrm{dR}}|_{\hat{\mathcal{T}}_{\mathfrak{p}}}$ is sent
isomorphically onto
$\mathcal{H}^{i}_{\mathfrak{p},\mathrm{cris}}(\hat{\mathcal{T}}_{\mathfrak{p}})$.
Moreover, the pair
$(\mathcal{H}^{i}_{\mathrm{cris}},\mathcal{H}^{i}_{\mathrm{dR}}|_{\hat{\mathcal{T}}_{\mathfrak{p}}})$
is a constant filtered F-crystal for $i=0,1$.

  4. (d)

Let $s$ be an alg-geometric point over $\mathcal{T}_{\mathfrak{p}}$ for some
finite prime $\mathfrak{p}\subseteq\mathcal{O}_{L}[1/N]$, and
$\xi\in\mathrm{Pic}(\mathcal{X}_{s})$. Then $\mathscr{P}(\mathcal{T},s,\xi)$
contains a point over the generic point of $\mathcal{T}$ (resp.
$\mathcal{T}_{\mathfrak{p}}$) if and only if
$c_{1,\mathrm{cris}}(\xi)\in\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}|_{s}$
(resp.
$c_{1,\mathrm{cris}}(\xi)\in(\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}\oplus\mathcal{H}^{1}_{\mathfrak{p},\mathrm{cris}})|_{s}$).

Recall that $\mathscr{P}(\mathcal{T},s,\xi)$ denotes the localization of
$\mathscr{P}_{\mathcal{X}/\mathcal{T}}$ at $(s,\xi)^{\flat}$ (cf. (2.1.3)).

###### Proof.

By Hironaka’s resolution of singularities in characteristic $0$, we can find a
smooth projective $L$-variety $\overline{\mathcal{X}}_{L}$ such that
$\mathcal{X}_{L}$ embeds as an open subvariety into
$\overline{\mathcal{X}}_{L}$. By enlarging $N$, we may assume that
$\overline{\mathcal{X}}_{L}$ extends to a smooth projective scheme
$\overline{\mathcal{X}}$ over $\mathcal{O}_{L}[1/N]$ and $\mathcal{X}$ is open
in $\overline{\mathcal{X}}$. Recall that in (5.1.1)(v) we chose a section
$b:\mathcal{O}_{L}[1/N]\to\mathcal{T}$. By the theorem of the fixed part, the
natural map induced by $\iota_{\mathbb{C}}:=\iota{\otimes}\mathbb{C}$

$\iota_{\mathbb{C},B}:\mathrm{H}^{2}(\overline{\mathcal{X}}_{\mathbb{C}},\mathbb{Q})\to\mathrm{H}^{2}(\mathcal{X}_{b_{\mathbb{C}}},\mathbb{Q})$
(10)

is a surjection onto the $\pi_{1}(\mathcal{T}_{\mathbb{C}},b)$-invariant part,
i.e., onto
$\mathbb{V}^{0}_{B,b_{\mathbb{C}}}\oplus\mathbb{V}^{1}_{B,b_{\mathbb{C}}}$.

(a) It suffices to show that
$\mathbb{V}^{i}_{b_{\mathbb{C}}}{\otimes}\mathbb{Q}_{\ell}$ is
$\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{T}[1/\ell],b_{\mathbb{C}})$-stable for
$i=0,1$. We further reduce to showing that they are
$\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{T}_{L},b_{\mathbb{C}})$-stable because
the natural map
$\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{T}_{L},b_{\mathbb{C}})\to\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{T}[1/\ell],b_{\mathbb{C}})$
is surjective. Since
$\mathrm{NS}(\mathcal{X}_{\eta})_{\mathbb{Q}}=\mathrm{NS}(\mathcal{X}_{\bar{\eta}})_{\mathbb{Q}}=\mathrm{H}^{0}(\mathbb{V}^{0})$,