# Integral canonical models of exceptional Shimura varieties

Benjamin Bakker, Ananth N. Shankar, Jacob Tsimerman

###### Abstract

We prove that Shimura varieties admit integral canonical models for
sufficiently large primes. In the case of abelian-type Shimura varieties, this
recovers work of Kisin-Kottwitz for sufficiently large primes. We also prove
the existence of integral canonical models for images of period maps
corresponding to geometric families. We deduce several consequences from this,
including an unramified rigid analogue of Borel’s extension theorem, a version
of Tate semisimplicity, CM lifting theorems, and a version of Tate’s isogeny
theorem for ordinary points.

###### Contents

  1. 1 Introduction
    1. 1.1 The extension property
    2. 1.2 Main Results for Shimura Varieties
    3. 1.3 Comparison with the extension property of Milne-Moonen
    4. 1.4 Integral canonical models for images of geometric period maps
    5. 1.5 A purity property for Fontaine–Laffaille modules
    6. 1.6 Comparison with past work
    7. 1.7 Acknowledgements
    8. 1.8 Outline
    9. 1.9 Notation
  2. 2 The extension property and natural boundary
    1. 2.1 The extension property
    2. 2.2 Natural boundary and extension in codimension 1
    3. 2.3 Natural boundary and local systems
    4. 2.4 Integral canonical models of smooth varieties
    5. 2.5 Integral canonical models of normal varieties
    6. 2.6 Descending and extending arithmetic local systems
    7. 2.7 Relation to Hodge theory over $\mathbb{C}$
  3. 3 Weak stability of Fontaine–Laffaille modules on nodal curves
    1. 3.1 The inverse Cartier transform
    2. 3.2 An inverse Cartier transform on nodal curves
    3. 3.3 Applications to Fontaine–Laffaille modules
  4. 4 Criteria for local integral canonical models
    1. 4.1 Extension in codimension $\geq 2$
    2. 4.2 Local models of images of period maps
    3. 4.3 Making period maps proper
  5. 5 Existence of integral canonical models
    1. 5.1 Proof of Theorem 1.3
    2. 5.2 Proof of Theorem 1.9
  6. 6 Application to unramified Borel-extension
  7. 7 Tate-semisimplicity and special points
    1. 7.1 Special points
    2. 7.2 Tate semisimplicity
  8. 8 Complete local rings for large primes
    1. 8.1 The Fontaine-Laffaille module on complete local rings
    2. 8.2 Versality of Filtrations
  9. 9 The $\mu$-ordinary locus: CM lifts and Tate’s isogeny theorem
    1. 9.1 Non-emptiness of the $\mu$-ordinary locus
    2. 9.2 An analogue of Tate’s isogeny theorem for ordinary points

##  1 Introduction

This paper is motivated by the following question, raised by Langlands and
made precise by Milne: Given a Shimura variety $S_{K}(G,X)$ over its reflex
field $E$, does $S_{K}(G,X)$ admit a “good model” $\mathscr{S}_{K}(G,X)$, over
completions $\mathcal{O}_{E_{v}}$?

For the moduli space of principally polarized abelian varieties
$\mathcal{A}_{g}$, for instance, the moduli problem provides a natural model.
As not all Shimura varieties are known to have an associated geometric moduli
problem (notably the exceptional Shimura varieties), it is preferable to have
a model which is uniquely determined by certain intrinsic properties, possibly
only for sufficiently large primes. At the very least, we would like our model
to be smooth over $\mathcal{O}_{E_{v}}$, which already may not exist at
certain primes. Indeed, even in the case of the modular curve, the level
structure at $p$ needs to be maximal in order for there to exist a smooth
model at $p$.

Smoothness alone however does not a priori guarantee a unique model. Milne
defined an “extension property” (and this notion was refined by Moonen in
[Moonen]) which formally guarantees uniqueness. Following Milne and Moonen, an
integral model is said to be an _integral canonical model_ if it is smooth and
if it satisfies their extension property. The moduli space of principally
polarized abelian varieties $\mathcal{A}_{g}$, for instance, is known to admit
integral canonical models over $\mathbb{Z}_{p}$ whenever the level structure
is prime to $p$, i.e. when the level structure is hyperspecial at $p$. Indeed,
smoothness follows from the fact that the deformation theory of polarized
abelian varieties with a prime-to-$p$-polarization is unobstructed (proved by
Grothendieck), and Chai and Faltings [faltingschai] prove an extension result
for abelian schemes over certain bases (see [Moonen, Section 3] for the
precise results they prove) from which the Milne–Moonen extension property for
$\mathcal{A}_{g}$ follows directly.

Langlands suggested more generally that integral canonical models should exist
whenever the compact open $K_{p}$ at $p$ is hyperspecial. In the case of PEL
Shimura varieties (i.e. Shimura varieties parameterizing polarized abelian
varieties with extra endomorphisms), Kottwitz in [Ko] constructs integral
models when the level structure at $p$ is hyperspecial. Kisin generalizes
Kottwitz’ work to the setting of Shimura varieties of abelian type
([Kisinintegral]). In [PappasRapoport], Pappas and Rapoport lay out a
framework for integral models for all Shimura varieties, and amongst various
other things, they show that the integral models constructed by Kisin satisfy
their framework. In [Lovering], Lovering constructs integral canonical models
for automorphic data associated to Shimura varieties of abelian type. The
extension property proven in most of these works ultimately relies on the work
of Chai and Faltings, which uses the moduli intepretation of
$\mathcal{A}_{g}$, and therefore does not carry over to exceptional Shimura
varieties.

###  1.1 The extension property

We now define our extension property. We note that our extension property is
the equivalent to the one formulated by Milne-Moonen, and we include a
discussion about this fact below. In the proper case, the extension property
is very simple:

######  Definition 1.1.

Let $F$ be a local field. We say that an integral model
$\mathscr{S}_{K}(G,X)/\mathcal{O}_{F}$ of a proper Shimura variety
$S_{K}(G,X)/F$ has the extension property if for every smooth scheme
$\mathcal{T}/\mathcal{O}_{F}$, every map over the generic fiber
$\phi\in\operatorname{Hom}(\mathcal{T}_{F},S_{K}(G,X))$ extends to a map from
$\mathcal{T}$ to $\mathscr{S}_{K}(G,X)$.

In the non-proper case, one must take care to ensure that the special fiber
doesn’t map to the “boundary” of $\mathscr{S}_{K}(G,X)$. For primes $\ell\neq
p$, we fix an $\ell$-adic automorphic local system
${}_{{\mathrm{et}}}\mathbb{V}_{\ell}$ on $\mathscr{S}_{K}(G,X)$ corresponding
to a faithful representation $V$ of $G^{\operatorname{ad}}$ (see Section 5.1
for precise definitions of ${}_{{\mathrm{et}}}\mathbb{V}_{\ell}$).

######  Definition 1.2.

We say that a model $\mathscr{S}_{K}(G,X)/\mathcal{O}_{F}$ of $S_{K}(G,X)$
satisfies the extension property if for every smooth scheme
$\mathcal{T}/\mathcal{O}_{F}$, and every map over the generic fiber
$\phi\in\operatorname{Hom}(\mathcal{T}_{F},S_{K}(G,X))$ such that
$\phi^{*}\mathbb{V}_{\ell}$ extends to $\mathcal{T}$ for some $\ell\neq p$,
the map $\phi$ extends to a map from $\mathcal{T}$ to $\mathscr{S}_{K}(G,X)$.

###  1.2 Main Results for Shimura Varieties

Our main theorem is to prove the existence of integral canonical models for
arbitrary Shimura varieties $S$, for sufficiently large primes. Concretely, we
prove the following:

######  Theorem 1.3.

Let $S$ be a Shimura variety over its reflex field $E$. For almost all finite
places $v$, there exists a model $\mathcal{S}$ over $\mathcal{O}_{v}$ which
has the extension property, and which admits a log-smooth compactification
over $\mathcal{O}_{v}$.

Moreover, for the places $v$ as above, $\mathcal{S}$ is the unique smooth
model of $S$ over $\mathcal{O}_{v}$ which admits a log-smooth compactification
over $\mathcal{O}_{v}$ and to which the local system
${}_{{\mathrm{et}}}\mathbb{V}_{\ell}$ extends.

Note that the uniqueness claim implies that any global model of $S_{K}(G,X)$
will specialize to the integral canonical model at almost all places.

While our theorem applies to all Shimura varieties, the fact that it only
applies for almost all finite places means that much stronger results
([Kisinintegral]) are known in the abelian setting. Therefore, the case of
greatest interest is when $S$ is a Shimura variety of exceptional type.

####  1.2.1 Applications

We deduce several applications for mod $v$ (and $v$-adic) Shimura varieties.

Firstly, we prove the following unramified $p$-adic analogue of Borel’s
theorem, which we will state in the proper case for ease of exposition (see
Section 6 for a more general statement)

######  Theorem 1.4.

Let $S_{K}(G,X)$ be an exceptional proper Shimura variety. Let $\mathsf{D}$
denote the closed unit rigid-analytic disc, and let
$\mathsf{D}^{\times}=\mathsf{D}\setminus\\{0\\}$. Let
$f:\mathsf{D}^{\times}\rightarrow S_{K}(G,X)^{{\mathrm{an}}}$ denote a rigid-
analytic map defined over $W(\overline{\mathbb{F}}_{p})[1/p]$. Then, $f$
extends to a map $\mathsf{D}\rightarrow S_{K}(G,X)^{{\mathrm{an}}}$.

In fact we also prove a version of Theorem 1.4 in the context of integral
models of images of period maps (see Theorem 1.9).

We also prove results pertaining to the mod $v$ arithmetic and geometry of
$\mathscr{S}_{K}(G,X)$. The main theorem implies that Hecke correspondences
away from $p$ extend to $\mathscr{S}_{K}(G,X)/\mathcal{O}_{E_{v}}$. We use
these Hecke correspondences to prove the following theorem:

######  Theorem 1.5.

Let $x\in\mathscr{S}_{K}(G,X)(\mathbb{F}_{q})$ be any point. Then, the
Frobenius endomorphism on ${}_{{\mathrm{et}}}\mathbb{V}_{\ell,\bar{x}}$ is
semi-simple.

Work of Esnault-Groechenig shows that $\mathbb{V}_{p}/S_{K}(G,X)$ is a
crystalline local system, which yields a relative Faltings-Fontaine-Laffaille
on the $p$-adic completion of $\mathscr{S}_{K}(G,X)$. Analogous to
[Kisinintegral, Section 1.5], we give an explicit description of the this
Faltings-Fontaine-Laffaille module restricted to complete local rings of
$\mathscr{S}_{K}(G,X)$ at closed points (see Theorem 8.9 for the precise
statement). We then use this in conjunction with the semi-simplicity of
$\ell$-adic Frobenii, to prove the following theorem (which generalizes the
Serre-Tate canonical lift).

######  Theorem 1.6.

The $\mu$-ordinary locus in $\mathscr{S}_{K}(G,X)_{\mathbb{F}_{v}}$ is open
dense. Let $x\in\mathscr{S}_{K}(G,X)(\mathbb{F}_{q})$ denote a $\mu$-ordinary
point. Then, there exists a canonical special point
$\tilde{x}\in\mathscr{S}_{K}(G,X)(W(\mathbb{F}_{q}))$ that lifts $x$.

The above theorem then immediately has the following consequence.

######  Theorem 1.7.

Let $x\in\mathscr{S}_{K}(G,X)(\mathbb{F}_{q})$ denote a $\mu$-ordinary point,
and let ${}_{{}_{\mathrm{cris}}}\mathbb{V}_{x}$ denote the $F$-crystal at $x$
(associated to the Fontaine-Laffaille module defined by
${}_{{\mathrm{et}}}\mathbb{V}_{p}/S_{K}(G,X)$). Then, the crystalline
$q$-Frobenius endomorphism on ${}_{{}_{\mathrm{cris}}}\mathbb{V}_{x}$ is
semisimple.

Finally, we also have an analogue of Tate’s isogeny theorem for ordinary
points.

######  Theorem 1.8.

Let $x$ and $y$ be ordinary mod $p$ points whose $\ell$-adic Galois
representations are isomorphic. Then a rational Hodge structures underlying
the canonical lifts $\tilde{x}$ and $\tilde{y}$ are isomorphic.

This result should be thought of as an exceptional analogue (in the ordinary
case) of Tate’s isogeny theorem, which states that two abelian varieties over
finite fields are isogenous if the $\ell$-adic Galois representations
associated to the abelian varieties are isomorphic.

###  1.3 Comparison with the extension property of Milne-Moonen

The usual (Milne-Moonen) phrasing of the extension property is to let the
level $K$ be of the form $K=K^{p}K_{p}$ where $K_{p}\subset G(\mathbb{Q}_{p})$
and $K^{p}\subset G(\mathbf{A}^{p}_{f})$, and consider the tower of varieties
$\mathscr{S}_{K}(G,X)$ where $K_{p}$ is fixed and $K^{p}$ varies. One then
asks for a tower of integral models with finite etale transition maps, and
that a map from the generic fiber of any scheme $\mathcal{T}$ to this tower
extends. This however entails defining a category of admissible test schemes
to be able to map to this entire tower, which is something like “pro-smooth
schemes over $\mathbf{Z}_{p}$”, and this requires some care to do precisely.
Geometrically these are more complicated (pro-) objects, which is why we
choose to work with our definition.

These two extension properties are very close to each other. Indeed, the
Milne-Moonen extension property implies ours as follows: Suppose $\mathcal{T}$
is a smooth scheme over $\mathcal{O}_{v}$, and
$\phi:\mathcal{T}_{F_{v}}\rightarrow S_{K}(G,X)$ such that all of the
$\phi^{*}_{{\mathrm{et}}}\mathbb{V}_{\ell}$ extend. Then we may define a tower
$\mathcal{T}_{K^{p}}$ with finite transition maps using the local systems
$\phi^{*}_{{\mathrm{et}}}\mathbb{V}_{\ell}$, and a corresponding map from
$\varprojlim_{K^{p}}Y_{K^{p},F_{v}}\rightarrow\varprojlim_{K^{p}}\mathscr{S}_{K}(G,X)_{F_{v}}$.
The Milne-Moonen extension property implies that this map extends and hence so
does our original one.

Conversely, we first point out that our construction in fact gives models for
the entire tower $\varprojlim_{K_{p}}\mathscr{S}_{K}(G,X)_{F_{v}}$. Suppose
one has an admissible test $\mathcal{T}$ over $\mathcal{O}_{v}$ and a map from
$\mathcal{T}_{F_{v}}$ to $\varprojlim_{K_{p}}\mathscr{S}_{K}(G,X)_{F_{v}}$. By
definition, $\mathcal{T}$ is a pro-etale cover of a smooth scheme over some
unramified extension $R$ of $\mathcal{O}_{v}$, and hence is an inverse limit
of smooth schemes over $R$. Each of the schemes in this inverse limit must
therefore map to some finite level Shimura variety in the tower, in a way that
limits to the original map. Moreover, the pullbacks of all the automorphic
local systems ${}_{{\mathrm{et}}}\mathbb{V}_{\ell}$ will extend, and thus our
extension property will apply at each finite level, and hence to all of
$\mathcal{T}$.

###  1.4 Integral canonical models for images of geometric period maps

Our methods in fact provide integral canonical models at sufficiently large
primes for many varieties which are interpretable as moduli spaces of Hodge
structures. We describe the case of variations coming from geometry, but first
give some general background from Hodge theory (see Section 2.7 for more
details).

Suppose an irreducible normal complex variety $P$ (which we can without loss
of generality assume smooth) is equipped with a polarizable integral variation
of Hodge structures $(\mathbb{W}_{\mathbb{Z}},F^{\bullet})$. The associated
period map $P^{\mathrm{an}}\to[\mathbf{G}(\mathbb{Z})\backslash D]$ will
factor as $P^{\mathrm{an}}\to
Y^{\mathrm{an}}\to[\mathbf{G}(\mathbb{Z})\backslash D]$ where $Y$ is a
generically inertia-free Deligne–Mumford stack with quasiprojective coarse
moduli space, $P\to Y$, has connected generic geometric fiber, and
$Y^{\mathrm{an}}\to[\mathbf{G}(\mathbb{Z})\backslash D]$ is finite [bbt]. This
map is obtained by taking a resolution $\tilde{P}\to P$ and then a partial
compactification $\tilde{P}\subset P^{\prime}$ for which the period map
extends to a proper period map and forming the Stein factorization. In
particular, the variation $(\mathbb{W}_{\mathbb{Z}},F^{\bullet})$ is pulled
back from $(\mathbb{V}_{\mathbb{Z}},F^{\bullet})$ on $Y$. In general
$\tilde{P}$ may be a stack as well, but the inertia of $Y$ acts faithfully on
$\mathbb{V}_{\mathbb{Z}}$, so $Y$ is a global quotient of a normal
quasiprojective variety by the action of a finite group, and after passing to
a finite étale cover of $P$ (for instance by adjoining level structure) we may
assume $Y$ (and therefore $P$) is a quasiprojective variety. We call such a
$Y$ the Stein factorization of the period map.

Now let $P$ be an irreducible normal variety defined over a number field $E$,
$f:Z\to P$ a smooth projective $F$-morphism, and for any fixed $k$ let
${}_{et}\mathbb{W}_{\ell}:=R^{k}f_{*}\mathbb{Z}_{\ell}$ be the arithmetic
local system—or more generally a subquotient obtained from maps between such
local systems induced by $P$-morphisms of families111We lay out the precise
conditions for a variation of Hodge structures to “come from geometry” in the
sense we need it in Section 4.. We let $Y$ be the Stein factorization of the
period map of the variation with underlying geometric local system
$\mathbb{V}_{\mathbb{Z}}=R^{k}f^{\mathrm{an}}_{*}\mathbb{Z}$ for some
embedding $E\subset\mathbb{C}$. Then $Y$ is defined over $E$, does not depend
on the embedding, and the local system ${}_{et}\mathbb{W}_{\ell}$ descends to
$Y$.

For example, for any moduli space $P$ admitting an infinitesimal Torelli
theorem (including moduli spaces of Calabi–Yau varieties and most moduli
spaces of complete intersections in $\mathbb{P}^{n}$, to name a couple), $Y$
is a canonical partial compactification (namely, the natural “Hodge–theoretic”
compactification) of $P$.

######  Theorem 1.9.

Let $E$ be a number field. Let $Y/F$ be the Stein factorization of the period
map of a geometric variation of Hodge structures which we assume to be
inertia-free (and therefore a quasiprojective variety). Let
${}_{et}\mathbb{V}_{\ell}$ be the descent of the natural $\ell$-adic
arithmetic local system to $Y$.

  1. 1.

There is a finite set of primes $\Sigma_{Y}$ of $\mathcal{O}_{E}$ such that,
for all finite places $v$ of $\mathcal{O}_{E,\Sigma_{Y}}$, there is a model
$\mathcal{Y}_{v}$ over $\mathcal{O}_{v}$ such that:

    1. (a)

Each ${}_{et}\mathbb{V}_{\ell}$ extends to $\mathcal{Y}_{v}$ for $\ell\neq p$,
where $p$ is the integral prime below $v$.

    2. (b)

For any irreducible smooth $\mathcal{X}/\mathcal{O}_{v}$ with generic point
$\eta_{\mathcal{X}}$, any morphism $\phi_{0}:\eta_{\mathcal{X}}\to Y_{E_{v}}$
extends to $\phi:\mathcal{X}\to\mathcal{Y}_{v}$ if and only if
$\phi_{0}^{*}(_{et}\mathbb{V}_{\ell})$ extends to $\mathcal{X}$ for some
$\ell\neq p$.

    3. (c)

There exists a global model $\mathcal{Y}$ of $Y$ which restricts to
$\mathcal{Y}_{v}$ for all places $v$ of $\mathcal{O}_{E,\Sigma_{Y}}$.

  2. 2.

For any $\mathcal{O}_{v}$-model $\mathcal{Z}_{v}$ of $Y$ that admits a uniform
log smooth resolution (see Definition 2.12) and to which
${}_{et}\mathbb{V}_{\ell}$ extends for some $\ell\neq p$, the identity map
over the generic fiber extends to a $\mathcal{O}_{v}$-morphism
$\mathcal{Z}_{v}\to\mathcal{Y}_{v}$.

  3. 3.

Let $Y,Y^{\prime}/E$ be the Stein factorizations of two possibly different
period maps arising as above, and for a finite place $v$ of
$\mathcal{O}_{E,\Sigma_{Y}\cup\Sigma_{Y^{\prime}}}$ let
$\mathcal{Y}_{v},\mathcal{Y}^{\prime}_{v}$ be models as in (1). Then any
morphism $\phi_{0}:\eta_{\mathcal{Y}}\to\mathcal{Y}^{\prime}_{v}$ extends to
$\phi:\mathcal{Y}_{v}\to\mathcal{Y}^{\prime}_{v}$ if and only if
$\phi_{0}^{*}(_{et}\mathbb{V}_{\ell})$ extends to
$\mathcal{Y}_{\mathcal{O}_{v}}$ for some $\ell\neq p$. In particular, the
local model $\mathcal{Y}_{v}$ of $Y$ is canonically determined.

In general, the image of a period map has no reason to be smooth (already over
$\mathbb{C}$), so it is important to allow for singularities. Even starting
with a smooth moduli space $S$ parametrizing varieties with an infinitesimal
Torelli theorem so that the period map
$S^{\mathrm{an}}\to[\mathbf{G}(\mathbb{Z})\backslash D]$ is immersive, it will
often not be proper; the “Hodge-theoretic” compactification is essentially the
closure of the image of $S^{\mathrm{an}}\to\mathbf{G}(\mathbb{Z})\backslash D$
(up to a finite map) and will therefore usually be singular anyway. This is
already true for the image of the Torelli map for the moduli space of curves
(see 2.11).

###  1.5 A purity property for Fontaine–Laffaille modules

The extension property in the settings of Theorem 1.3 and Theorem 1.9 boils
down to a purity property of a $p$-torsion Fontaine–Laffaille module pulled
back from a variety on which it has immersive Kodaira–Spencer map. It is
motivated by classical Hodge theory and the arguments of [dejongoort,
moretbailly, faltingschai]. Briefly, let $F$ be a mixed characteristic local
field with ring of integers $\mathcal{O}$ and residue field $k$,
$\mathcal{X}/\mathcal{O}$ a smooth scheme, and $f_{F}:X\to Y$ a morphism over
the generic point to a model of an image of a period map $\mathcal{Y}$. There
are three steps to proving the extension of $f_{F}$ to a morphism
$f:\mathcal{X}\to\mathcal{Y}$:

  1. 1.

$\mathcal{Y}$ will support an $\ell$-adic local system
${}_{et}\mathbb{V}_{\ell}$ with large local monodromy at the boundary. It then
follows (assuming the existence of a reasonable resolution and log smooth
compactification) that the map extends in codimension one as soon as
$f_{F}^{*}(_{et}\mathbb{V}_{\ell})$ extends to $\mathcal{X}$. This in
particular establishes the claim if $\dim\mathcal{X}=1$.

  2. 2.

If $\dim\mathcal{X}>2$, then assuming the result for $\dim\mathcal{X}=2$, the
conclusion follows by induction. An extension is unique if it exists, and by
assumption exists on a general hyperplane section $\mathcal{H}$. Deformation
theory can then be used to locally lift the extension to the completion of
$\mathcal{X}$ along $\mathcal{H}$, which is enough to deduce the existence of
the extension to $\mathcal{X}$. This is the same argument used by [dejongoort,
moretbailly, faltingschai].

  3. 3.

The main step is then to handle the case $\dim\mathcal{X}=2$, and here we must
use that $\mathcal{Y}$ supports a crystalline local system whose associated
Fontaine–Laffaille module has immersive Kodaira–Spencer map mod $p$. In this
case, $\mathcal{X}/\mathcal{O}$ is a family of smooth curves, and can be
extended on a blow-up $\mathcal{X}^{\prime}$ at points in the special fiber.
It is proven in [OV] that the strong divisibility property of $p$-torsion
Fontaine–Laffaille modules implies a stability property: they cannot contain
positive degree subsheaves. Fontaine–Laffaille modules cannot be pulled back
along maps from nodal curves if the lift to $W_{2}$ smooths out some nodes,
since there are no longer local lifts of Frobenius and therefore the inverse
Cartier transform no longer makes sense. We show however it exists in a weaker
sense up to twisting by the nodes, and this is enough to rule out nodal curves
in $\mathcal{Y}$ that lift to $W_{2}$ and smooth in a controlled way. In
particular it follows that the exceptional curves must be
contracted—otherwise, the positive degree tangent bundle of a rational curve
would contradict the stability property we prove. For a precise statement, see
Proposition 3.9.

###  1.6 Comparison with past work

In the Abelian case which Kisin handles, proving the existence of smooth
models is difficult, whereas the extension property follows directly from the
extension property for $\mathcal{A}_{g}$. Unlike [Kisinintegral], we don’t
directly address the question of for which places $v$ the Shimura variety
$S_{K}(G,X)$ admits smooth models over $\mathcal{O}_{v}$. Instead, we “spread
out” the global model $S_{K}(G,X)$ over $\mathcal{O}_{E}[1/N]$ for some large
$N$, which in particular guarantees smoothness at primes $v\nmid N$ (amongst
other good properties). However, the lack of an interpretation of our Shimura
varieties as a moduli space means we need an entirely different argument for
establishing the extension property.

Regarding the applications to the mod $p$ and $p$-adic geometry of Shimura
varieties, Kisin ([Kisinmodppoints]) proves that every
$\overline{\operatorname{\mathbb{F}}}_{p}$-point of abelian Shimura varieties
admit CM lifts up to isogeny. Note that the up-to-isogeny requirement is
crucial as work of Oort [oortJAG] shows (for a more general statement, see
also [KLSS]). Noot in [Noot] proves the existence of CM lifts (on the nose)
for points of Hodge-type Shimura varieties that intersect the ordinary locus
of $\mathcal{A}_{g}$. Noot’s theorem was generalized to $\mu$-ordinary points
by [MoonenSerreTate] in the PEL-type case to the and [ShankarZhou] in the
Hodge type case. In [padicBorel], the authors prove the $p$-adic extension and
algebraicity theorems for maps to Shimura varieties (and Rapoport-Zink spaces)
of abelian type defined over discretely valued $p$-adic fields and where the
source of the map can have any dimension.

###  1.7 Acknowledgements

B. B. benefited from many conversations with Yohan Brunebarbe throughout, and
first learned of the work of Ogus–Vologodsky from Raju Krishnamoorthy. We are
also grateful to Michael Groechenig, Kiran Kedlaya, Mark Kisin, Arthur Ogus,
Abhishek Oswal, and Vadim Vologodsky for helpful conversations. B. B. was
partially supported by NSF DMS-2131688. A. S. was partially supported by the
NSF grant DMS-2100436, and would like thank MSRI for their hospitality during
the spring of 2023.

###  1.8 Outline

In Section 2 we formulate the definition of an integral canonical model in
terms of the extension property, and show that extension in codimension 1 can
be controlled in many cases (including those of interest) by a local system.
We also discuss the descent and extension of arithmetic local systems to
integral models of period images, and the relation of the extension property
to purity results in classical Hodge theory. In Section 3 we prove some weak
semistability results for the pull-backs of Fontaine–Laffaille modules to
nodal curves. In Section 4 we formulate precise conditions under which the
presence of a Fontaine–Laffaille module on a local model implies the extension
property, and in Section 5 we apply it to the case of Shimura varieties and
period images to prove the existence of integral canonical models as in
Theorem 1.3 and Theorem 1.9. In Section 6 we use the same semistability
results to prove Theorem 1.4. In Section 7 we prove Hecke correspondences
extend and use this to prove Theorem 1.5. In Sections 8 and 9, we analyze the
complete local rings of the integral canonical models of Shimura varieties at
closed points and prove Theorems 1.6, 1.7, and 1.8.

###  1.9 Notation

With the exception of Section 3, we use unadorned Roman letters $S,Y$ to
denote algebraic varieties over a field $F$ (typically a number field or a
characteristic 0 local field), script letters $\mathcal{S},\mathcal{Y}$ to
denote their integral models over a ring of integers $\mathcal{O}_{F}$ in $F$,
and a subscript (either $\mathcal{S}_{v},\mathcal{Y}_{v}$ or
$\mathcal{S}_{0},\mathcal{Y}_{0}$) to denote the special fiber—the base-change
to the residue field $k$ of $\mathcal{O}_{F}$—at a finite place $v$ (or $0$ in
the local situation, when there is no ambiguity). In section Section 3, we
denote by unadorned Roman letters $S,Y$ varieties over a perfect
characteristic $p$ field $k$, and by $\tilde{S},\tilde{Y}$ a lift to
$W_{2}(k)$.

##  2 The extension property and natural boundary

In this section we single out a class of integral models for which extension
on the level of fundamental groups is the only obstruction to extending
morphisms defined over the generic point. In particular, such models will be
uniquely determined when they exist. The key property these models will have
is the extension of morphisms from smooth schemes, just as in the work of
Milne and Moonen. In the Shimura case (or whenever the model is smooth) this
property obviously uniquely determines the model; in the general case it will
be important to allow for singularities, as even for smooth moduli spaces, the
extension property will often force a partial compactification which is not
smooth—for instance, see 2.11. We also discuss how the extendability on the
level of fundamental groups can be detected by the extendability of local
systems.

###  2.1 The extension property

######  Definition 2.1.

Let $\mathcal{O}$ be a complete DVR with fraction field $F$ and residue field
$k$.

  1. 1.

Let $\mathcal{Y},\mathcal{Y}^{\prime}/\mathcal{O}$ be finite type flat
$\mathcal{O}$-schemes. An $F$-morphism $f_{F}:Y\to Y^{\prime}$ of the generic
fibers is $\mathcal{O}$-admissible if the maps on fundamental groups fits into
a diagram

${\pi_{1}^{{\mathrm{et}}}(Y,\bar{y})}$${\pi_{1}^{{\mathrm{et}}}(Y^{\prime},\bar{y}^{\prime})}$${\pi_{1}^{{\mathrm{et}}}(\mathcal{Y},\bar{y})}$${\pi_{1}^{{\mathrm{et}}}(\mathcal{Y}^{\prime},\bar{y}^{\prime})}$

for compatibly chosen basepoints.

  2. 2.

We say a finite type flat $\mathcal{O}$-scheme $\mathcal{Y}/\mathcal{O}$
satisfies the smooth extension property over $\mathcal{O}$ if for any smooth
$\mathcal{O}$-scheme $\mathcal{X}/\mathcal{O}$, any $\mathcal{O}$-admissible
$F$-morphism $f_{F}:X\to Y$ on the generic fiber extends to an
$\mathcal{O}$-morphism $f:\mathcal{X}\to\mathcal{Y}$.

###  2.2 Natural boundary and extension in codimension 1

In the presence of a sufficiently well-behaved log smooth compactification of
a resolution (one with “natural boundary” as defined below), admissible
morphisms with smooth source will extend over codimension 1 points, and this
will be the first step of the proof of the extension property in the setting
of section 4.

The following definition is made with a view towards the general case where
$\mathcal{Y}$ is not smooth, but for Shimura varieties we will have
$\mathcal{S}=\mathcal{Y}$.

######  Definition 2.2.

Let $\mathcal{O}$ be a Dedekind domain with fraction field $F$.

  1. 1.

Let $\mathcal{Y}/\mathcal{O}$ be a normal finite type flat
$\mathcal{O}$-scheme. A resolution of $\mathcal{Y}/\mathcal{O}$ is a smooth
$\mathcal{O}$-scheme $\mathcal{S}/\mathcal{O}$ with a proper dominant map
$\pi:\mathcal{S}\to\mathcal{Y}$ such that
$\mathcal{O}_{\mathcal{Y}}\xrightarrow{\cong}\pi_{*}\mathcal{O}_{\mathcal{S}}$
via the natural map.

  2. 2.

We say $(\bar{\mathcal{S}},\mathcal{D})/\mathcal{O}$ is log smooth if it is
étale locally isomorphic over $\mathcal{O}$ to
$(\mathbb{A}^{n}_{\mathcal{O}},\mathcal{D})$, where $\mathcal{D}$ is a union
of coordinate hyperplanes.

  3. 3.

Let $\mathcal{S}/\mathcal{O}$ be a smooth $\mathcal{O}$-scheme. By a log
smooth compactification we mean a log smooth proper $\mathcal{O}$-scheme
$(\bar{\mathcal{S}},\mathcal{D})/\mathcal{O}$ with
$\bar{\mathcal{S}}\setminus\mathcal{D}=\mathcal{S}$ (as an
$\mathcal{O}$-scheme).

  4. 4.

Let $\mathcal{Y}/\mathcal{O}$ be a normal finite type flat
$\mathcal{O}$-scheme, $\pi:\mathcal{S}\to\mathcal{Y}$ a resolution, and
$\bar{\mathcal{S}}/\mathcal{O}$ a log smooth compactification of
$\mathcal{S}$. We say that the boundary of $\bar{\mathcal{S}}$ is natural
(with respect to $\mathcal{Y}$) if for every generic point $\eta$ of a stratum
of $\mathcal{D}$, the geometric inertia of $\eta$ injects in
$\pi_{1}^{{\mathrm{et}}}(\mathcal{Y},\bar{y})$ for a chosen geometric
basepoint $\bar{y}$ of $\mathcal{Y}$.

######  Lemma 2.3.

Let $\mathcal{O}$ be a characteristic 0 complete DVR with fraction field $F$.
Let $\mathcal{Y}/\mathcal{O}$ be a normal finite type flat
$\mathcal{O}$-scheme, $\pi:\mathcal{S}\to\mathcal{Y}$ a resolution, and
$\bar{\mathcal{S}}/\mathcal{O}$ a log smooth compactification of $\mathcal{S}$
with natural boundary. Let $\mathcal{X}/\mathcal{O}$ be a normal finite type
flat $\mathcal{O}$-scheme and $f_{F}:X\to Y$ an $\mathcal{O}$-admissible
morphism over the generic point. Then there is a finite type flat
$\mathcal{X}^{\prime}/\mathcal{O}$, a proper dominant generically finite
morphism $\mathcal{X}^{\prime}\to\mathcal{X}$, and a morphism
$\mathcal{X}^{\prime}\to\mathcal{S}$ such that the following diagram on the
generic fibers commutes:

${X^{\prime}}$${S}$${X}$${Y.}$$\scriptstyle{\pi_{F}}$$\scriptstyle{f_{F}}$

Moreover, if the image of $f_{F}$ meets the locus of $Y$ where $\pi$ is an
isomorphism, then $\mathcal{X}^{\prime}\to\mathcal{X}$ can be taken to be an
isomorphism over a dense open set.

###### Proof.

The map $f_{F}:X\to Y$ lifts to $S$ over a dense open subset of $X$; let
$\mathcal{W}\subset\mathcal{X}\times_{\mathcal{O}}\bar{\mathcal{S}}$ be the
closure of the graph. We then have that $\mathcal{W}\to\mathcal{X}$ is proper
dominant with a morphism $\mathcal{W}\to\bar{\mathcal{S}}$, and taking an
appropriate complete intersection of ample hypersurfaces, we may find
$\mathcal{X}^{\prime}\subset\mathcal{W}$ which is proper dominant generically
finite over $\mathcal{X}$. If $X$ meets the locus where $\pi$ is an
isomorphism, then we take $\mathcal{X}^{\prime}=\mathcal{W}$ and the morphism
$\mathcal{X}^{\prime}\to\mathcal{X}$ is a generic isomorphism. Either way, if
the boundary of $\bar{\mathcal{S}}$ intersects $\mathcal{X}^{\prime}$
nontrivially, then there is a finite extension $\mathcal{O}^{\prime}$ of
$\mathcal{O}$ an $\mathcal{O}^{\prime}$-point of
$\mathcal{X}^{\prime}/\mathcal{O}$ whose closed point maps to the boundary.
Because the boundary is natural, by the argument of [PST, Thm 4.4] this
contradicts the admissibility assumption.

∎

######  Corollary 2.4.

In the setting of the lemma, assuming the image of $f_{F}$ meets the locus of
$Y$ where $\pi$ is an isomorphism, there is a codimension $\geq 2$ subset
$\mathcal{Z}\subset\mathcal{X}$ and an extension of $f_{F}$ to a morphism
$f:\mathcal{X}\setminus\mathcal{Z}\to\mathcal{Y}$.

###### Proof.

$\mathcal{X}^{\prime}\to\mathcal{X}$ is proper so every codimension 1 point
$\eta$ lifts to $\mathcal{X}^{\prime}$ and maps to $\mathcal{S}$, so $f_{F}$
extends over $\eta$. ∎

###  2.3 Natural boundary and local systems

For $\mathcal{S}/\mathcal{O}$ smooth over a Dedekind domain and
$\bar{\mathcal{S}}$ a log smooth compactification, both the naturality of the
boundary and the admissibility condition on $\mathcal{O}$-morphisms to
$\mathcal{S}$ can be checked using an $\ell$-adic local system
$\mathbb{V}_{\ell}$ on $\mathcal{S}$. The key property such a local system
must have is “nonextendability” over the generic fiber, meaning it does not
extend to any nontrivial partial compactification. This notion is treated
carefully in [brunebarbe, §3] (where it is referred to as “maximality”); the
following definition is easily seen to be equivalent to the definition
therein.

######  Definition 2.5.

Let $Y/F$ be a normal variety with an $\ell$-adic local system
$\mathbb{V}_{\ell}$. We say $\mathbb{V}_{\ell}$ has faithful geometric local
monodromy if for some (hence any) resolution $\pi:S\to Y$ and log smooth
compactification $S\subset\bar{S}$, the geometric inertia at the generic point
of every stratum of the boundary $D$ acts faithfully on
$\pi^{*}\mathbb{V}_{\ell}$. We say an $\ell$-adic local system on an integral
model has faithful geometric local monodromy if the restriction to the generic
fiber does.

Note that faithfulness of geometric local monodromy can be checked over the
algebraic closure $\bar{F}$, or even over $\mathbb{C}$ for a choice of
embedding $F\subset\mathbb{C}$. Faithfulness of geometric local monodromy
restricts well:

######  Lemma 2.6 ([brunebarbe, §3]).

Let $Y/\mathbb{C}$ be a normal variety and $\mathbb{V}_{\ell}$ an $\ell$-adic
local system on $Y$. Then the following are equivalent:

  1. 1.

$\mathbb{V}_{\ell}$ has faithful geometric local monodromy.

  2. 2.

For any curve $C/\mathbb{C}$ and any proper morphism $f:C\to Y$,
$f^{*}\mathbb{V}_{\ell}$ has faithful geometric local monodromy.

  3. 3.

For any normal variety $X/\mathbb{C}$ and any proper morphism $f:X\to Y$,
$f^{*}\mathbb{V}_{\ell}$ has faithful geometric local monodromy.

The next lemma interpets naturality of the boundary and admissibility in terms
of the local system as promised.

######  Lemma 2.7.

Suppose $\mathcal{Y}/\mathcal{O}$ is a normal finite type flat
$\mathcal{O}$-scheme, $\pi:\mathcal{S}\to\mathcal{Y}$ a resolution,
$\bar{\mathcal{S}}/\mathcal{O}$ a log smooth compactification of
$\mathcal{S}$, and $\mathbb{V}_{\ell}$ an $\ell$-adic local system on
$\mathcal{Y}$ such that $\mathbb{V}_{\ell}|_{Y}$ has faithful geometric local
monodromy. Then

  1. 1.

$\bar{\mathcal{S}}$ has natural boundary.

  2. 2.

For $\mathcal{X}/\mathcal{O}$ regular, a morphism $f_{K}:X\to Y$ over the
generic point which meets the locus of $Y$ where $\pi$ is an isomorphism is
$\mathcal{O}$-admissible if and only if $f_{K}^{*}\mathbb{V}_{\ell}$ extends
to $\mathcal{X}$.

###### Proof.

For 1, the geometric inertia of every stratum acts faithfully on
$\pi_{K}^{*}\mathbb{V}_{\ell}$, and this action factors through
$\pi_{1}^{{\mathrm{et}}}(\mathcal{S},\bar{s})\to\pi_{1}^{{\mathrm{et}}}(\mathcal{Y},\bar{y})$,
so it follows that the boundary is natural.

For 2, the forward implication is obvious. For the reverse implication, the
same argument as in Lemma 2.3 shows that $f_{F}$ extends to
$f:\mathcal{X}\setminus\mathcal{Z}\to\mathcal{Y}$ for a codimension $\geq 2$
subset $\mathcal{Z}$, but by purity of the branch locus
$\pi_{1}^{{\mathrm{et}}}(\mathcal{X}\setminus\mathcal{Z},\bar{x})\to\pi_{1}^{{\mathrm{et}}}(\mathcal{X},\bar{x})$
is an isomorphism. ∎

######  Corollary 2.8.

For $\mathcal{Y},\mathbb{V}_{\ell}$ as in the lemma, $\mathcal{X}/\mathcal{O}$
a normal finite type flat $\mathcal{O}$-scheme, and $f_{F}:X\to Y$ a morphism
over the generic point such that $f_{F}^{*}\mathbb{V}_{\ell}$ extends to
$\mathcal{X}$. Then there is a codimension $\geq 2$ subset
$\mathcal{Z}\subset\mathcal{X}$ and an extension of $f_{F}$ to a morphism
$f:\mathcal{X}\setminus\mathcal{Z}\to\mathcal{Y}$.

###  2.4 Integral canonical models of smooth varieties

For simplicity, we first define integral canonical models in the smooth case
(to be generalized in the next section).

######  Definition 2.9.

Let $\mathcal{O}$ be a Dedekind domain with fraction field $F$ and
$\mathcal{Y}/\mathcal{O}$ a quasiprojective smooth $\mathcal{O}$-scheme. We
say $\mathcal{Y}/\mathcal{O}$ is an integral canonical model if for every
closed point $v$ of $\operatorname{Spec}\mathcal{O}$, the base-change
$\mathcal{Y}_{v}/\mathcal{O}_{v}$ satisfies the extension property over
$\mathcal{O}_{v}$.

The following is immediate:

######  Lemma 2.10.

Let $\mathcal{O}$ be a Dedekind domain with fraction field $F$. Let
$\mathcal{S},\mathcal{S}^{\prime}/\mathcal{O}$ be two smooth integral
canonical models and $f_{F}:S\to S^{\prime}$ a morphism over the generic
point.

  1. 1.

$f_{F}$ extends to a morphism $f:\mathcal{S}\to\mathcal{S}^{\prime}$ if and
only if there is a commutative diagram

${\pi_{1}^{{\mathrm{et}}}(S,\bar{s})}$${\pi_{1}^{{\mathrm{et}}}(S^{\prime},\bar{s}^{\prime})}$${\pi_{1}^{{\mathrm{et}}}(\mathcal{S},\bar{s})}$${\pi_{1}^{{\mathrm{et}}}(\mathcal{S}^{\prime},\bar{s}^{\prime})}$

for compatibly chosen basepoints.

  2. 2.

Suppose further that $\mathcal{S}^{\prime}$ admits a log smooth
compactification and an $\ell$-adic local system $\mathbb{V}_{\ell}$ such that
$\mathbb{V}_{\ell}|_{S^{\prime}}$ has faithful geometric local monodromy. Then
$f_{F}$ extends to a morphism $f:\mathcal{S}\to\mathcal{S}^{\prime}$ if and
only if $f_{F}^{*}(\mathbb{V}_{\ell})$ extends to $\mathcal{S}$.

###  2.5 Integral canonical models of normal varieties

We begin with a simple example to demonstrate that, even for a smooth variety,
a model which satisfies the smooth extension property might necessarily
involve a partial compactification with singularities.

######  Example 2.11.

Consider the moduli stacks $\mathcal{A}_{g}$ and $\mathcal{M}_{g}$ of
principally polarized $g$-dimensional abelian varieties and genus $g$ curves,
say over $\mathbb{Z}_{p}$ for $p\gg 0$, and let
$\mathcal{M}_{g}\to\mathcal{A}_{g}$ be the Torelli map. After taking
sufficient level structure, both of these stacks are schemes, and the Torelli
map is generically immersive. $\mathcal{A}_{g}$ satisfies the smooth extension
property, so the closure of the image of $\mathcal{M}_{g}$ in
$\mathcal{A}_{g}$ will as well. This closure is quite singular however: the
map $\mathcal{M}_{g}\to\mathcal{A}_{g}$ extends to the partial
compactification $\tilde{\mathcal{M}}_{g}\to\mathcal{A}_{g}$ parametrizing
compact-type nodal curves (that is, nodal curves whose dual graph is a tree),
but the boundary gets contracted with positive-dimensional (non-uniruled)
fibers, as the Jacobian does not record the attaching nodes.

In the normal case, to deduce the extension of arbitrary morphisms from the
case of morphisms with smooth source, we will require the existence of an
integral resolution with the property that the subvarieties contracted in a
special fiber are limits of subvarieties contracted in the generic fiber.

######  Definition 2.12.

Let $\mathcal{O}$ be a Dedekind domain with fraction field $F$. Let
$\mathcal{Y}/\mathcal{O}$ be a normal finite type flat $\mathcal{O}$-scheme.

  1. 1.

A resolution of $\mathcal{Y}/\mathcal{O}$ is a smooth $\mathcal{O}$-scheme
$\mathcal{S}/\mathcal{O}$ with a proper dominant map
$\pi:\mathcal{S}\to\mathcal{Y}$ such that
$\mathcal{O}_{\mathcal{Y}}\xrightarrow{\cong}\pi_{*}\mathcal{O}_{\mathcal{S}}$
via the natural map.

  2. 2.

A resolution $\pi:\mathcal{S}\to\mathcal{Y}$ is uniform if there is a
flattening stratification of $\pi$ by locally closed subsets $\mathcal{Y}^{j}$
of $\mathcal{Y}$, each of which is flat over $\operatorname{Spec}\mathcal{O}$.

######  Lemma 2.13.

Let $\mathcal{O}$ be a Dedekind domain with fraction field $F$, $Y/F$ a normal
variety, $\pi_{F}:S\to Y$ a resolution, and $\pi:\mathcal{S}\to\mathcal{Y}$ a
model of $\pi_{F}$ defined over $\mathcal{O}$. Then $\pi$ is a uniform
resolution over a (nonempty) open set of $\mathcal{O}$.

###### Proof.

Since $\mathcal{O}_{\mathcal{Y}}\to\pi_{*}\mathcal{O}_{\mathcal{S}}$ is an
isomorphism over the generic point, it is over an open set of
$\operatorname{Spec}\mathcal{O}_{F}[1/N]$. Taking a flattening stratification
$\mathcal{Y}^{j}$, the strata which do not dominate map to finitely many
primes. ∎

######  Definition 2.14.

Let $\mathcal{O}$ be a Dedekind domain with fraction field $F$ and
$\mathcal{Y}/\mathcal{O}$ a normal quasiprojective flat $\mathcal{O}$-scheme.
We say $\mathcal{Y}/\mathcal{O}$ is an integral canonical model if:

  1. 1.

$\mathcal{Y}$ admits a uniform resolution $\pi:\mathcal{S}\to\mathcal{Y}$.

  2. 2.

For every closed point $v$ of $\operatorname{Spec}\mathcal{O}$, the base-
change $\mathcal{Y}_{v}/\mathcal{O}_{v}$ satisfies the extension property over
$\mathcal{O}_{v}$.

Note that if $\mathcal{Y}/\mathcal{O}$ is smooth, the condition on the
existence of a uniform resolution is automatic, so this definition generalizes
Definition 2.9. The next proposition generalizes Lemma 2.10 to the normal
case.

######  Proposition 2.15.

Let $\mathcal{O}$ be a Dedekind domain with fraction field $F$. Let
$\mathcal{Y}/\mathcal{O}$ be a normal quasiprojective flat
$\mathcal{O}$-scheme admitting a uniform resolution,
$\mathcal{Y}^{\prime}/\mathcal{O}$ an integral canonical model, and
$f_{F}:Y\to Y^{\prime}$ a morphism over the generic point.

  1. 1.

$f_{F}$ extends to a morphism $f:\mathcal{Y}\to\mathcal{Y}^{\prime}$ if and
only if there is a commutative diagram

${\pi_{1}^{{\mathrm{et}}}(Y,\bar{y})}$${\pi_{1}^{{\mathrm{et}}}(Y^{\prime},\bar{y}^{\prime})}$${\pi_{1}^{{\mathrm{et}}}(\mathcal{Y},\bar{y})}$${\pi_{1}^{{\mathrm{et}}}(\mathcal{Y}^{\prime},\bar{y}^{\prime})}$

for compatibly chosen basepoints.

  2. 2.

Suppose further that $\mathcal{Y}^{\prime}$ admits a uniform resolution
$\pi^{\prime}:\mathcal{S}^{\prime}\to\mathcal{Y}^{\prime}$, a log smooth
compactification of $\mathcal{S}^{\prime}$, and that $\mathcal{Y}^{\prime}$
has an $\ell$-adic local system $\mathbb{V}_{\ell}$ such that
$\mathbb{V}_{\ell}|_{Y^{\prime}}$ has faithful geometric local monodromy. Then
$f_{F}$ extends to a morphism $f:\mathcal{Y}\to\mathcal{Y}^{\prime}$ if and
only if $f_{F}^{*}(\mathbb{V}_{\ell})$ extends to $\mathcal{Y}$.

For the proof we need the following:

######  Lemma 2.16 (Rigidity Lemma, [debarre, Lemma 1.15]).

Let $X,Y,Z$ be integral finite type schemes, $f:X\to Y$ a proper morphism with
$\mathcal{O}_{Y}\xrightarrow{\cong}f_{*}\mathcal{O}_{X}$ via the natural map,
and $g:X\to Z$ a morphism which is constant on $f^{-1}(y)$ for some point $y$
of $Y$. Then for an open neighborhood $y\in U\subset Y$,
$g|_{f^{-1}(U)}:f^{-1}(U)\to Z$ factors through $f|_{f^{-1}(U)}:f^{-1}(U)\to
Y$.

###### Proof.

The proof is exactly the same as in [debarre]. Consider the (proper) morphism
$h=f\times g:X\to Y\times Z$, and let $W$ be the image with projection map
$p:W\to Y$. Since $\mathcal{O}_{Y}\subset p_{*}\mathcal{O}_{W}\subset
f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$, the Stein factorization of $p$ is
trivial. Since $h^{-1}((p^{-1}(y)))=f^{-1}(y)$ maps to a point in $Z$,
$p^{-1}(y)$ is a point, hence the fibers of $p$ are 0-dimensional in a
neighborhood of $y$, and therefore $p:W\to Y$ is an isomorphism in a
neighborhood of $y$. ∎

######  Corollary 2.17.

Let $\mathcal{O}$ be a Dedekind domain with fraction field $F$ Let
$\mathcal{Y},\mathcal{Y}^{\prime}/\mathcal{O}$ be normal finite type flat
$\mathcal{O}$-schemes, and $\pi:\mathcal{S}\to\mathcal{Y}$ a uniform
resolution. Assume $\mathcal{Y}^{\prime}/\mathcal{O}$ is quasiprojective. Then
any morphism $f:\mathcal{S}\to\mathcal{Y}^{\prime}$ which factors through
$\pi_{F}:S\to Y$ over the generic point factors through $\pi$.

###### Proof.

Let $L^{\prime}$ be a very ample bundle on $\mathcal{Y}^{\prime}$, let
$\mathcal{Y}^{j}\subset\mathcal{Y}$ be an $\mathcal{O}$-flat stratum in a
flattening stratification of $\mathcal{S}\to\mathcal{Y}$, let
$\mathcal{S}^{j}\to\mathcal{Y}^{j}$ be the base-change to $\mathcal{Y}^{j}$,
and consider the composition
$\mathcal{S}^{j}\to\mathcal{S}\to\mathcal{Y}^{\prime}$. For any fiber $F$ of
$\mathcal{S}^{j}\to\mathcal{Y}^{j}$, $L|_{F}$ is globally generated, hence has
a section which is generically nonzero on every geometric component. On the
other hand, $L^{\vee}$ is trivial on every geometric fiber of
$\mathcal{S}^{j}\to\mathcal{Y}^{j}$ over the generic point, hence by
semicontinuity $L^{\vee}|_{F}$ has at least one nonzero section. The product
is a section of $\mathcal{O}_{F}$ which is not identically zero, so since the
fibers of $\mathcal{S}^{j}\to\mathcal{Y}^{j}$ are geometrically connected it
must be nowhere zero. Thus, $L|_{F}$ is trivial, and so $f$ contracts every
fiber of $\mathcal{S}\to\mathcal{Y}$. By the rigidity lemma, $f$ factors
through $\mathcal{S}\to\mathcal{Y}$. ∎

###### Proof of Proposition 2.15.

The forward implication of part 1 is trivial; for the reverse implication, by
the smooth extension property we have an extension of the composition $S\to
Y\to Y^{\prime}$ to $\mathcal{S}_{v}\to\mathcal{Y}^{\prime}_{v}$ for every
finite place, and therefore an extension $\mathcal{S}\to\mathcal{Y}^{\prime}$.
Then by Corollary 2.17, the map $\mathcal{S}\to\mathcal{Y}^{\prime}$ factors
through the desired extension $f:\mathcal{Y}\to\mathcal{Y}^{\prime}$. Part 2
follows from part 1 by Lemma 2.7.

∎

######  Corollary 2.18.

Assume $Y$ admits a local system $\mathbb{V}_{\ell}$ with faithful geometric
local monodromy. Then:

  1. 1.

There is at most one integral canonical model $\mathcal{Y}/\mathcal{O}$ to
which $\mathbb{V}_{\ell}$ extends and which admits a uniform resolution with a
log smooth compactification.

  2. 2.

Assume such an integral canonical model $\mathcal{Y}$ exists. Then for any
normal finite type flat model $\mathcal{Z}/\mathcal{O}$ to which
$\mathbb{V}_{\ell}$ extends and which admits a uniform resolution, the
identity map over the generic fiber extends to a morphism
$\mathcal{Z}\to\mathcal{Y}$.

###  2.6 Descending and extending arithmetic local systems

Let $\mathcal{O}$ be a Dedekind domain with fraction field $F$,
$\mathcal{P}/\mathcal{O}$ smooth with geometrically irreducible generic fiber,
and $\mathcal{Y}/\mathcal{O}$ normal finite type flat with geometrically
irreducible generic fiber. Let $f:\mathcal{P}\to\mathcal{Y}$ be a proper
morphism with
$\mathcal{O}_{\mathcal{Y}}\xrightarrow{\cong}f_{*}\mathcal{O}_{\mathcal{P}}$.
In this section we describe how arithmetic local systems can be extended to
integral models and descended along $f$.

We start with the following:

######  Lemma 2.19.

Let $f:X\to Y$ be a proper morphism of schemes with $Y$ integral and assume
$f$ has geometrically connected fibers and geometrically reduced generic
fiber. Then over a dense open $U\subset Y$ we have the following:

  1. 1.

The base-change $X_{U}\to U$ induces an exact sequence

$\pi_{1}^{{\mathrm{et}}}(X_{\bar{\eta}},\bar{x})\to\pi_{1}^{{\mathrm{et}}}(X_{U},\bar{x})\to\pi_{1}^{{\mathrm{et}}}(U,\bar{u})\to
1$

for compatibly chosen basepoints, where $\eta$ is the generic point of $Y$.

  2. 2.

For every geometric point $\bar{u}$ of $U$, the specialization map
$sp:\pi_{1}^{{\mathrm{et}}}(X_{\bar{\eta}},\bar{x})\to\pi_{1}^{{\mathrm{et}}}(X_{\bar{u}},\bar{x})$
is surjective.

###### Proof.

According to [stacks-project, Tag 0579], since the generic fiber is
geometrically reduced we may take $U$ to be the interior of the set of points
with geometrically reduced fibers. We may further assume $X_{U}\to U$ is flat
by generic flatness. Then the first claim follows from the homotopy exact
sequence [stacks-project, Tag 0C0J] and the second from base-change to the
henselization of the local ring. ∎

The following is standard but we include it for completeness. For a morphism
of schemes $X\to Y$ which induces a surjection
$\pi_{1}^{{\mathrm{et}}}(X,\bar{x})\to\pi_{1}^{{\mathrm{et}}}(Y,\bar{y})$ for
compatibly chosen basepoints, we denote by
$I_{X/Y}\subset\pi_{1}^{{\mathrm{et}}}(X,\bar{x})$ the kernel (somewhat
abusively, since we suppress the basepoints from the notation).

######  Lemma 2.20.

The map $f_{F}:P\to Y$ induces the following commutative diagram with exact
rows and columns for compatibly chosen basepoints.

${1}$${1}$${I_{P_{\bar{F}}/Y_{\bar{F}}}}$${I_{P/Y}}$${1}$${\pi_{1}^{{\mathrm{et}}}(P_{\bar{F}},\bar{p})}$${\pi_{1}^{{\mathrm{et}}}(P,\bar{p})}$${\pi_{1}^{{\mathrm{et}}}(F,\bar{F})}$${1}$${1}$${\pi_{1}^{{\mathrm{et}}}(Y_{\bar{F}},\bar{y})}$${\pi_{1}^{{\mathrm{et}}}(Y,\bar{y})}$${\pi_{1}^{{\mathrm{et}}}(F,\bar{F})}$${1}$${1}$${1}$

###### Proof.

The exactness of the rows is standard and the surjectivity of the vertical
maps to the bottom row is a consequence of normality and the generic fiber
being geometrically connected. ∎

######  Corollary 2.21.

Let $\mathbb{V}_{\ell}$ be an $\mathbb{Z}_{\ell}$-local system on $P$ for
which the geometric local system $\bar{\mathbb{V}}_{\ell}$ on $P_{\bar{F}}$
descends to $Y_{\bar{F}}$. Then $\mathbb{V}_{\ell}$ descends to $Y$.

######  Lemma 2.22.

Assume $\mathcal{O}$ has perfect residue fields222Over perfect fields, a
scheme is geometrically reduced if and only if it is reduced.. Then after
inverting finitely many primes of $\mathcal{O}$, any $\ell$-adic local system
$\mathbb{V}_{\ell}$ on $\mathcal{P}$ which geometrically descends to
$Y_{\bar{F}}$ descends to $\mathcal{Y}$. Moreover, the same is true for any
further localization of $\mathcal{O}$.

###### Proof.

We first claim that after inverting finitely many primes, the following is
true: for every geometric point $\bar{y}_{0}$ of $\mathcal{Y}$ there is a
geometric point $\bar{y}$ of $Y$ specializing to it such that the
specialization map
$\pi_{1}^{{\mathrm{et}}}(\mathcal{P}_{\bar{y}},\bar{y})\to\pi_{1}^{{\mathrm{et}}}(\mathcal{P}_{\bar{y}_{0}},\bar{y}_{0})$
is surjective. Using Lemma 2.19, there is a stratification $\mathcal{Y}^{i}$
of $\mathcal{Y}$ such that this is true for
$(\mathcal{P}_{\mathcal{Y}^{i}})^{\mathrm{red}}\to\mathcal{Y}^{i}$, provided
$\mathcal{Y}^{i}$ is flat over $\mathcal{O}$. After shrinking $\mathcal{O}$,
we may assume this is the case for all $\mathcal{Y}^{i}$, and therefore the
claim is proven.

By Corollary 2.21, we may assume $\mathbb{V}_{\ell}$ descends to $Y$. To
finish it suffices to show any finite étale cover
$\mathcal{P}^{\prime}\to\mathcal{P}$ which descends to $Y$ over $P$ descends
to $\mathcal{Y}$. The descent clearly patches, so we may assume
$\mathcal{Y}=\operatorname{Spec}R$ for a local ring $R$. Base-changing to the
henselization $\bar{R}$, the cover
$\mathcal{P}^{\prime}_{R}\to\mathcal{P}_{R}$ descends to $R$ by [stacks-
project, Tag 0A48], since it is trivial on every geometric fiber of
$\mathcal{P}_{R}\to\operatorname{Spec}R$ by the above surjection of
specialization maps. The descent to $R$ comes equipped with descent data which
gives a finite étale cover of $\operatorname{Spec}R$ which pulls back to
$\mathcal{P}^{\prime}$.

∎

######  Corollary 2.23.

Assume $\mathcal{O}$ has perfect residue fields. Then after inverting finitely
primes of $\mathcal{O}$, the following is true. For any $\ell$ and any $n$,
any $\mathbb{Z}_{\ell}$-local system $\mathbb{V}_{\ell}$ on $\mathcal{P}[1/n]$
which geometrically descends to $Y_{\bar{F}}$ descends to $\mathcal{Y}[1/n]$.

###  2.7 Relation to Hodge theory over $\mathbb{C}$

We end this section by summarizing some classical extension results in Hodge
theory using the above language. This is both motivation for the using
extension properties to determine integral models and is needed to prove the
characteristic 0 part of the extension in Theorem 1.9.

######  Lemma 2.24.

Let $P$ be a smooth complex algebraic variety equipped with a polarizable
integral variation of Hodge structures $(W_{\mathbb{Z}},F^{\bullet}W)$, and
let $\phi:P^{\mathrm{an}}\to[\mathbf{G}(\mathbb{Z})\backslash D]$ be the
period map.

  1. 1.

For any partial log smooth compactification $(P^{\prime},D)$,
$(W_{\mathbb{Z}},F^{\bullet})$ extends to $P^{\prime}$ if and only if the
monodromy of $W_{\mathbb{Z}}$ around each component of $D$ is trivial.

  2. 2.

$\phi$ is proper if and only if and only if $V_{\mathbb{Z}}$ has faithful
geometric local monodromy.

  3. 3.

There is a finite étale cover $\pi:\tilde{P}\to P$ and a partial log smooth
compactification $\tilde{P}^{\prime}$ of $\tilde{P}$ such that
$\pi^{*}W_{\mathbb{Z}}$ extends to $\tilde{P}^{\prime}$ with faithful
geometric local monodromy.

  4. 4.

If the monodromy of $W_{\mathbb{Z}}$ is unipotent, then there is a partial log
smooth compactification $P^{\prime}$ of $P$ such that $W_{\mathbb{Z}}$ extends
to $P^{\prime}$ with failtful geometric local monodromy.

###### Proof.

(1) is a result of Griffiths [griffithsiii, Theorem 9.5]. For (2), (3), and
(4), see [brunebarbe, §3]. ∎

For simplicity, we will assume $W_{\mathbb{Z}}$ has unipotent local monodromy.
In this case, it follows from the lemma that for any period map
$\phi:P\to[\mathbf{G}(\mathbb{Z})\backslash D]$, there is a partial log smooth
compactification $P^{\prime}$ of $P$, a proper extension
$\phi^{\prime}:P^{\prime{\mathrm{an}}}\to[\mathbf{G}(\mathbb{Z})\backslash
D]$, and an algebraic map $f:P^{\prime}\to Y$ to a normal $Y$ with
$\mathcal{O}_{Y}\to f_{*}\mathcal{O}_{P^{\prime}}$ such that $\phi^{\prime}$
factors as
$P^{\prime{\mathrm{an}}}\xrightarrow{f^{\mathrm{an}}}Y^{\mathrm{an}}\xrightarrow{\psi^{\mathrm{an}}}$.
We call $Y$ the Stein factorization of the period map $\phi$. Observe that the
local system $W_{\mathbb{Z}}$ as well as the associated filtered flat vector
bundle $(W,\nabla,F^{\bullet}W)$ are pulled back from $V_{\mathbb{Z}}$ and
$(V,\nabla,F^{\bullet}V)$ on $Y$.

######  Corollary 2.25.

Let $Y$ be the Stein factorization of a period map
$(W_{\mathbb{Z}},F^{\bullet}W)$ on $P$ with unipotent local monodromy. Let $X$
be a smooth complex algebraic variety and $U\subset X$ a dense Zariski open
set. Then a morphism $\phi_{U}:U\to Y$ extends to $\phi:X\to Y$ if and only if
$\pi_{U}^{*}\mathbb{W}_{\mathbb{Z}}$ extends to $X$.

######  Lemma 2.26.

Suppose $P$ is defined over a number field $F$.

  1. 1.

If the filtered flat vector bundle $(W,\nabla,F^{\bullet}W)$ underling
$(W_{\mathbb{Z}},F^{\bullet}W)$ is defined over $F$, so is the Stein
factorization of the (compactified) period map $f:P^{\prime}\to Y$. Moreover,
the descent $(V,\nabla,F^{\bullet}V)$ is also defined over $F$.

  2. 2.

If $W_{\mathbb{Z}_{\ell}}$ extends to an arithmetic local system
${}_{et}\mathbb{W}_{\ell}$ on $P$, then $V_{\mathbb{Z}_{\ell}}$ extends to an
arithmetic local system $\mathbb{V}_{\ell}$ on $Y$.

##  3 Weak stability of Fontaine–Laffaille modules on nodal curves

In this section we prove some semistability properties of Fontaine–Laffaille
modules over nodal curves (see Section 3.3). These results will be critical in
the main step in the proof of the extension of morphisms from smooth sources
in codimenison $\geq 2$ in Section 4.

###  3.1 The inverse Cartier transform

We begin by recalling the nonabelian Hodge correspondence in characteristic
$p$ of Ogus–Vologodsky [OV] via exponential twisting [LSZtwist].

Let $k$ be a perfect field of characterstic $p>2$. We denote by
$W_{2}=W_{2}(k)$ the length 2 Witt vectors. Throughout, by a lift of a scheme
(usually from $k$ to $W_{2}$) we mean a flat lift. For a variety $Y/k$, we
denote by $F_{Y}:Y\to Y$ the absolute Frobenius, dropping the subscript if
there’s no chance of confusion. We have a natural lift of Frobenius
$F_{W}:W_{2}\to W_{2}$, and in the sequel by a lift of Frobenius to a lift of
$Y$ we always mean a lift commuting with $F_{W}$.

####  3.1.1 Flat sheaves and Higgs sheaves on varieties

We first recall the definitions of Higgs sheaves and flat sheaves on smooth
$k$-varieties. As $p\neq 2$ we may unambiguously speak of $\Omega_{X}^{2}$.
Throughout we denote by
$T_{X}:={\mathcal{H}\operatorname{om}}_{\operatorname{\mathcal{O}}_{X}}(\Omega_{X},\operatorname{\mathcal{O}}_{X})$.

By a flat sheaf on $X$ we mean a locally free
$\operatorname{\mathcal{O}}_{X}$-module $V$ with a $k$-linear map $\nabla:V\to
V\otimes\Omega_{X}$ which satisfies the Leibniz rule and for which the
$\operatorname{\mathcal{O}}_{X}$-linear map $\nabla\wedge\nabla:V\to
V\otimes\Omega_{X}^{2}$ vanishes. By a Higgs sheaf on $X$ we mean a locally
free $\operatorname{\mathcal{O}}_{X}$-module $V$ with an
$\operatorname{\mathcal{O}}_{Y}$-linear map $\theta:V\to V\otimes\Omega_{X}$
such that $\theta\wedge\theta:V\to V\otimes\Omega^{2}_{X}$ vanishes. Observe
that since differentials pull back, both flat sheaves and Higgs sheaves pull
back under arbitrary morphisms $f:X\to Y$. We denote by $MIC_{m}(X)$ (resp.
$HIG_{m}(X)$) the category of flat sheaves $(V,\nabla)$ (resp. Higgs sheaves
$(V,\theta)$) with $p$-curvature $\psi(\nabla):=\nabla^{p}-\nabla_{(-)^{p}}$
(resp. Higgs field $\theta$) nilpotent of exponent $\leq m$, meaning
$\psi(\nabla)^{m}=0$ (resp. $\theta^{m}=0$) with the natural notion of
morphisms. We will always assume $m\leq p$.

Recall that for any locally free $\mathcal{O}_{X}$-module $V$ on $X$, the
pull-back along absolute Frobenius $F^{*}V$ admits a canonical flat connection
$\nabla_{can}$ whose flat sections are the pull-back sections $F^{-1}V$.
Explicitly, if we write
$F^{*}V=\operatorname{\mathcal{O}}_{X}\otimes_{\operatorname{\mathcal{O}}_{X}}V$
where the tensor is taken with respect to the $p$-power algebra structure
$\operatorname{\mathcal{O}}_{X}\to\operatorname{\mathcal{O}}_{X}$, then
$\nabla_{can}=d\otimes\operatorname{id}$.

####  3.1.2 Nonabelian Hodge correspondence à la Lan–Sheng–Zuo

Let $Y/k$ be smooth and $\tilde{Y}/W_{2}$ a flat lift. Let $Y_{i}$ be a
Zariski cover of $Y$ by affines with overlaps $Y_{ij}:=Y_{i}\cap Y_{j}$ and
$\tilde{F}_{i}:\tilde{Y}_{i}\to\tilde{Y}_{i}$ lifts of absolute Frobenius. By
lifting to $\operatorname{\mathcal{O}}_{\tilde{Y}}$ and applying
$p^{-1}(\tilde{F}_{i}-\tilde{F}_{j})$ (resp. $p^{-1}d\tilde{F}_{i}$) we obtain
$F$-derivations of $\operatorname{\mathcal{O}}_{Y_{ij}}$ (resp.
$\operatorname{\mathcal{O}}_{Y_{i}}$) valued in
$\operatorname{\mathcal{O}}_{Y_{ij}}$ (resp. $\Omega_{Y_{i}}$), which we may
think of as $\operatorname{\mathcal{O}}_{Y}$-linear morphisms

$\displaystyle s_{ij}$
$\displaystyle:=p^{-1}(\tilde{F}_{i}-\tilde{F}_{j}):F^{*}\Omega_{Y_{ij}}\to\mathcal{O}_{Y_{ij}}$
$\displaystyle\zeta_{i}$
$\displaystyle:=p^{-1}d\tilde{F}_{i}:F^{*}\Omega_{Y_{i}}\to\Omega_{Y_{i}}$

which satisfy $ds_{ij}=\zeta_{i|j}-\zeta_{j|i}$. For a Higgs sheaf
$(V,\theta)\in HIG_{m}(Y)$, it is shown in [LSZtwist] that the inverse Cartier
transform $C_{\tilde{Y}}^{-1}(V,\theta)$ is the flat sheaf obtained by locally
forming $(F^{*}V|_{Y_{i}},\nabla_{can}+\zeta_{i}(F^{*}\theta))$ on $Y_{i}$ and
gluing via

$\exp(s_{ij}(F^{*}\theta)):(F^{*}V|_{Y_{ij}},\nabla_{can}+\zeta_{i}(F^{*}\theta))\to(F^{*}V|_{Y_{ij}},\nabla_{can}+\zeta_{j}(F^{*}\theta))$

up to negating the sign of the connection. Observe that for any Higgs subsheaf
$(V^{\prime},\theta^{\prime})\subset(V,\theta)$ we have a natural inclusion of
flat sheaves $C_{\tilde{Y}}^{-1}(V^{\prime},\theta^{\prime})\subset
C^{-1}_{\tilde{Y}}(V,\theta)$. In particular, if $(G,0)$ is a Higgs subsheaf
of $(V,\theta)$, then $(F^{*}G,\nabla_{can})$ is a flat subsheaf of
$C^{-1}_{\tilde{Y}}(V,\theta)$.

###  3.2 An inverse Cartier transform on nodal curves

If $X$ is not smooth, the construction from the previous section fails because
a given lift $\tilde{X}$ may not admit local lifts of Frobenius. If $X$ is
normal, one may get some mileage out of restricting to the smooth locus as in
[langer], and there is a logarithmic version due to Schepler [schepler].
However, the case of interest for us will be that of nodal curves whose nodes
smooth out. In this section we construct two types of inverse Cartier
transforms: the first for Higgs sheaves pulled-back from a smooth variety $Y$,
and the second for Higgs subsheaves of such pull-backs to nodal curves after
twisting sufficiently by the ideal of the nodes.

####  3.2.1 The inverse Cartier transform of pullback Higgs sheaves

Let $\tilde{Y}/W_{2}$ be a smooth $W_{2}$-scheme, $X/k$ a smooth variety, and
$f:X\to Y$ a morphism. Let $(V_{Y},\theta_{Y})\in HIG_{m}(Y)$ be a Higgs sheaf
on $Y$. Denote $V_{X}=f^{*}V_{Y}$ and by
$\rho_{Y|X}:f^{*}\Omega_{Y}\to\Omega_{X}$ the natural map. We have two sorts
of Higgs fields on $V_{X}$, each satisfying the obvious integrability
condition: $\theta_{Y|X}:V_{X}\to V_{X}\otimes f^{*}\Omega_{Y}$ and
$\theta_{X}:V_{X}\to V_{X}\otimes\Omega_{X}$, where
$\theta_{X}=(\operatorname{id}\otimes\rho_{Y|X})\circ\theta_{Y|X}$.

######  Lemma 3.1.

The pullback as a flat sheaf $f^{*}C_{\tilde{Y}}^{-1}(V_{Y},\theta_{Y})$ is
obtained by locally forming
$(F^{*}V_{X},\nabla_{can}+\rho_{Y|X}\circ(f^{*}\zeta_{i})(F^{*}\theta_{Y|X}))$
and gluing via $\exp((f^{*}s_{ij})(F^{*}\theta_{Y|X}))$.

###### Proof.

Frobenius commutes with $f$. ∎

In fact the pullback twisting data may be used to define an inverse Cartier
transform of a more general kind of object which we call a pullback Higgs
sheaf on $X$: precisely, this is a pair $(V,\theta)$ consisting of a locally
free $\operatorname{\mathcal{O}}_{X}$-module $V$ on $X$ and an
$\operatorname{\mathcal{O}}_{X}$-linear map $\theta:V\to V\otimes
f^{*}\Omega_{Y}$ satisfying the integrability condition. Indeed, by the same
computations as in [LSZtwist], we define a flat sheaf
$C^{-1}_{\tilde{Y}|X}(V,\theta)$ by locally forming
$(F^{*}V,\nabla_{can}+\rho_{Y\mid X}\circ(f^{*}\zeta_{i})(F^{*}\theta))$ and
gluing via $\exp((f^{*}s_{ij})(F^{*}\theta))$. If we define morphisms of
pullback Higgs sheaves in the natural way, and $HIG_{m}(Y|X)$ the category of
pullback Higgs sheaves of exponent $\leq m$, we obtain the following result,
whose details we leave to the reader:

######  Lemma 3.2.

The above definition yields a functor $C^{-1}_{\tilde{Y}|X}:HIG_{p}(Y|X)\to
MIC_{p}(X)$ and there is a natural isomorphism
$C^{-1}_{\tilde{Y}|X}(V_{X},\theta_{Y|X})\cong
f^{*}C^{-1}_{\tilde{Y}}(V_{Y},\theta_{Y})$.

######  Corollary 3.3.

For $(V_{Y},\theta_{Y})\in HIG_{m}(Y)$ as above, if
$(G,0)\to(V_{X},\theta_{Y|X})$ is a morphism of restricted Higgs sheaves, then

$(F^{*}G,\nabla_{can})\cong C^{-1}_{\tilde{Y}|X}(G,0)\to
f^{*}C_{\tilde{Y}}^{-1}(V_{Y},\theta_{Y})$

is a morphism of flat sheaves.

####  3.2.2 The case of nodal curves

Now take $\tilde{X}/W_{2}$ to be a flat nodal curve and
$\tilde{\iota}:\tilde{X}\to\tilde{Y}$ a locally closed embedding over $W_{2}$,
with base-change $\iota:X\to Y$ to $k$. The case of an embedding will be
enough to deduce the case of a general $W_{2}$-liftable morphism (see the
proof of Proposition 3.7). Let $\nu:X^{\prime}\to X$ be the normalization of
the fiber over $k$, $\iota^{\prime}=\nu\circ\iota$, and let $D$ be the divisor
of points of $X^{\prime}$ mapping to nodes of $X$, which we call prenodes. Let
$Y_{i}$ be an affine open covering of $Y$. Set $X_{i}=X\cap Y_{i}$ and
$X_{ij}=X_{i}\cap X_{j}$; likewise let $X^{\prime}_{i}=\nu^{-1}(X_{i})$ and
$X^{\prime}_{ij}=\nu^{-1}(X_{ij})$. We can and will assume that no overlap
$X_{ij}$ contains a node.

If $I$ is the ideal of $X$ in $Y$, $N_{X/Y}^{\vee}=I/I^{2}$ is the conormal
bundle of $X$, which is locally free. We have a natural exact sequence

$0\to
N^{\vee}_{X/Y}\xrightarrow{\delta}\iota^{*}\Omega_{Y}\xrightarrow{\rho_{Y|X}}\Omega_{X}\to
0.$ (3.2.1)

Let $\tilde{I}$ be the ideal of $\tilde{X}$ in $\tilde{Y}$. By lifting to
$\tilde{I}$ we have morphisms

$t_{i}:=p^{-1}\tilde{\iota}^{*}\tilde{F}_{i}:F^{*}N_{X_{i}/Y_{i}}^{\vee}\to\mathcal{O}_{X_{i}}.$

since the map $p^{-1}\tilde{F}_{i}:\tilde{I}\to F_{*}\mathcal{O}_{X_{i}}$
kills $pI+\tilde{I}^{2}$. Note that $\rho_{Y|X}\circ\iota^{*}\zeta_{i}\circ
F^{*}\delta=dt_{i}$ on $X_{i}$ and $\iota^{*}s_{ij}\circ
F^{*}\delta=t_{i|j}-t_{j|i}$ on the overlap $X_{ij}$.

The pullback of (3.2.1) to $X^{\prime}$ fits into a commutative diagram with
exact rows and columns

${0}$${0}$${0}$${Q}$${\nu^{*}\Omega_{X}}$${\Omega_{X^{\prime}}}$${0}$${0}$${N_{X^{\prime}/Y}^{\vee}}$${\iota^{\prime*}\Omega_{Y}}$${\Omega_{X^{\prime}}}$${0}$${\nu^{*}N_{X/Y}^{\vee}}$${\nu^{*}N_{X/Y}^{\vee}}$${0}$${0}$$\scriptstyle{\rho_{Y|X^{\prime}}}$$\scriptstyle{\nu^{*}\rho_{Y|X}}$$\scriptstyle{\nu^{*}\delta}$

The locally free sheaf $N_{X^{\prime}/Y}^{\vee}$ is the conormal bundle of the
individual branches of $X^{\prime}$ in $Y$. Note that for any locally free
sheaf $V$ on a curve and any subspace $U\subset k(x)\otimes V$ of the fiber at
$x$, we may define the subsheaf $F$ of $V(x)$ of sections with simple poles in
$U$ at $x$ by taking $F(-x)$ to be the kernel of the composition $V\to
k(x)\otimes V\to(k(x)\otimes V)/U$. For any prenode $x_{1}$, there is a
canonical subspace of the fiber $k(x_{1})\otimes N_{X^{\prime}/Y}$ given by
the tangent direction of the curve at the other prenode $x_{2}$ which is
attached to $x_{1}$ in $X$. We define $M_{X^{\prime}/Y}\subset
N_{X^{\prime}/Y}(D)$ to be the subsheaf of sections of $N_{X^{\prime}/Y}$ with
simple poles at each prenode $x_{1}$ in this subspace.

######  Lemma 3.4.

  1. 1.

$Q$ has length one at each prenode. In particular, there is a natural
inclusion $\nu^{*}N_{X/Y}\subset N_{X^{\prime}/Y}(D)$.

  2. 2.

Via this inclusion, $M_{X^{\prime}/Y}=\nu^{*}N_{X/Y}$.

###### Proof.

We may prove the claim étale locally, and therefore assume we have coordinates
$y_{1},\ldots,y_{n}$ of $Y$ and that $X$ has ideal
$(y_{1}y_{2},y_{3},\ldots,y_{n})$. On the branch cut out by
$(y_{2},y_{3},\ldots,y_{n})$, $N^{\vee}_{X^{\prime}/Y}$ is generated by
$dy_{2},\ldots,dy_{n}$, while $\nu^{*}N^{\vee}_{X/Y}$ is generated by
$y_{1}dy_{2},dy_{3},\ldots,dy_{n}$. This gives the first claim. Since
$M_{X^{\prime}/Y}$ is generated by $y_{1}^{-1}\frac{\partial}{\partial
y_{2}},\frac{\partial}{\partial y_{3}},\ldots,\frac{\partial}{\partial
y_{n}}$, the second claim is established as well. ∎

Let $W_{X^{\prime}/Y}$ be the subsheaf of $(\iota^{\prime*}T_{Y})(D)$ of
sections with simple poles at each prenode in the tangent direction of the
attached branch. By the lemma, there is then a natural exact sequence

$0\to\nu^{*}N_{X/Y}^{\vee}\to W^{\vee}_{X^{\prime}/Y}\to\Omega_{X^{\prime}}\to
0.$ (3.2.2)

Choose a splitting of (3.2.2) on each $X_{i}^{\prime}$ and use it to obtain
$\hat{t}_{i}$ such that the following diagram commutes.

${F^{*}\nu^{*}N^{\vee}_{X_{i}/Y_{i}}}$${\mathcal{O}_{X^{\prime}_{i}}}$${F^{*}W^{\vee}_{X^{\prime}_{i}/Y_{i}}}$$\scriptstyle{\nu^{*}t_{i}}$$\scriptstyle{\hat{t}_{i}}$

The lifts $\hat{t}_{i}$ yield
$\eta_{i}:F^{*}\Omega_{X_{i}^{\prime}}\to\Omega_{X_{i}^{\prime}}$ such that
$\eta_{i}\circ
F^{*}\rho_{Y_{i}|X^{\prime}_{i}}=\rho_{Y_{i}|X^{\prime}_{i}}\circ\iota^{\prime*}\zeta_{i}-d\hat{t}_{i}$
as maps $F^{*}W_{X^{\prime}_{i}/Y_{i}}^{\vee}\to\Omega_{X_{i}^{\prime}}$, and
moreover on overlaps we may write $u_{ij}\circ
F^{*}\rho_{Y_{ij}|X^{\prime}_{ij}}=\iota^{\prime*}s_{ij}-(\hat{t}_{i|j}-\hat{t}_{j|i})$
for $u_{ij}:F^{*}\Omega_{X_{ij}}\to\mathcal{O}_{X_{ij}}$. We have
$du_{ij}=\eta_{i|j}-\eta_{j|i}$.

Let $(V,\theta)\in HIG_{m}(Y|X^{\prime})$ be a pullback Higgs sheaf. Let $J$
be the ideal of the (reduced) set of prenodes of $X^{\prime}$ at which
$\theta$ is nonzero in the tangent direction of the attached branch. We write
$\hat{t}_{i}(F^{*}\theta)$ for the resulting composition

$F^{*}(J\otimes
V|_{X_{i}^{\prime}})\xrightarrow{F^{*}\theta}F^{*}(V|_{X_{i}^{\prime}}\otimes
W_{X_{i}^{\prime}/Y_{i}}^{\vee})\xrightarrow{\operatorname{id}\otimes\hat{t}_{i}}F^{*}V|_{X_{i}^{\prime}}$

and likewise for $\iota^{\prime*}s_{ij}(F^{*}\theta)$ and $u_{ij}\circ
F^{*}\rho_{Y|X^{\prime}}(F^{*}\theta)$. We therefore have local morphisms

$\exp(-\hat{t}_{i}(F^{*}\theta)):F^{*}(J^{m-1}\otimes V)|_{X_{i}^{\prime}}\to
V|_{X_{i}^{\prime}}$

and, because the overlaps contain no nodes, commutative diagrams on the
overlaps

${F^{*}(J^{m-1}\otimes
V)|_{X_{ij}^{\prime}}}$${F^{*}V|_{X_{ij}^{\prime}}}$${F^{*}(J^{m-1}\otimes
V)|_{X_{ij}^{\prime}}}$${F^{*}V|_{X_{ij}^{\prime}}.}$$\scriptstyle{\exp(-\hat{t}_{i}(F^{*}\theta))}$$\scriptstyle{\exp(u_{ij}\circ
F^{*}\rho_{Y_{ij}|X^{\prime}_{ij}}(F^{*}\theta))}$$\scriptstyle{\exp(\iota^{\prime*}s_{ij}(F^{*}\theta))}$$\scriptstyle{\exp(-\hat{t}_{j}(F^{*}\theta))}$

Denote by $(V,\theta_{X})$ the Higgs sheaf associated to $(V,\theta)$ and let
$(G,0)\subset(V,\theta_{X})$ be a Higgs subsheaf with vanishing Higgs field.
By restricting the above local endomorphisms, we obtain local maps

$\exp(-\hat{t}_{i}(F^{*}\theta)):F^{*}(J^{m-1}\otimes G)|_{X_{i}^{\prime}}\to
F^{*}V|_{X_{i}^{\prime}}$ (3.2.3)

and commutative diagrams on the overlaps

${F^{*}V|_{X_{ij}^{\prime}}}$${F^{*}(J^{m-1}\otimes
G)|_{X_{ij}^{\prime}}}$${F^{*}V|_{X_{ij}^{\prime}}}$$\scriptstyle{\exp(\iota^{\prime*}s_{ij}(F^{*}\theta))}$$\scriptstyle{\exp(-\hat{t}_{i}(F^{*}\theta))}$$\scriptstyle{\exp(-\hat{t}_{j}(F^{*}\theta))}$
(3.2.4)

again because there are no nodes in the overlaps.

######  Lemma 3.5.

The local maps (3.2.3) give morphisms of flat sheaves

$\exp(-\hat{t}_{i}(F^{*}\theta)):(F^{*}(J^{m-1}\otimes
G)|_{X_{i}^{\prime}},\nabla_{can})\to(F^{*}V|_{X_{i}^{\prime}},\nabla_{can}+\rho_{Y_{i}|X^{\prime}_{i}}\circ\iota^{\prime*}\zeta_{i}(F^{*}\theta))$

and glue via (3.2.4) to a global morphism of flat sheaves
$(F^{*}(J^{m-1}\otimes G),\nabla_{can})\to
C_{\tilde{Y}|X^{\prime}}^{-1}(V,\theta)$. The morphism does not depend on the
choice of splittings of (3.2.2).

###### Proof.

We first show that the local map (3.2.3) respects the connections. Let $s$ be
a local section of $J^{m-1}\otimes G$, so that $\nabla_{can}F^{*}s=0$. We have

$\displaystyle\nabla_{can}\exp(-\hat{t}_{i}(F^{*}\theta))F^{*}s$
$\displaystyle=\exp(-\hat{t}_{i}(F^{*}\theta))(-d\hat{t}_{i}(F^{*}\theta))F^{*}s$
$\displaystyle=\exp(-\hat{t}_{i}(F^{*}\theta))(-\rho_{Y_{i}|X^{\prime}_{i}}\circ\iota^{\prime*}\zeta_{i}(F^{*}\theta))F^{*}s$
$\displaystyle=-\rho_{Y_{i}|X_{i}^{\prime}}\circ\iota^{\prime*}\zeta_{i}(F^{*}\theta)\exp(-\hat{t}_{i}(F^{*}\theta))F^{*}s$

which implies the compatibility of the connections. According to the
description preceding Lemma 3.2, these maps glue as morphisms of flat sheaves
since the vertical map of (3.2.4) is the gluing map for
$C_{\tilde{Y}|X^{\prime}}^{-1}(V,\theta)$.

A different choice of splittings of (3.2.2) yields $\hat{t}_{i}^{\prime}$ for
which $\hat{t}_{i}-\hat{t}_{i}^{\prime}=r_{i}\circ
F^{*}\rho_{Y_{i}|X^{\prime}_{i}}$ for
$r_{i}:F^{*}\Omega_{X_{i}^{\prime}}\to\mathcal{O}_{X_{i}^{\prime}}$. Since
$r_{i}\circ F^{*}\rho_{Y_{i}|X^{\prime}_{i}}(F^{*}\theta)$ kills
$F^{*}(J^{m-1}\otimes G)$, the maps in (3.2.3) do not depend on the choice of
lift. ∎

######  Remark 3.6.

Its reasonable to ask whether there is a version of Lemma 3.5 for a Higgs
subsheaf $(G,\psi)\subset(V,\theta_{X})$ for which the Higgs field is nonzero.
The maps (3.2.3) glue via the maps $\exp(u_{ij}(F^{*}\psi))$, and the natural
local connection which glues is $\nabla_{can}+\eta_{i}(F^{*}\psi)$, which must
be interpreted as a connection with log poles. It is no longer clear that
either the resulting flat sheaf with log poles or the morphism is independent
of the choice of splittings of (3.2.2) (even up to isomorphism).

In summary, we have the following:

######  Proposition 3.7.

Let $\tilde{Y}/W_{2}$ be smooth, $\tilde{X}/W_{2}$ a nodal curve,
$\tilde{f}:\tilde{X}\to\tilde{Y}$ a $W_{2}$-morphism, $\nu:X^{\prime}\to X$
the normalization of the fiber over $k$, and $f^{\prime}=f\circ\nu$. Let
$(V,\theta)\in HIG_{m}(Y)$ with $m\leq p$, and let $J$ be the ideal of the
(reduced) set of prenodes in $X^{\prime}$ at which $\theta$ is nonzero in the
tangent direction of the attached branch. Then for any Higgs subsheaf
$(G,0)\subset f^{\prime*}(V,\theta)$ there is a flat subsheaf
$(F^{*}(J^{m-1}\otimes G),\nabla_{can})\subset
f^{\prime*}C^{-1}_{\tilde{Y}}(V,\theta)$. The morphism of the underlying
sheaves is locally isomorphic to the natural inclusion $F^{*}(J^{m-1}\otimes
G)\subset F^{*}f^{\prime*}V$.

###### Proof.

By [OV], the inverse Cartier transform commutes with pullback under morphisms
of smooth $k$-varieties which lift to $W_{2}$. Thus we may assume $f$ is an
embedding $\iota$ by replacing $Y$ with $\mathbb{P}^{N}\times Y$. The claim
then follows from Lemma 3.2 and Lemma 3.5. ∎

###  3.3 Applications to Fontaine–Laffaille modules

In this subsection we prove some degree bounds on Higgs subsheaves of the
graded object of a Fontaine–Laffaille module pulled-back to a nodal curve
(Propositions 3.9 and 3.10 below). Recall [OV, §4.6] that we may think of a
(strict) $p$-torsion Fontaine–Laffaille module
$\mathbf{M}=(M,\nabla,F^{\bullet},\Phi)$ on a lift $\tilde{Y}$ of a smooth
variety $Y/k$ as a locally free flat sheaf $(M,\nabla)$ with a locally split
filtration $F^{\bullet}$ satisfying Griffiths transversality and equipped with
an isomorphism of flat sheaves

$\Phi:C^{-1}_{\tilde{Y}}\operatorname{gr}_{F}\mathbf{M}\xrightarrow{\cong}(M,\nabla).$

where
$\operatorname{gr}_{F}\mathbf{M}:=(\operatorname{gr}_{F}M,\operatorname{gr}_{F}\nabla)$
is the associated Higgs bundle. The category of $p$-torsion Fontaine–Laffaille
modules $\mathbf{M}$ whose graded object $\operatorname{gr}_{F}M$ has nonzero
pieces in degrees $[a,w+a]$ is denoted
$\mathrm{MF}^{\nabla}_{[a,w+a],1}(\tilde{Y})$. Note that for $p$-torsion
Fontaine–Laffaille modules, the index of the Hodge filtration may be freely
shifted, as there is no integrality to force the degrees to be nonnegative.

We give the setup for these results now for convenience.

######  Setup 3.8.

Let $\tilde{Y}/W_{2}$ be a smooth $W_{2}$-scheme equipped with a strict
$p$-torsion Fontaine–Laffaille module
$\mathbf{M}=(M,\nabla,F^{\bullet}M,\Phi)\in\mathrm{MF}^{\nabla}_{[a,w+a],1}(\tilde{Y})$.
Assume $p$ is an odd prime with $p-1\geq w$.

Let $\tilde{X}/W_{2}$ be a nodal curve together with a $W_{2}$-morphism
$\tilde{f}:\tilde{X}\to\tilde{Y}$. Let $\nu_{B}:B\to X$ be the normalization
of a proper component of the special fiber $X$ and let $g:B\to Y$ be the
composition $g=f\circ\nu_{B}$. Let $D_{B,\mathbf{M}}$ the divisor of prenodes
on $B$ at which the Higgs field $\operatorname{gr}_{F}\nabla$ is nonzero in
the tangent direction of the attached branch, and $d_{B,\mathbf{M}}=\deg
D_{B,\mathbf{M}}$ its degree.

######  Proposition 3.9.

Assume Setup 3.8. Suppose $(V,\theta)\subset
g^{*}\operatorname{gr}_{F}\mathbf{M}$ is a Higgs subsheaf. Then $\mu(V)\leq
wd_{B,\mathbf{M}}$ provided $p-1>wd_{B,\mathbf{M}}\operatorname{rk}(V)$.

###### Proof.

We argue as in the proof of Ogus–Vologodsky [OV]. Note that the exponent of
$\operatorname{gr}_{F}\mathbf{M}$ is $m\leq w+1$. We first reduce to the case
$(V,\theta)=(L,0)$ with rank one by taking the $\operatorname{rk}(V)$-th wedge
power. We must then show that any Higgs sheaf $(L,0)\subset
g^{*}\operatorname{gr}_{F}\mathbf{M}$ with $\operatorname{rk}(L)=1$ and
vanishing Higgs field has $\deg(L)\leq wd_{B,\mathbf{M}}$. Suppose this is not
the case, and let $(L,0)$ be a rank one Higgs subsheaf with vanishing Higgs
field of maximal degree, $\deg(L)>wd_{B,\mathbf{M}}$. Let
$J_{B,\mathbf{M}}=\mathcal{O}_{B}(-D_{B,\mathbf{M}})$. By Proposition 3.7 we
have that

$(F^{*}(J_{B,\mathbf{M}}^{m-1}\otimes L),\nabla_{can})\to
g^{*}C^{-1}_{\tilde{Y}}\operatorname{gr}_{F}\mathbf{M}\xrightarrow{g^{*}\Phi}g^{*}(M,\nabla)$

is an injective morphism of flat sheaves. Let $i\geq 0$ be minimal such that

$F^{*}(J_{B,\mathbf{M}}^{m-1}\otimes L)\to g^{*}M/F^{i+1}g^{*}M$

is nonzero. We therefore have an injective map

$F^{*}(J_{B,\mathbf{M}}^{m-1}\otimes L)\to\operatorname{gr}^{i}_{F}g^{*}M.$
(3.3.1)

Since $(F^{*}(J_{B,\mathbf{M}}^{m-1}\otimes L),\nabla_{can})$ is a flat
subsheaf and the Higgs field of $\operatorname{gr}_{F}g^{*}M$ is
$\operatorname{gr}_{F}\nabla$, (3.3.1) yields an injective morphism of Higgs
sheaves

$(F^{*}(J_{B,\mathbf{M}}^{m-1}\otimes L),0)\to
g^{*}\operatorname{gr}_{F}\mathbf{M}.$

But

$\displaystyle\deg(F^{*}(J_{B,\mathbf{M}}^{m-1}\otimes L))$
$\displaystyle=p(\deg(L)-(m-1)d_{B,\mathbf{M}})$ $\displaystyle\geq
p(\deg(L)-wd_{B,\mathbf{M}})$
$\displaystyle\geq\deg(L)-wd_{B,\mathbf{M}}+(p-1)$ $\displaystyle>\deg(L)$

because of the assumption on $p$, and this contradiction finishes the proof. ∎

Let $(V,\theta)$ be a Higgs bundle on a smooth variety $Y/k$. There is a
natural endomorphism Higgs bundle
$(\operatorname{\mathsf{End}}(V),\operatorname{ad}\theta)$. The
Kodaira–Spencer map of $(V,\theta)$ is the Higgs field of
$(\operatorname{\mathsf{End}}(V),\operatorname{ad}\theta)$, thought of as a
map $\operatorname{ad}\theta:T_{Y}\to\operatorname{\mathsf{End}}(V)$. For
$\mathbf{M}\in\operatorname{MF}^{\nabla}_{[a,w+a],1}(Y)$, there is also a
natural endomorphism object
$\operatorname{\mathsf{End}}(\mathbf{M})=(\operatorname{\mathsf{End}}(M),\operatorname{ad}\nabla,F^{\bullet}\operatorname{\mathsf{End}}(M),\operatorname{ad}\Phi)\in\operatorname{MF}^{\nabla}_{[-w,w],1}(Y)$
in the category of Fontaine–Laffaille modules. The Kodaira–Spencer map of
$\mathbf{M}$ is the Kodaira–Spencer map of the associated Higgs sheaf
$\operatorname{gr}_{F}\mathbf{M}$.

######  Proposition 3.10.

Assume Setup 3.8 and let
$P=\operatorname{img}(g^{*}\operatorname{gr}_{F}\operatorname{ad}\nabla)$ be
the image of the pullback of the Kodaira–Spencer map
$g^{*}\operatorname{gr}_{F}\operatorname{ad}\nabla:g^{*}T_{Y}\to
g^{*}\operatorname{gr}_{F}\operatorname{\mathsf{End}}(\mathbf{M})$. Then $P$
does not contain a positive degree subsheaf of rank $r$ provided
$p>2rwd_{B,\wedge^{r}\operatorname{\mathsf{End}}(\mathbf{M})}$.

###### Proof.

For simplicity of notation, set $(V,\theta)=\operatorname{gr}_{F}\mathbf{M}$.
Observe that the Kodaira–Spencer map
$\operatorname{ad}\theta:(T_{Y},0)\to(\operatorname{\mathsf{End}}(V),\operatorname{ad}\theta)$
is a morphism of Higgs sheaves, as therefore is
$\wedge^{r}\operatorname{ad}\theta:(\wedge^{r}T_{Y},0)\to(\wedge^{r}\operatorname{\mathsf{End}}(V),\wedge^{r}\operatorname{ad}\theta)$.
Thus, we have morphisms of flat sheaves

$(F^{*}\wedge^{r}T_{Y},\nabla_{can})=C_{\tilde{Y}}^{-1}(\wedge^{r}T_{Y},0)\to
C_{\tilde{Y}}^{-1}(\wedge^{r}\operatorname{\mathsf{End}}(V),\wedge^{r}\operatorname{ad}\theta)\xrightarrow{\Phi}(\wedge^{r}\operatorname{\mathsf{End}}(M),\wedge^{r}\operatorname{ad}\nabla)$

and so morphisms of flat sheaves

$g^{*}(F^{*}\wedge^{r}T_{Y},\nabla_{can})\to
g^{*}C_{\tilde{Y}}^{-1}\operatorname{gr}_{F}\wedge^{r}\operatorname{\mathsf{End}}(\mathbf{M})\xrightarrow{g^{*}\Phi}g^{*}(\wedge^{r}\operatorname{\mathsf{End}}(M),\wedge^{r}\operatorname{ad}\nabla).$

Observe that the image of the first map is naturally identified with
$(F^{*}\wedge^{r}P,\nabla_{\mathrm{can}})$. Indeed, its enough to check this
on the level of sheaves. The first map is locally identified with
$g^{*}F^{*}\wedge^{r}\operatorname{ad}\theta$, while
$(F^{*}\wedge^{r}P,\nabla_{\mathrm{can}})$ is the image of
$F^{*}\wedge^{r}g^{*}\operatorname{ad}\theta$, since $F^{*},\wedge^{r}$
preserve injections and surjections of locally free sheaves. The functors
$F^{*},g^{*},\wedge^{r}$ all naturally commute, so the claim follows.

If $G\subset P$ is subsheaf with $\operatorname{rk}(G)=r$, then setting
$L=\wedge^{r}G$ by the same argument as in the previous proof we get an
injective morphism of Higgs sheaves

$(F^{*}L,0)\to
g^{*}\operatorname{gr}_{F}\wedge^{r}\operatorname{\mathsf{End}}(\mathbf{M})$

but then by Proposition 3.9 we would have

$2rwd_{B,\wedge^{r}\operatorname{\mathsf{End}}(\mathbf{M})}\geq\deg(F^{*}L)=p\deg(L)$

which given the assumption on $p$ means we must have $\deg(L)=\deg(G)\leq 0$.

∎

######  Remark 3.11.

The same proof shows for instance that in Setup 3.8,
$g^{*}\operatorname{gr}^{0}_{F}M$ does not contain a positive degree subsheaf
of rank $r$ provided $p>rwd_{B,\wedge^{r}\mathbf{M}}$.

Proposition 3.10 will allow us to control the liftability of rational
components of nodal curves with boundedly many nodes. This will be the key
ingredient for the surface case of Proposition 4.4. The normal case requires a
bit more care, so we only treat the smooth case now, and postpone the general
setup until then.

######  Definition 3.12.

Given a Higgs bundle $(V,\theta)$ on $Y$, we say the Kodaira–Spencer map is
quasi-finite if there is a partition of $Y$ into locally closed smooth
subschemes $Y^{i}$ on which the Kodaira–Spencer map
$T_{Y^{i}}\to\operatorname{\mathsf{End}}(V)|_{Y^{i}}$ is immersive, meaning it
is injective with locally-free cokernel.

Of course, if the Kodaira–Spencer map of $(V,\theta)$ is already immersive,
then $(V,\theta)$ has quasi-finite Kodaira–Spencer map, and this will be the
relevant case for Shimura varieties.

######  Corollary 3.13.

Assume Setup 3.8, and further assume that the Kodaira–Spencer map of
$\operatorname{gr}_{F}(M,\nabla)$ is quasifinite. Then if $B$ is rational,
$g:B\to Y$ is constant provided
$p>2wd_{B,\operatorname{\mathsf{End}}(\mathbf{M})}$

###### Proof.

Let $B\xrightarrow{\alpha}B^{\prime}\xrightarrow{\beta}Y$ be the factorization
of $g$ where $\beta$ is the normalization of the image. By assumption, we have
$T_{B^{\prime}}\subset\beta^{*}\operatorname{gr}_{F}(\operatorname{\mathsf{End}}(M),\nabla)$
is contained in the image of the Kodaira–Spencer map, and therefore the same
is true for $\alpha^{*}T_{B^{\prime}}\subset
g^{*}\operatorname{gr}_{F}(\operatorname{\mathsf{End}}(M),\nabla)$, which
contradicts Proposition 3.10.

∎

######  Corollary 3.14.

Assume Setup 3.8, and further assume that the Kodaira–Spencer map of
$\operatorname{gr}_{F}(M,\nabla)$ is quasifinite. Finally, assume either

  1. 1.

$\tilde{X}$ is obtained by successively blowing up a smooth curve
$\tilde{X}^{\prime}/W_{2}$ at closed points and $p>2w$.

  2. 2.

$\tilde{X}$ is obtained by successively blowing up a nodal curve
$\tilde{X}^{\prime}/W_{2}$ at closed points and $p>4w$.

Then $X\to Y$ factors through $X\to X^{\prime}$.

###### Proof.

The dual graph of the exceptional curves is a tree and the components that are
not contracted form a subtree. The extremal components have
$d_{B,\mathbf{M}}\leq 1$ in the first case and $\leq 2$ in the second, so the
claim follows form the previous corollary. ∎

##  4 Criteria for local integral canonical models

The goal of this section is to give a general setting in which local integral
canonical models exists, in the sense of Definition 2.14. For simplicity, we
first give the smooth case.

######  Theorem 4.1.

Let $F$ be a mixed characteristic local field with ring of integers
$\mathcal{O}$ and residue field $k$ of characteristic $p$. Let
$\mathcal{S}/\mathcal{O}$ be a smooth $\mathcal{O}$-scheme equipped with:

  1. 1.

A log smooth compactification $\bar{\mathcal{S}}/\mathcal{O}$ of $\mathcal{S}$
with natural boundary.

  2. 2.

A strict $p$-torsion Fontaine-Laffaille module
$\mathbf{M}\in\mathrm{MF}^{\nabla}_{[a,w+a],1}(\mathcal{S})$ whose Higgs
bundle $\operatorname{gr}_{F}\mathbf{M}$ has quasifinite Kodaira–Spencer map.

Then if $p>2w$, $\mathcal{S}$ is an integral canonical model.

See Definition 2.2 and Definition 3.12 for the definitions of natural boundary
and quasifinite Kodaira–Spencer map. In the normal case, we will additionally
need a form of stratified resolution.

######  Definition 4.2.

Let $\mathcal{O}$ be a Dedekind domain and $\mathcal{Y}/\mathcal{O}$ a normal
finite type flat $\mathcal{O}$-scheme. A stratified resolution of
$\mathcal{Y}$ consists of:

  1. 1.

An increasing sequence of closed subspaces
$\mathcal{Y}^{1}\subset\ldots\subset\mathcal{Y}^{m}=\mathcal{Y}$, each flat
over $\mathcal{O}$, such that $\mathcal{Y}^{j}\setminus\mathcal{Y}^{j-1}$ is
smooth over $\mathcal{O}$ for each $j$. Let $\mathcal{Y}^{\prime
j}\to\mathcal{Y}^{j}$ be the normalization.

  2. 2.

For each $j$, a resolution $\pi^{\prime
j}:\mathcal{S}^{j}\to\mathcal{Y}^{\prime j}$ and a log smooth compactification
$\bar{\mathcal{S}}^{j}/\mathcal{O}$ with the property that the composition
$\mathcal{S}^{j}\to\mathcal{Y}^{\prime j}\to\mathcal{Y}^{j}$ (which we call
$\pi^{j}:\mathcal{S}^{j}\to\mathcal{Y}^{j}$) is an isomorphism over
$\mathcal{Y}^{j}\setminus\mathcal{Y}^{j-1}$.

We say the stratified resolution:

  1. 3.

is _uniform_ if the resolution of the top stratum
$\pi^{m}:\mathcal{S}^{m}\to\mathcal{Y}^{m}=\mathcal{Y}$ is uniform.

  2. 4.

_has natural boundary_ if each $\bar{\mathcal{S}}^{j}$ has natural boundary.

######  Theorem 4.3.

Let $F$ be a mixed characteristic local field with ring of integers
$\mathcal{O}$ and residue field $k$ of characteristic $p$. Let
$\mathcal{Y}/\mathcal{O}$ be a normal finite type flat $\mathcal{O}$-scheme
equipped with:

  1. 1.

A uniform stratified resolution
$\\{\pi^{j}:\mathcal{S}^{j}\to\mathcal{Y}^{j}\\}$ with natural boundary.

  2. 2.

A Higgs bundle $(V,\theta)$ on $\mathcal{Y}_{0}$ with quasifinite
Kodaira–Spencer map.

  3. 3.

For each $j$ a strict $p$-torsion Fontaine-Laffaille module
$\mathbf{M}^{j}\in\mathrm{MF}^{\nabla}_{[a^{j},w^{j}+a^{j}],1}(\mathcal{S}^{j})$
and a nonzero torsion-free coherent sheaf $G^{j}$ on $\mathcal{S}^{j}$ such
that

$\displaystyle(G^{j},0)\otimes\operatorname{gr}_{F}\mathbf{M}^{j}$
$\displaystyle\cong(G^{j},0)\otimes\pi^{j*}(V,\theta)$
$\displaystyle(G^{j},0)\otimes\operatorname{gr}_{F}\operatorname{\mathsf{End}}(\mathbf{M}^{j})$
$\displaystyle\cong(G^{j},0)\otimes\pi^{j*}(\operatorname{\mathsf{End}}(V),\operatorname{ad}\theta)$

as Higgs sheaves.

Then if $p>2\max(w^{j})$, $\mathcal{Y}$ is an integral canonical model.

Note that Theorem 4.3 directly implies Theorem 4.1 by taking the trivial
stratification, $(V,\theta)=\operatorname{gr}_{F}\mathbf{M}$, and
$(G^{m},0)=(\mathcal{O}_{\mathcal{S}_{0}},0)$.

###  4.1 Extension in codimension $\geq 2$

We now use the results of Section 3.3 to show that, in the presence of a
Fontaine–Laffaille module with quasi-finite Kodaira–Spencer map, morphisms
from smooth sources which are defined outside of a set of codimension $\geq 2$
extend.

######  Proposition 4.4.

Assume the setup of Theorem 4.3. Let $\mathcal{X}/\mathcal{O}$ be a smooth
$\mathcal{O}$-scheme and $f:\mathcal{X}\setminus\mathcal{Z}\to\mathcal{Y}$ a
morphism for $\mathcal{Z}\subset\mathcal{X}$ a subset of codimension $\geq 2$
which is contained in the special fiber. Then there is an extension
$f:\mathcal{X}\to\mathcal{Y}$.

###### Proof.

If $j$ is minimal such that $Y^{j}$ contains the image of the generic fiber
$X$, then the image of $X$ meets $Y^{j}\setminus Y^{j-1}$, so there is a dense
open set of $X$ which lifts to $S^{j}$. By purity of the branch locus,
$\mathcal{X}\setminus\mathcal{Z}\to\mathcal{X}$ induces an isomorphism on
fundamental groups, so the map $X\to Y$ is $\mathcal{O}$-admissible. It
follows by Lemma 2.3 that there is a finite type flat
$\mathcal{X}^{\prime}/\mathcal{O}$, a proper morphism
$\mathcal{X}^{\prime}\to\mathcal{X}$ which is an isomorphism on a dense open
subset, and a morphism $\mathcal{X}^{\prime}\to\mathcal{S}^{j}$ whose generic
fiber lifts the morphism $X\to Y^{j}$.

There is nothing to prove for $\dim\mathcal{X}\leq 1$. For
$\dim\mathcal{X}\geq 2$, we argue as in [dejongoort] (see also [moretbailly,
faltingschai]) by induction on $\dim\mathcal{X}$ starting with the base case
$\dim\mathcal{X}=2$ and using deformation theory for the induction step.

Assume first that $\dim\mathcal{X}=2$. By resolution of singularities for
surfaces ([lipman], specifically [stacks-project, Tag 0AHI]), there is a
composition of blow-ups at closed points
$\mathcal{X}_{n}\to\cdots\to\mathcal{X}_{1}\to\mathcal{X}$ whose composition
factors as $\mathcal{X}_{n}\to\mathcal{X}^{\prime}\to\mathcal{X}$. To show
that $\mathcal{X}^{\prime}\to\mathcal{S}^{j}\to\mathcal{Y}$ factors through
$\mathcal{X}^{\prime}\to\mathcal{X}$ (and hence the existence of the required
extension), it suffices (for instance by Lemma 2.16) to show the following:

######  Claim.

The composition
$\mathcal{X}_{n}\to\mathcal{X}^{\prime}\to\mathcal{S}^{j}\to\mathcal{Y}$
contracts all exceptional curves.

###### Proof.

Note that by the quasifiniteness assumption on the Kodaira–Spencer map of
$(V,\theta)$, a component $B$ of the special fiber of $\mathcal{X}_{n}$ is
contracted if and only if the Higgs field of $(V,\theta)$ vanishes on the
image of $B$ in $\mathcal{Y}_{0}$. Moreover, for any such $B$, the Higgs field
of $\operatorname{gr}_{F}\operatorname{\mathsf{End}}(\mathbf{M}^{j})$ also
vanishes on $B$, by assumption (3). The exceptional curves form a tree and the
components $B$ which are not contracted form a sub-tree whose extremal
components have $d_{B,\operatorname{\mathsf{End}}(\mathbf{M}^{j})}\leq 1$, so
by Proposition 3.9 it suffices to show that
$g^{*}\operatorname{gr}_{F}\operatorname{\mathsf{End}}(\mathbf{M}^{j})$
contains a sub-Higgs sheaf of slope larger than $w^{j}$, where
$g:B\to\mathcal{S}^{j}_{0}$ is the map on the special fiber restricted to $B$.

Let $B\xrightarrow{\alpha}B^{\prime}\xrightarrow{\beta}\mathcal{Y}_{0}$ be the
factorization of $h:=\pi^{j}_{0}\circ g:B\to\mathcal{Y}_{0}$, where $\beta$ is
the normalization of the image. The image meets some stratum
$\mathcal{Y}^{i}_{0}$ in the open set where $\pi^{i}$ is an isomorphism, so
there is a lift $\gamma:B^{\prime}\to\mathcal{S}^{i}_{0}$ of $\beta$ for some
$i$. Note that the assumption on the Kodaira–Spencer map of $(V,\theta)$ and
assumption (3) imply that the Kodaira–Spencer map of
$\gamma^{*}\operatorname{gr}_{F}\mathbf{M}^{i}$ is also quasifinite, so
$(T_{B^{\prime}},0)$ is a Higgs subsheaf of
$\gamma^{*}\operatorname{gr}_{F}\operatorname{\mathsf{End}}(\mathbf{M}^{i})$.
As in the proof of Proposition 3.9, it follows that $(F^{*}T_{B^{\prime}},0)$
is a Higgs subsheaf of
$\gamma^{*}\operatorname{gr}_{F}\operatorname{\mathsf{End}}(\mathbf{M}^{i})$.
Finally, assumption (3) for the $i$th and $j$th strata mean we have
isomorphisms of Higgs sheaves

$(G,0)\otimes
g^{*}\operatorname{\mathsf{End}}(\mathbf{M}^{j})\cong(G,0)\otimes
h^{*}(\operatorname{\mathsf{End}}(V),\operatorname{ad}\theta)\cong(G,0)\otimes(\alpha\circ\gamma)^{*}\operatorname{\mathsf{End}}(\mathbf{M}^{i})$

where $G=g^{*}G^{j}\otimes(\gamma\circ\alpha)^{*}G^{i}$. Thus,
$(G,0)\otimes\alpha^{*}(F^{*}T_{B^{\prime}},0)$ is a Higgs subsheaf of
$(G,0)\otimes g^{*}\operatorname{\mathsf{End}}(\mathbf{M}^{j})$. Since $B$ is
rational, $G$ splits333Possibly after a field extension. as a sum of factors
$\mathcal{O}_{B}(d)$, and taking the largest degree factor we find that
$((\alpha^{*}F^{*}T_{B^{\prime}})(e),0)$ is a Higgs subsheaf of
$g^{*}\operatorname{gr}_{F}\operatorname{\mathsf{End}}(\mathbf{M}^{j})$ for
some $e\geq 0$. Since $\deg(\alpha^{*}F^{*}T_{B^{\prime}})(e)\geq 2p$, the
conclusion follows. ∎

We now treat the general case $\dim\mathcal{X}>2$ by induction via deformation
theory. It will be sufficient to show that for any $x\in\mathcal{Z}$, $f$
extends on an étale neighborhood of $x$. In particular we may assume
$\mathcal{X}$ is affine and $\mathcal{Y}$ is a closed subscheme of a smooth
affine $\mathcal{O}$-scheme $\mathcal{V}$. Let $\mathcal{H}\subset\mathcal{X}$
be a divisor containing $x$ such that $\mathcal{H}\cap\mathcal{Z}$ is
codimension $\geq 2$ in $\mathcal{H}$. By induction, the map
$\mathcal{H}\setminus\mathcal{Z}\to\mathcal{Y}$ extends to
$f_{\mathcal{H}}:\mathcal{H}\to\mathcal{Y}$. Let
$g_{\mathcal{H}}:\mathcal{H}\to\mathcal{V}$ be the composition with the
embedding $\mathcal{Y}\subset\mathcal{V}$. Let $\mathcal{H}_{i}$ be the $i$th
order thickening of $\mathcal{H}$ in $\mathcal{X}$. Given a lift
$\mathcal{H}_{i}\to\mathcal{Y}$ of
$f_{\mathcal{H}}:\mathcal{H}\to\mathcal{Y}$, lifts to a morphism
$\mathcal{H}_{i+1}\to\mathcal{V}$ are unobstructed since the obstruction lies
in
$H^{1}(\mathcal{H},\operatorname{\mathcal{O}}_{\mathcal{H}}(-i\mathcal{H})\otimes
g_{\mathcal{H}}^{*}T_{\mathcal{V}})=0$; the set of such lifts is naturally a
torsor under
$H^{0}(\mathcal{H}_{i},\operatorname{\mathcal{O}}_{\mathcal{H}}(-i\mathcal{H})\otimes
g_{\mathcal{H}}^{*}T_{\mathcal{V}})$. Likewise, lifts of
$\mathcal{H}_{i}\setminus\mathcal{Z}\to\mathcal{Y}$ to
$\mathcal{H}_{i+1}\setminus\mathcal{Z}\to\mathcal{V}$ are a torsor under
$H^{0}(\mathcal{H}\setminus\mathcal{Z},\operatorname{\mathcal{O}}_{\mathcal{H}}(-i\mathcal{H})\otimes
g_{\mathcal{H}\setminus\mathcal{Z}}^{*}T_{\mathcal{S}_{v}})\cong
H^{0}(\mathcal{H},\operatorname{\mathcal{O}}_{\mathcal{X}}(-i\mathcal{H})\otimes
g^{*}_{\mathcal{H}}T_{\mathcal{V}})$, since
$\operatorname{\mathcal{O}}_{\mathcal{H}}(-i\mathcal{H})\otimes
g^{*}_{\mathcal{H}}T_{\mathcal{V}}$ is locally free and
$\operatorname{codim}_{\mathcal{H}}\mathcal{Z}\geq 2$. In particular, there is
a lift $\mathcal{H}_{i+1}\to\mathcal{V}$ which agrees with the given map
$\mathcal{H}_{i+1}\setminus\mathcal{Z}\to\mathcal{Y}\to\mathcal{V}$. Since the
ideal of $\mathcal{H}_{i}$ in $\mathcal{H}_{i+1}$ is locally free (over
$\mathcal{H}$), this lift factors through a morphism
$\mathcal{H}_{i+1}\to\mathcal{Y}$. Letting
$\mathcal{X}^{\wedge}_{\mathcal{H}}$ denote the completion of $\mathcal{X}$
along $\mathcal{H}$, it follows that there is a formal map
$\mathcal{X}^{\wedge}_{\mathcal{H}}\to\mathcal{Y}$ whose restriction to
$(\mathcal{X}\setminus\mathcal{Z})_{\mathcal{H}\setminus\mathcal{Z}}^{\wedge}$
agrees with the completion of
$f:\mathcal{X}\setminus\mathcal{Z}\to\mathcal{Y}$ along
$\mathcal{H}\setminus\mathcal{Z}$.

Let $\mathcal{W}$ be the closure of the graph of
$\mathcal{X}\setminus\mathcal{Z}\to\mathcal{Y}$ in
$\mathcal{X}\times\mathcal{Y}$; note that since
$\mathcal{X}^{\prime}\to\mathcal{X}$ is proper, it follows that
$\mathcal{W}\to\mathcal{X}$ is as well. Since the ideal sheaf of
$\mathcal{H}_{i}$ in $\mathcal{H}_{i+1}$ is locally free (over $\mathcal{H}$)
for each $i$, the natural formal map
$\mathcal{X}_{\mathcal{H}}^{\wedge}\to\mathcal{X}\times\mathcal{Y}$ factors
through $\mathcal{W}$ as it does so over
$(\mathcal{X}\setminus\mathcal{Z})_{\mathcal{H}\setminus\mathcal{Z}}^{\wedge}$.
Thus, the projection $\mathcal{W}\to\mathcal{X}$ admits a section over
$\mathcal{X}_{\mathcal{H}}^{\wedge}$. Say the image of $w$ is $x$; then the
map
$\operatorname{Spec}(\operatorname{\mathcal{O}}_{\mathcal{W},w})\to\operatorname{Spec}(\operatorname{\mathcal{O}}_{\mathcal{X},x})$
admits a formal section. Since $\mathcal{W}\to\mathcal{X}$ is proper and
birational, the following lemma implies it is an isomorphism above $x$.

######  Lemma 4.5.

Let $q:\mathcal{W}\to\mathcal{X}$ be a proper birational morphism of finite-
type integral $\mathcal{O}$-schemes, $w$ a closed point of $\mathcal{W}$ and
$x=q(w)$. If the induced map
$\operatorname{Spec}(\operatorname{\mathcal{O}}_{\mathcal{W},w})\to\operatorname{Spec}(\operatorname{\mathcal{O}}_{\mathcal{X},x})$
admits a formal section then it is an isomorphism at $w$.

###### Proof.

The claim is étale-local on $\mathcal{X}$. By Artin approximation
[artinapprox, Corollary (2.5)] (see also [stacks-project, Tag 0CAU]), after
replacing $\mathcal{X}$ with an étale cover there is an algebraic map
$\mathcal{X}\to\mathcal{W}$ agreeing with the formal section to first order.
But the composition $\mathcal{X}\to\mathcal{W}\to\mathcal{X}$ agrees with the
identity to first order, and therefore is an isomorphism. We may assume it is
in fact the identity, in which case $\mathcal{W}\to\mathcal{X}$ admits an
algebraic section. It follows that the short exact sequence of
$\operatorname{\mathcal{O}}_{\mathcal{X},x}$-modules

$0\to\operatorname{\mathcal{O}}_{\mathcal{X},x}\to\operatorname{\mathcal{O}}_{\mathcal{W},w}\to\operatorname{\mathcal{O}}_{\mathcal{W},w}/\operatorname{\mathcal{O}}_{\mathcal{X},x}\to
0$

splits. Since $q$ is birational,
$\operatorname{\mathcal{O}}_{\mathcal{W},w}/\operatorname{\mathcal{O}}_{\mathcal{X},x}$
is torsion, hence must be zero since
$\operatorname{\mathcal{O}}_{\mathcal{W},w}$ is a domain. ∎

∎

###### Proof of Theorem 4.3.

By assumption, $\mathcal{Y}$ admits a uniform resolution. Given smooth
$\mathcal{X}/\mathcal{O}$ and an $\mathcal{O}$-admissible $f_{F}:X\to Y$, by
Corollary 2.4 $f_{F}$ extends in codimension 1 and by Proposition 4.4 the rest
of the way.

∎

######  Remark 4.6.

The proof of the claim in Proposition 4.4 in the setting of Theorem 4.1 is
more easily summarized as follows. If $B$ is an extremal component of the tree
of exceptional curves of $\mathcal{X}_{n}$, and
$B\xrightarrow{\alpha}B^{\prime}\xrightarrow{\beta}\mathcal{S}_{0}$ the
factorization through the normalization, then $\alpha^{*}T_{B^{\prime}}$ is a
positive degree sheaf embedded in
$g^{*}\operatorname{gr}_{F}\operatorname{\mathsf{End}}(\mathbf{M})$ via the
Kodaira–Spencer map, hence contradicts Proposition 3.10.

###  4.2 Local models of images of period maps

In this section we give a slightly different formulation of Theorem 4.3 which
deduces the existence of the $\mathbf{M}^{j}$ at each level of the resolution
from a single global $\mathbf{M}$.

######  Proposition 4.7.

Let $F$ be a mixed characteristic local field with ring of integers
$\mathcal{O}$ and residue field $k$ of characteristic $p$. Let
$\mathcal{Y}/\mathcal{O}$ be a normal finite type flat $\mathcal{O}$-scheme
equipped with:

  1. 1.

A uniform stratified resolution
$\\{\pi^{j}:\mathcal{S}^{j}\to\mathcal{Y}^{j}\\}$ with natural boundary.

  2. 2.

A smooth $\mathcal{P}/\mathcal{O}$ with a proper dominant
$\phi:\mathcal{P}\to\mathcal{Y}$ and a Fontaine-Laffaille module
$\mathbf{M}\in\mathrm{MF}^{\nabla}_{[a,w+a]}(\mathcal{P})$ with mod $p$
reduction $\mathbf{M}_{0}$ such that:

    1. (a)

The filtered flat vector bundle $(M_{0},\nabla,F^{\bullet}M_{0})$ underlying
$\mathbf{M}_{0}$ is pulled back from $(V,\nabla,F^{\bullet}V)$ on
$\mathcal{Y}_{0}$.

    2. (b)

The Higgs bundle $(\operatorname{gr}_{F}V,\operatorname{gr}_{F}\nabla)$ has
quasifinite Kodaira–Spencer map.

    3. (c)

The local system ${}_{et}\mathbb{V}_{p}$ on $P$ corresponding to $\mathbf{M}$
under the Fontaine–Laffaille correspondence descends to $Y$.

  3. 3.

We finally suppose for each $j$ there is a smooth
$\mathcal{T}^{j}/\mathcal{O}$ and a commutative diagram

${\mathcal{T}^{j}}$${\mathcal{P}}$${\mathcal{S}^{j}}$${\mathcal{Y}^{\prime
j}}$${\mathcal{Y}}$$\scriptstyle{q^{j}}$$\scriptstyle{t^{j}}$$\scriptstyle{h}$$\scriptstyle{\pi^{j}}$

where $q^{j}$ is proper dominant and étale over a dense open subset of
$\mathcal{S}^{j}$.

Then if $p>2w$, $\mathcal{Y}$ is an integral canonical model.

Before the proof we need the following:

######  Lemma 4.8.

Let $F$ be a mixed characteristic local field with ring of integers
$\mathcal{O}$ and residue field $k$ of characteristic $p$. Let
$\mathcal{S},\mathcal{T}/\mathcal{O}$ be smooth.

  1. 1.

If $f:\mathcal{T}\to\mathcal{S}$ is finite étale of prime to $p$ degree and
${}_{et}\mathbb{V}_{p}$ is a $\mathbb{Z}_{p}$-local system on $S$ such that
$f^{*}_{et}\mathbb{V}_{p}$ is crystalline, then ${}_{et}\mathbb{V}_{p}$ is
crystalline, and the inclusion ${}_{et}\mathbb{V}_{p}\to
f_{*}f^{*}(_{et}\mathbb{V}_{p})$ respects the lift of Frobenius.

  2. 2.

Suppose given:

    1. (a)

$f:\mathcal{T}\to\mathcal{S}$ a proper dominant morphism, with base-change
$f_{0}:\mathcal{T}_{0}\to\mathcal{S}_{0}$ to the special fiber;

    2. (b)

$\mathcal{U}_{0}\subset\mathcal{S}_{0}$ a dense open set with preimage
$\mathcal{V}_{0}=f_{0}^{-1}(\mathcal{U}_{0})$;

    3. (c)

strict $p$-torsion Fontaine–Laffaille modules
$\mathbf{M}=(M,\nabla,F^{\bullet}M,\Phi_{\mathbf{M}})$ on $\mathcal{T}$ and
$\mathbf{N}_{\mathcal{U}}=(N_{\mathcal{U}},\nabla,F^{\bullet}N_{\mathcal{U}},\Phi_{\mathbf{N}_{\mathcal{U}}})$
on $\mathcal{U}$ as well as an extension $(N,\nabla,F^{\bullet}N)$ of the
underlying filtered flat vector bundle of $\mathbf{N}_{\mathcal{U}}$ to
$\mathcal{S}_{0}$;

    4. (d)

compatible isomorphisms
$f^{*}\mathbf{N}_{\mathcal{U}}\xrightarrow{\cong}\mathbf{M}|_{\mathcal{V}}$
and
$f_{0}^{*}(N,\nabla,F^{\bullet}N)\xrightarrow{\cong}(M,\nabla,F^{\bullet}M)$.

Then there is a $p$-torsion Fontaine–Laffaille module $\mathbf{N}$ extending
$\mathbf{N}_{\mathcal{U}}$ and an isomorphism
$f_{0}^{*}\mathbf{N}\to\mathbf{M}$ compatible with the above isomorphisms.

###### Proof.

The category of Fontaine–Laffaille modules is a full subcategory of the
category of local systems, so the descent datum of $f^{*}_{et}\mathbb{V}_{p}$
also descends the corresponding Fontaine–Laffaille modules [faltings]. Since
${}_{et}\mathbb{V}_{p}\to f_{*}f^{*}(_{et}\mathbb{V}_{p})$ is a direct factor
of local systems, it follows it comes from a direct factor of
Fontaine–Laffaille modules as well.

We now prove (2). Observe that the $p$-curvature of $(N,\nabla)$ is nilpotent
of the same index as $\mathbf{N}$ (and $\mathbf{M}$). Unwrapping the
definitions, we have isomorphisms

$\Phi_{\mathbf{N}_{\mathcal{U}}}:C_{\mathcal{U}}^{-1}\operatorname{gr}_{F}\mathbf{N}|_{\mathcal{U}_{0}}\xrightarrow{\cong}(N,\nabla)|_{\mathcal{U}_{0}}$
$\Phi_{\mathbf{M}}:C_{\mathcal{T}}^{-1}\operatorname{gr}_{F}\mathbf{M}\xrightarrow{\cong}(M,\nabla)$

such that $f_{0}^{*}\Phi_{\mathbf{N}_{\mathcal{U}}}=\Phi_{\mathbf{M}}$ over
$\mathcal{V}_{0}$, using the natural identification
$f^{*}_{0}C^{-1}_{\mathcal{S}}\operatorname{gr}_{F}\mathbf{N}\cong
C^{-1}_{\mathcal{T}}f_{0}^{*}\operatorname{gr}_{F}\mathbf{N}\cong
C^{-1}_{\mathcal{T}}\operatorname{gr}_{F}\mathbf{M}$. Thus, we have solid
commutative diagrams

${f_{0*}f^{*}_{0}C_{\mathcal{S}}^{-1}\operatorname{gr}_{F}\mathbf{N}}$${f_{0*}f_{0}^{*}(N,\nabla)}$${f_{0*}f^{*}_{0}C_{\mathcal{S}}^{-1}\operatorname{gr}_{F}\mathbf{N}|_{\mathcal{U}_{0}}}$${f_{0*}f_{0}^{*}(N,\nabla)|_{\mathcal{U}_{0}}}$restricting
to${C_{\mathcal{S}}^{-1}\operatorname{gr}_{F}\mathbf{N}}$${(N,\nabla)}$${C_{\mathcal{S}}^{-1}\operatorname{gr}_{F}\mathbf{N}|_{\mathcal{U}_{0}}}$${(N,\nabla)|_{\mathcal{U}_{0}}.}$$\scriptstyle{f_{0*}\Phi_{\mathbf{M}}}$$\scriptstyle{f_{0*}\Phi_{\mathbf{M}}|_{\mathcal{U}_{0}}}$$\scriptstyle{\Phi_{\mathbf{N}}}$$\scriptstyle{\Phi_{\mathbf{N}_{\mathcal{U}}}}$
(4.2.1)

Since $\mathcal{T},\mathcal{S}$ are smooth and $f$ is proper dominant (and
therefore the same for $f_{0}$), $\mathcal{O}_{\mathcal{S}_{0}}\to
f_{0*}\mathcal{O}_{\mathcal{T}_{0}}$ has torsion-free cokernel. As $N$ and
$C^{-1}_{\mathcal{S}}\operatorname{gr}_{F}\mathbf{N}$ are locally free, the
vertical arrows in the left square of (4.2.1) have torsion-free cokernel as
well, and it follows that $f_{0*}\Phi_{\mathbf{M}}$ restricts to an extension
$\Phi_{\mathbf{N}}:C^{-1}_{\mathcal{S}}\operatorname{gr}_{F}\mathbf{N}\xrightarrow{\cong}(N,\nabla)$
of $\Phi_{\mathbf{N}_{\mathcal{U}}}$. ∎

###### Proof of Proposition 4.7.

By assumption, for each $j$, we have a Fontaine–Laffaille module
$t^{j*}\mathbf{M}$ on $\mathcal{T}^{j}$ whose corresponding local system is
pulled back from $\mathcal{Y}$. Moreover, the underlying filtered flat vector
bundle of $t^{j*}\mathbf{M}_{0}$ is pulled back from $(V,\nabla,F^{\bullet}V)$
on $\mathcal{Y}_{0}$. Applying part 1 of Lemma 4.8 to the open set
$\mathcal{U}^{j}$ of $\mathcal{S}^{j}$ over which $q^{j}$ is finite étale, we
conclude that there is a Fontaine–Laffaille module
$\mathbf{N}_{\mathcal{U}^{j}}$ which pulls back to
$(t^{j*}\mathbf{M})|_{(q^{j})^{-1}(\mathcal{U}^{j})}$. Applying part 2, we
find that $(\mathbf{N}_{\mathcal{U}^{j}})_{0}$ extends to a Fontaine–Laffaille
module $\mathbf{N}^{j}_{0}$ which pulls back to $t^{j*}\mathbf{M}_{0}$. In
particular, we have $q^{j*}\operatorname{gr}_{F}\mathbf{N}^{j}_{0}\cong
q^{j*}\pi^{j*}(\operatorname{gr}_{F}V,\operatorname{gr}_{F}\nabla)$, so

$(q^{j}_{*}\mathcal{O}_{\mathcal{T}^{j}_{0}},0)\otimes\operatorname{gr}_{F}\mathbf{N}^{j}_{0}\cong(q^{j}_{*}\mathcal{O}_{\mathcal{T}_{0}^{j}},0)\otimes\pi^{j*}(\operatorname{gr}_{F}V,\operatorname{gr}_{F}\nabla)$

as Higgs sheaves. By the same argument, we have

$(q^{j}_{*}\mathcal{O}_{\mathcal{T}^{j}_{0}},0)\otimes\operatorname{gr}_{F}\operatorname{\mathsf{End}}(\mathbf{N}^{j})\cong(q^{j}_{*}\mathcal{O}_{\mathcal{T}_{0}^{j}},0)\otimes\pi^{j*}(\operatorname{gr}_{F}\operatorname{\mathsf{End}}(V),\operatorname{gr}_{F}\operatorname{ad}\nabla)$

as Higgs sheaves. Thus, all the conditions of Theorem 4.3 are satisfied. ∎

###  4.3 Making period maps proper

We will need a result allowing us to extend geometric Fontaine–Laffaille
modules when the underlying filtered flat vector bundle extends.

######  Proposition 4.9.

Let $F$ be a mixed charactersitic local field with ring of integers
$\mathcal{O}$ and residue field $k$. Let $\mathcal{P}/\mathcal{O}$ be smooth
with a partial log smooth compactification
$(\mathcal{P}^{\prime},\mathcal{D})/\mathcal{O}$. Let
$f:\mathcal{Z}\to\mathcal{P}$ be a smooth projective morphism with
geometrically connected fibers of relative dimension $<\textrm{char}(k)$. Let
$\mathbf{M}=H^{k}_{DR}(\mathcal{Z}/\mathcal{P})$ be the $p$-torsion
Fontaine–Laffaille module with underlying filtered flat vector bundle the De
Rham cohomology of the special fiber
$(M,\nabla,F^{\bullet}M)=H^{k}_{DR}(\mathcal{Z}_{0}/\mathcal{P}_{0})$. Assume:

  1. 1.

There is a log smooth curve (possibly not irreducible)
$(\mathcal{C}^{\prime},\Delta)/\mathcal{O}$ and a morphism
$g^{\prime}:(\mathcal{C}^{\prime},\Delta)\to(\mathcal{P}^{\prime},\mathcal{D})$
with
$\Delta=(\mathcal{C}^{\prime}\times_{\mathcal{P}^{\prime}}\mathcal{D})^{\mathrm{red}}$
such that $g^{\prime}(\Delta)$ meets every component of $\mathcal{D}$ and the
base-change $\mathcal{Z}_{\mathcal{C}}$ (where
$\mathcal{C}=\mathcal{C}^{\prime}\setminus\Delta$) has a log smooth
compactification
$(\mathcal{Z}^{\prime}_{\mathcal{C}},\mathcal{E})/\mathcal{O}$ and a morphism
$f^{\prime}:\mathcal{Z}^{\prime}_{\mathcal{C}}\to\mathcal{C}^{\prime}$
extending the natural one.

  2. 2.

The filtered flat vector bundle $(M,\nabla,F^{\bullet}M)$ extends to
$(M^{\prime},\nabla^{\prime},F^{\bullet}M^{\prime})$ on
$\mathcal{P}^{\prime}$.

  3. 3.

$g^{\prime*}(M^{\prime},\nabla^{\prime},F^{\bullet}M^{\prime})\cong
H^{k}_{DR}((\mathcal{Z}^{\prime}_{\mathcal{C},0},\mathcal{E}_{0})/(\mathcal{C}_{0}^{\prime},\Delta_{0}))$
via an isomorphism extending the natural one, where
$H^{k}_{DR}((\mathcal{Z}^{\prime}_{\mathcal{C},0},\mathcal{E}_{0})/(\mathcal{C}_{0}^{\prime},\Delta_{0}))$
is the relative log De Rham cohomology.

Then $(M^{\prime},\nabla^{\prime},F^{\bullet}M^{\prime})$ underlies an
extension $\mathbf{M}^{\prime}$ of $\mathbf{M}$ to $\mathcal{P}^{\prime}$ as a
Fontaine–Laffaille module.

###### Proof.

The main step is the following straightforward claim. Throughout by
“$\tilde{\mathcal{C}}$” we mean the base-change to $W_{2}(k)$.

######  Claim.

There is an isomorphism

$F^{*}\operatorname{gr}_{F}\mathbb{H}^{k}(\Omega^{\bullet}_{\mathcal{Z}^{\prime}_{\mathcal{C},0}/\mathcal{C}_{0}^{\prime}}(\log\mathcal{E}_{0}))\xrightarrow{\Phi}\mathbb{H}^{k}(\Omega^{\bullet}_{\mathcal{Z}_{\mathcal{C},0}/\mathcal{C}_{0}^{\prime}}(\log\mathcal{E}_{0}))$

extending the usual Frobenius morphism of Deligne–Illusie over
$\mathcal{C}_{0}$.

###### Proof.

By log smoothness, both $\tilde{\mathcal{C}}^{\prime}$ and
$\tilde{\mathcal{Z}}^{\prime}_{\mathcal{C}}$ étale locally444In fact, the
lifts of Frobenius preserving the boundary exist Zariski locally. To simplify
notation, if $(\tilde{X},\tilde{D})/W_{2}$ is log smooth and $(X,D)$ the
reduction mod $p$, then there is a natural exact sequence  $0\to
F^{*}T_{X}(-\log D)\to
F^{*}T_{X}\to\bigoplus_{i}F^{*}\mathcal{O}_{D_{i}}(D_{i})\to 0$ where the
$D_{i}$ are the components of $D$ with defining equations $f_{i}$, and the
last map sends $1\otimes\delta$ to $1\otimes\frac{\delta f_{i}}{f_{i}}\mod
f_{i}$ on the $i$th factor. Lifts of Frobenius to $\tilde{X}$ are a torsor for
$H^{0}(F^{*}T_{X})$, so if $\tilde{F}$ is a local lift of Frobenius to
$\tilde{X}$, and $\tilde{f}_{i}$ local defining equations for $\tilde{D}_{i}$,
then $\tilde{F}(\tilde{f}_{i})=\tilde{f}^{p}_{i}+pg_{i}$, and Zariski locally
we may modify $\tilde{F}$ so that each $\tilde{F}(\tilde{f}_{i})$ is divisible
by $\tilde{f}_{i}$. admit compatible lifts of Frobenius fixing each component
of the boundary. Thus, as in [deligneillusie], we obtain a morphism of
filtered complexes

$(\Omega_{\mathcal{Z}^{\prime}_{\mathcal{C},0}/\mathcal{C}_{0}^{\prime}}^{\bullet}(\log\mathcal{E}_{0}),0)\to(F_{*}\Omega_{\mathcal{Z}^{\prime}_{\mathcal{C},0}/\mathcal{C}_{0}^{\prime}}^{\bullet}(\log\mathcal{E}_{0}),F_{*}d)$

which restricts to the usual quasi-isomorphism over $\mathcal{C}_{0}$ and is
easily checked to be a quasi-isomorphism. ∎

To prove the proposition, we must show the Frobenius
$\Phi:C^{-1}_{\mathcal{P}}\operatorname{gr}_{F}\mathbf{M}\to\mathbf{M}$
extends to an isomorphism
$\Phi^{\prime}:C^{-1}_{\mathcal{P}^{\prime}}(\operatorname{gr}_{F}M^{\prime},\operatorname{gr}_{F}\nabla^{\prime})\to(M^{\prime},\nabla^{\prime})$.
Since both sides are locally free, it suffices to show that $\Phi$ extends in
codimension one, which may therefore be checked on the pullback
$g^{\prime*}(M^{\prime},\nabla^{\prime},F^{\bullet}M^{\prime})$. This is the
claim. ∎

##  5 Existence of integral canonical models

In this section, we deduce the existence of integral canonical models for
sufficiently large primes.

###  5.1 Proof of Theorem 1.3

In this section we show that models of Shimura varieties satisfy the
conditions of Theorem 4.1 for large primes, thereby proving Theorem 1.3. Let
$S=S_{K}(G,X)$ be a Shimura variety. We let $E$ denote the reflex field.
$S_{K}(G,X)$ carries a flat principal $G$-bundle $\mathcal{E}/S$, which admits
a canonical model over $E$. A choice of hodge co-character defines a parabolic
subgroup $P\subset G$, and gives a parabolic bundle $\mathcal{P}/S$, along
with a canonical morphism $\mathcal{P}\rightarrow\mathcal{E}$, where all this
data descends to $E_{G}$. We note that $\mathcal{P}$ is not flat for the
connection on $\mathcal{E}$.

Let $V$ denote a faithful rational representation of $G$, and
$\mathbb{V}\subset V$ denote a lattice. The choice of $\mathbb{V}$ specifies a
compact open subgroup of $G(\mathbb{A}_{f})$ by considering the stabilizers of
$\mathbb{V}_{\hat{\mathbb{Z}}}$. We choose the level structure $K$ to be a
subgroup of this compact open subgroup, such that it is neat, and such that it
acts trivially on $\mathbb{V}/3\mathbb{V}$. Corresponding to $V$ there is a
family of $\mathbb{Q}_{\ell}$ local systems ${}_{{\mathrm{et}}}V_{\ell}$ with
$\mathbb{Z}_{\ell}$ sub-systems ${}_{{\mathrm{et}}}\mathbb{V}_{\ell}$.
Moreover, there is a filtered flat vector bundle ${}_{{}_{\mathrm{dR}}}V$ over
$S_{K}(G,X)$ – indeed we have that ${}_{{}_{\mathrm{dR}}}V$ is defined by
$\mathcal{E}\times_{G}V$. The filtration is induced by the fact that
${}_{{}_{\mathrm{dR}}}V$ is also canonically isomorphic to
$\mathcal{P}\times_{G_{E}}V_{E}$. Now, let $p$ be a prime and let $v\mid p$ be
a prime of $E$. By [DLLZ], there is a canonical isomorphism between
$(_{{}_{\mathrm{dR}}}V)_{E,v}$ and ${}_{p-_{\mathrm{dR}}}V$, the filtered flat
bundle obtained by applying the $p$-adic Riemann-Hilbert correspondence to
${}_{{\mathrm{et}}}V_{p}|_{S_{K}(G,X)_{E,v}}$.

By choosing $N$ to be a large enough integer, we may assume that $S_{K}(G,X)$
spreads out to a smooth integral model
$\mathscr{S}_{K}(G,X)/\mathcal{O}_{E}[1/N]$ such that:

  * •

$G$ has a reductive model $\mathcal{G}/\mathbb{Z}[1/N]$. There are elements
$s_{\alpha}\in\mathbb{V}[1/N]^{\otimes}$ whose point-wise stabilizers define
the group $\mathcal{G}$.

  * •

For every $p\nmid N$, we have $K=\mathcal{G}(\mathbb{Z}_{p})\cdot K^{p}$,
where $K^{p}$ is the prime-to-$p$ level structure.

  * •

$P\subset G_{E}$ is induced by a parabolic subgroup of $\mathcal{G}$ defined
over $\mathcal{O}_{E}[1/N]$.

  * •

The map $\mathcal{P}\rightarrow\mathcal{E}$ is induced by a map
$\mathscr{P}\rightarrow\mathscr{E}$ over $\mathcal{S}$, where $\mathscr{E}$ is
a flat principal $\mathcal{G}$ bundle and $\mathscr{P}$ is a principal bundle
for the parabolic of $\mathcal{G}$ defined over $\mathcal{O}_{E}[1/N]$.

Note that this gives us an integral model
${}_{{}_{\mathrm{dR}}}\mathbb{V}/\mathscr{S}_{K}(G,X)$ for the filtered flat
bundle ${}_{{}_{\mathrm{dR}}}V$ (where the flat bundle is defined as
$\mathscr{E}\times_{\mathcal{G}}\mathbb{V}[\frac{1}{N}]$, and the filtration
is induced by $\mathscr{P}\rightarrow\mathscr{E}$).

######  Lemma 5.1.

After increasing $N$ we have the following.

  * •

$\mathscr{S}_{K}(G,X)$ admits a log-smooth compactification over
$\mathcal{O}_{E}[1/N]$.

  * •

The local systems ${}_{{\mathrm{et}}}\mathbb{V}_{\ell}$ extend to
$\mathscr{S}_{K}(G,X)[1/\ell]$.

  * •

${}_{{\mathrm{et}}}\mathbb{V}_{p}$ is crystalline on
$\mathscr{S}_{K}(G,X)_{E_{v}}$ for every finite place $v$ of
$\mathcal{O}_{E}[1/N]$.

  * •

For each finite place $v$ of $\mathcal{O}_{E}[1/N]$, let
${}_{{}_{\mathrm{cris}}}\mathbb{V}/\mathscr{S}_{K}(G,X)_{\mathcal{O}_{E_{v}}}$
denote the Fontaine-Laffaille module corresponding to
${}_{{\mathrm{et}}}\mathbb{V}_{p}$ under the Fontaine-Laffaille
correspondence. Then the Kodaira–Spencer map of the Higgs bundle
$\operatorname{gr}_{F}(_{{}_{\mathrm{dR}}}\mathbb{V})$ is immersive.

###### Proof.

The first condition follows by choosing a log smooth compactification over the
generic fiber.

For the second condition, we use the following result of [KP, Thm 1.3]:

######  Theorem 5.2 (Klevdal-Patrikis).

There is an integral model $\mathcal{S}$ of $S_{K}(G,X)$ over
$\mathbb{Z}[1/N]$ for some $N$ such that for each $\ell$ the local systems
${}_{{\mathrm{et}}}\mathbb{V}_{\ell}$ all extend to $\mathcal{S}[1/\ell]$.

The third condition is proven by Esnault-Groechenig [EG] for large enough
primes, and this in particular gives an integral structure to the filtered
flat bundle ${}_{p-_{\mathrm{dR}}}V$. Under the canonical isomorphism
${}_{p-_{\mathrm{dR}}}V\rightarrow_{{}_{\mathrm{dR}}}V|_{S_{K}(G,X)_{E_{v}}}$,
the integral structures induced by ${}_{{}_{\mathrm{cris}}}\mathbb{V}$ and
${}_{{}_{\mathrm{dR}}}\mathbb{V}$ are the same by [PST].

Finally, using this isomorphism, the final condition follows because the same
is true for the Higgs bundle of ${}_{{}_{\mathrm{dR}}}V$ over the generic
point. ∎

It follows from Theorem 4.1 that $\mathscr{S}_{K}(G,X)_{\mathcal{O}_{E_{v}}}$
is an integral canonical model. It remains to argue that
$\mathscr{S}_{K}(G,X)_{\mathcal{O}_{E_{v}}}$ is the unique model admitting a
log smooth compactification. Let $\mathcal{S}$ be another such model. The
extension property yields that the identity map over the generic fiber extends
to $f:\mathcal{S}\to\mathscr{S}_{K}(G,X)_{\mathcal{O}_{E_{v}}}$, and by purity
of the branch locus this map is an open immersion. The inverse image of the
boundary of $\bar{\mathcal{S}}$ in the closure
$\mathcal{W}\subset\bar{\mathcal{S}}\times\bar{\mathscr{S}}_{K}(G,X)_{\mathcal{O}_{E_{v}}}$
of the graph of $f$ must map to the boundary of
$\bar{\mathscr{S}}_{K}(G,X)_{\mathcal{O}_{E_{v}}}$, by the smooth extension
property. Thus, the extension is an isomorphism. ∎

###  5.2 Proof of Theorem 1.9

Suppose $F$ is a number field, $P/F$ a smooth variety, and $f:Z\to P$ a smooth
projective family. Let $f:P^{\prime}\to Y$ be the Stein factorization of the
(compactification) of the period map associated to
$\mathbb{W}_{\mathbb{Z}}:=Rf_{*}\mathbb{Z}_{Z^{\mathrm{an}}}$ as described in
2.7. Then $f$ is defined over $F$. By spreading out over
$\mathcal{O}:=\mathcal{O}_{F}[1/n]$ and inverting finitely many primes, we may
assume we have smooth $\mathcal{P},\mathcal{Z}/\mathcal{O}$, a log smooth
partial compactification $(\mathcal{P}^{\prime},\mathcal{D})/\mathcal{O}$ of
$\mathcal{P}$, and $\mathcal{Y}/\mathcal{O}$ normal finite type flat as well
as:

  1. 1.

$f:\mathcal{Z}\to\mathcal{P}$ smooth projective

  2. 2.

$\phi:\mathcal{P}^{\prime}\to\mathcal{Y}$ proper with
$\mathcal{O}_{\mathcal{Y}}\xrightarrow{\cong}\mathcal{O}_{\mathcal{P}^{\prime}}$.

  3. 3.

$\mathcal{Y}$ admits a uniform stratified resolution
$\\{\pi^{j}:\mathcal{S}^{j}\to\mathcal{Y}^{j}\\}$.

  4. 4.

For each $j$ there is a smooth $\mathcal{T}^{j}/\mathcal{O}$ and a commutative
diagram

${\mathcal{T}^{j}}$${\mathcal{P}^{\prime}}$${\mathcal{S}^{j}}$${\mathcal{Y}^{\prime
j}}$${\mathcal{Y}}$$\scriptstyle{q^{j}}$$\scriptstyle{t^{j}}$$\scriptstyle{\bar{\phi}}$$\scriptstyle{\pi^{j}}$

where $q^{j}$ is proper dominant and étale over a dense open subset of
$\mathcal{S}^{j}$.

  5. 5.

There is a log smooth curve (possibly not irreducible)
$(\mathcal{C}^{\prime},\Delta)/\mathcal{O}$ and a morphism
$g^{\prime}:(\mathcal{C}^{\prime},\Delta)\to(\mathcal{P}^{\prime},\mathcal{D})$
with
$\Delta=(\mathcal{C}^{\prime}\times_{\mathcal{P}^{\prime}}\mathcal{D})^{\mathrm{red}}$
such that $g^{\prime}(\Delta)$ meets every component of $\mathcal{D}$ and the
base-change $\mathcal{Z}_{\mathcal{C}}$ (where
$\mathcal{C}=\mathcal{C}^{\prime}\setminus\mathcal{E}$) has a log smooth
compactification
$(\mathcal{Z}^{\prime}_{\mathcal{C}},\mathcal{E})/\mathcal{O}$ and a morphism
$f^{\prime}:\mathcal{Z}^{\prime}_{\mathcal{C}}\to\mathcal{C}^{\prime}$
extending the natural one.

  6. 6.

The filtered flat vector bundle $(V,\nabla,F^{\bullet}V)$ on $Y$ extends to
$(\mathcal{V},\nabla,F^{\bullet}\mathcal{V})$ on $\mathcal{Y}$, and its
pullback to $\mathcal{P}$ is identified with the filtered flat vector bundle
underlying $H^{k}_{DR}(\mathcal{Z}/\mathcal{P})$. Moreover, the reduction
$(\mathcal{V},\nabla,F^{\bullet}\mathcal{V})_{v}$ has quasifinite
Kodaira–Spencer map for every finite place $v$.

  7. 7.

By Corollary 2.23, the local system
$H^{k}_{et}(\mathcal{Z}/\mathcal{P}[1/\ell],\mathbb{Z}_{\ell})$ extends to
$\mathcal{P}^{\prime}$ and descends to ${}_{et}\mathbb{V}_{\ell}$ on
$\mathcal{Y}[1/\ell]$. By Lemma 2.24 and Lemma 2.7, it follows that the
stratified resolution has natural boundary.

This yields part 1(a), and both 2 and 3 follow from 1(b) by Corollary 2.18 and
Proposition 2.15 (1(c) is clear from construction). Thus it remains to show
the extension property in 1(b), and since by Lemma 2.24 we have the extension
in characterstic 0, it suffices to show $\mathcal{Y}_{v}$ is an integral
canonical model for each finite place $v$. By Theorem 4.3 we must show for
each finite place $v$ of $\mathcal{O}$ that there is a Fontaine–Laffaille
module $\mathbf{M}$ on $\mathcal{P}^{\prime}_{v}$ with the required
properties. The filtered flat vector bundle underlying the $p$-torsion module
of $H^{k}_{DR}(\mathcal{Z}_{v}/\mathcal{P}_{v})$ extends to
$\mathcal{P}^{\prime}_{v}$, and so by Proposition 4.9 it extends as a
Fontaine–Laffaille module whose underlying filtered flat vector bundle is the
pullback of $(\mathcal{V},\nabla,F^{\bullet}\mathcal{V})_{\mathcal{O}_{v}}$ on
$\mathcal{Y}_{v}$.∎

##  6 Application to unramified Borel-extension

In this section, we will apply the extension property for Shimura varieties
and images of geometric period maps. To that end, let $F$ be a number field,
and let $(Y,\overline{Y},\mathbb{V})$ denote one of the following:

  1. 1.

$Y\subset\overline{Y}$ is a Shimura variety embedded inside a log smooth
toroidal compactification, and $\mathbb{V}$ is
${}_{{\mathrm{et}}}\mathbb{V}_{\ell}$ associated to the adjoint representation
of the underlying group $G$ as in Section 5.1. We pick $\ell=3$.

  2. 2.

$Y\subset\overline{Y}$ is as in Section 5.2. Recall that there is a smooth
proper family over a smooth variety $P$, and $Y$ is the image of the period
morphism from $P$. For every prime $\ell$, there is an etale $\ell$-adic local
system on $P$ induced by the smooth proper family, and this $\ell$-adic local
system descends to an $\ell$-adic local system $\mathbb{V}$ on $Y$. As in the
Shimura case, we will choose $\ell=3$.

In the non-Shimura setting, we assume the existence of stratified resolution
of $\mathcal{Y}/\mathcal{O}_{F}[1/N]$ as in Definition 4.2 where the
resolution is also defined over $\mathcal{O}_{F}[1/N]$ where $N$ is some large
enough integer. We suppose that $p\nmid N$ is a large enough prime. In
particular, we suppose that $p$ is large enough such that the main theorems of
Section 5 hold, and such that
$p\nmid\\#\operatorname{GL}_{n}(\mathbb{Z}/3\mathbb{Z})$, where $n$ is the
rank of $\mathbb{V}$.

######  Definition 6.1.

Let $K$ denote a finite extension of $W(\overline{\mathbb{F}}_{p})[1/p]$ or
$\mathbb{F}_{p}((t))$. We say that a point $x\in Y(K)$ has good reduction if
its specialization lies in the interior
$\mathcal{Y}(\overline{\mathbb{F}}_{p})$. We say that $x$ has bad reduction
otherwise.

The content of the following lemma is that a point having good reduction or
bad reduction can be tested by considering the fiber of $\mathbb{V}$ at that
point.

######  Lemma 6.2.

Let $K$ be as in Definition 6.1. Let $x\in Y(K)$. Then $x$ has bad reduction
if and only if the inertial action of
$I_{x}\subset\operatorname{Gal}(\overline{K}/K)$ on $\mathbb{V}_{x}$ is quasi-
unipotent of infinite order.

###### Proof.

By arguments similar to those in [PST] (which we may use, as $\mathbb{V}$ has
prime-to-$p$ monodromy given our choice of $p$), we have that a point $x\in
Y(K)$ has bad reduction if and only if the inertial action on the fiber at $x$
is quasi-unipotent of infinite order. Indeed, in the Shimura case, this is
just [PST, Theorem 4.4] (the result in _loc. cit._ is for finite extensions of
$W(\overline{\operatorname{\mathbb{F}}}_{p})$, but the identical argument goes
through in the case of finite extensions of
$\overline{\operatorname{\mathbb{F}}}_{p}((t))$). In the geometric case, at
the cost of replacing $K$ by a finite extension, there is a point $y\in P(K)$
lifting $x$. The same argument as in [PST, Theorem 4.4] yields that $y$
specializes to the boundary of $P$ if and only if the inertial action is
trivial. The claim follows for $x$ as $y$ specializes to the boundary of $Y$
if and only $x$ specializes to the boundary of $P$. ∎

Let $\mathsf{D}$ denote the rigid-analytic closed unit disc (defined over
$\mathbb{Q}_{p}$) and let $\mathsf{D}^{\times}=\mathsf{D}\setminus\\{0\\}$.
Recall that a _thin annulus_ is a rigid analytic annulus whose outer and inner
radii are the same, and a _thick annulus_ is one whose inner and outer radii
are different. We are now ready to state the main theorem of this section –
this is a one-dimensional analogue of Borel’s extension theorem in the
unramified setting (see the main result of [padicBorel]).

######  Theorem 6.3.

Let $Y$ be as above, and let $f:\mathsf{D}^{\times}\rightarrow
Y^{{\mathrm{an}}}$ denote a rigid-analytic map defined over $W[1/p]$, where
$W=W(\overline{\mathbb{F}}_{p})$. Suppose that the image of $f$ contains a
point with good reduction. Then, $f$ extends to a rigid-analytic map
$\mathsf{D}\rightarrow Y^{{\mathrm{an}}}$.

We first need the following lemma:

######  Lemma 6.4.

Let $f:\mathsf{D}^{\times}\rightarrow Y^{{\mathrm{an}}}$ be a rigid-analytic
map defined over a finite extension of $W[1/p]$, such that the image of $f$
contains a point with good reduction. Then the image of $f$ is entirely
contained in the good reduction locus.

###### Proof.

Note that the local system $f^{-1}\mathbb{V}$ has prime-to-$p$ monodromy. The
proof of this lemma is essentially the same as [padicBorel, Theorem 3.3]. As
in _loc. cit._ , it suffices to prove the lemma for maps from closed annuli.
Proposition 3.6 of _loc. cit._ proves that an $\ell$-adic local system on a
thin annulus must be unramified everywhere if it is at a single point.

To deduce the case of thick annuli from thin annuli, we proceed as in
[padicBorel, Section 3.1.2]. The argument applies verbatim to reduce to
proving that any map
$f:\mathbb{G}_{m}\rightarrow\bar{\mathcal{Y}}_{\mathbb{F}_{p}}$ extends to a
map from $\mathbb{P}^{1}$. Note that the local system
$\mathbb{V}/\mathcal{Y}_{\mathbb{F}_{p}}$ is point-wise pure. Indeed, in the
Shimura case, $\mathbb{V}$ is a direct sum of irreducible local systems with
trivial determinant (this follows from the fact that the local system is
induced by the adjoint representation of $G$). In the case of geometric period
images, it suffices to prove point-wise-purity with $P$ in place of $Y$. But
this is true as $\mathbb{V}$ is induced by a geometric family over $P$.

Suppose now that $x\in\mathbb{P}^{1}\setminus\mathbb{G}_{m}$ maps to the
boundary. We have that $f^{-1}(\mathbb{V})$ has quasi-unipotent local
monodromy of infinite order. We have that $f$ descends to some finite field.
In both the cases, the local system originally arises from an irreducible
local system over a smooth base, and is therefore point-wise pure over that
pure base. As this is a point-by-point condition, it is also point-wise pure
over $Y$, and therefore over when pulled back to $\mathbb{G}_{m}$ by $f$. By
[DeligneWeil2, Corollaire 3.4.12], the local system has semi-simple geometric
monodromy over $\mathbb{G}_{m}$. As the prime-to-$p$ geometric fundamental
group of $\mathbb{G}_{m}$ is abelian, the local system pulled back to
$\mathbb{G}_{m}$ must have toric geometric monodromy. Therefore, the local
monodromy around the boundary of $\mathbb{G}_{m}$ is not quasi-unipotent with
infinite order in the first place, and the result follows. ∎

We are now ready to prove Theorem 6.3.

###### Proof.

By the lemma above, we may assume that the image of $\mathsf{D}^{\times}$ is
contained in the good reduction locus of $Y^{{\mathrm{an}}}$. Let
$Y^{j}\subset Y$ denote the smallest stratum that contains the image of
$\mathsf{D}^{\times}$. We claim that the map
$f:\mathsf{D}^{\times}\rightarrow{Y^{j}}^{{\mathrm{an}}}$ lifts to a map
$g:\mathsf{D}^{\times}\rightarrow{S^{j}}^{{\mathrm{an}}}$. Indeed,
$\mathsf{D}^{\times}\cap{Y^{j-1}}^{{\mathrm{an}}}$ is a analytic closed
subset, and so consists of a sequence of points $Q_{i}\in\mathsf{D}^{\times}$
with the property that only finitely many of the $Q_{i}$ lie in any closed
sub-annulus of $\mathsf{D}^{\times}$. Pick $D_{i}$ to be a small closed disc
centered around $Q_{i}$ such that $D_{i}\cap Y^{j-1}=Q_{i}$. It suffices to
show that the map $D_{i}\rightarrow{Y^{j}}^{{\mathrm{an}}}$ lifts to
$D_{i}\rightarrow S^{j}$. But this follows from the fact that
$S^{j}\rightarrow Y^{j}$ is proper.

Let $\mathsf{A}_{n}$ denote the rigid-analytic annulus with outer radius 1 and
inner radius $\frac{1}{p^{n}}$, and let $g_{n}=g|_{\mathsf{A}_{n}}$.
Therefore, the map $f_{n}$ is induced by a map of formal schemes
$\mathscr{X}_{n}\rightarrow\widehat{\mathcal{S}^{j}}^{p}$, where
$\mathscr{X}_{n}$ is a formal scheme with rigid generic fiber
$\mathsf{A}_{n}$, and $\widehat{\mathcal{S}^{j}}^{p}$ is the $p$-adic
completion of $\mathcal{S}^{j}$. By [padicBorel, Appendix], we may assume that
the special fiber of $\mathscr{X}_{n}$ is a union of projective lines and two